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continuation on page 95

Lecture Notes in Economics and Mathematical Systems Manag~ng Ed~tors: M. Beckmann and H P. Kinzl

Systems Theory

B. D. 0. Anderson . M. A. Arbib E. G. Manes

Foundations of System Theory: Finitary and lnfinitary Conditions

Springer-Verlag Berlin . Heidelberg . New York 1976

Editorial Board H. Albach . A. V. Balakrishnan . M. Beckmann (Managing Editor) P. Dhrymes . J. Green . W. Hildenbrand - W. Krelle H. P. Kiinzi (Managing Editor) . K. Ritter . R Sato . H. Schelbert P. Schonfeld

Managing Editors Prof. Dr. M. Beckmann Prof. Dr. H. P. Kinzi Brown University Universitat Ziirich Providence, RI 02912/USA 8090 ZiirichlSchweiz

Authors Brian D. 0. Anderson Department of Electrical Engineering University of Newcastle New South Wales 2308lAustralia

Michael A Arbib Department of Computer and Information Science University of Massachusetts Amherst, Massachusetts, 01002lUSA

Ernest G. Manes Department of Mathematics University of Massachusetts Amherst; Massachusetts, 01002lUSA

~ i b r t l y or coag-s camlagtn~ o p ~ b u a l i r ~ 8 1 s

W i b , ?llchael A Poundations or system theory.

(Ieeturr notes in ecrmdcs and rrthematfca systems-; 115)

B i b L i a @ q i w D. ID-s ioder.. 1 mtem theory. I. Kanes, Bmest G., 1943-

60ult B(LMo1. . Ande~nderm, Brian D. O., joint Andetho?.

m. 1YUe. Iv. series. 4295.A7 003 76-1967

AMS Subject Classifications (1970): 18620, 18C10, 93A99, 93615

ISBN 3-540-07611-5 Springer-Verlag Berlin . Heidelberg . NewYork ISBN 0-387-0761 1-5 Springer-Verlag New York . Heidelberg . Berlin

This wow .s s~bject to copyr.ght. All rlgnts are reserved, whetner the whole or part of the mater al is concerned, spec~fically those of translation, re- printing, reuse of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 5 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 8 by ~ ~ r i n ~ e r - ~ i r l a ~ Berlin . Heidelberg 1976 Printed in Germany Printing and binding: Beltr Olfsetdruok, . HemsbachIBergstr. . ',

This paper is one ofa series in which the ideas of category theory

are applied to problems of system theory. As with the three principal

earlier papers, [I-31, the emphasis is on study of the realization problem,

or the problem of associating with an input-oucput description of a system

an internal description with something analogous to a stata-space..

In this paper, several sorts of machines will be discussed, which

arrange th-elves in the following hierarchy:

(Sequential Machine)

Input process Machine

(Tree automaton)

Decomposable Machine

(Linear System,

Group Machine)

1

Output process

Machine

Each member of the hierarchy includes members below it; examples are included

- State-behavior

Machine

I

in parentheses, and each example is at its lowest possible point in the

hierarchy. There are contrived examples of output process machines and

state-behavior machines which are not adjoint machines [ 3 ] , but as yet,

no examples with the accepted stature of linear systems [ 4 ] , group machines

[ 5 , 61, sequential machines [7, Ch. 21, and tree automata [7, Ch. 4 1 .

To grasp in very grear-generality what we attempt to do in this paper,

we recall several facts concerning discrete-time linear systems:

(1) One can take an external description of a linear system, in the

A form of a map f taking past input sequences into future output sequences

and, using results of module theory, construct an internal realization

A . essentially by factoring f rnto an onto linear map followed by a one-to-one

linear map [ 4 ] . The codomain of the onto linear map is the scate-space, and

A is (Space of Input 5equences)lKer f . (It is actually possible to do all

this in matrix terms, working with Hankel matrices of Markov parameters, and

this viewpoint may be more Pamillar to some.)

(2) One can use a Nerode equivalence class theory [a], originally

developed for sequential machines, to obtain the set of reachable and

A observable states associated with a map f . Then one can observe that the

set is really a linear space, and the procedure is equivalent to (1).

(3 ) In a finite-dimensional reachable and observable linear system,

all states can be reached in a bounded time, and all can be observed in a

bounded time 191.

In [ 2 ] , we extended (1) by recalling from category theory the concept

of E-nl factorization which generalizes the idea of factoring into an onto

or epi map followed by a one-to-one or mono map.

In this paper, we attempt to expand the application of the other ideas

to various classes of machines. To extend (2 ) , we attempt to force a Nerode

equivalence structure into the more general situations. To extend (3) and

obtain a notion of finiteness in a more general context, we generalize the

idea of reachable (and observable) in a finite time.

As background, we require familiarity with 121, but not necessarily [l]

or [3], though we should be dishonest were we to claim that knowledge of [I]

and [3] would be of no extra help. Likewise, though lmowledge of some

category theory, as per say [lo, 11, or 121, would be helpful, we only

strictly require knowledge of those category theory ideas used in 121,

introducing other category theory concepts as required.

The body of the paper is organized into a structure best depicted as

follows:

Reference 121 u Section 1

Review of linear systems ideas, Category theory results, and

Definition of classes of machines I I

Section 2 4.1-4.11, 4.19, 4.20 1 Nerode equivalence approaches to realization, including special results for state

behavior machines. Relation with & -?il realization approach

Generalization of the finite-dimensionality idea, via E - T factorization, and

with emphasis on adjoint machines

d Section 3

Wee automata, as example of Nerode

approach applied to input process machine -

4.12-4.18

Generalization of the finite-dimensionality

idea via Nerode equivalence, for --

-1 -- adjoint machines (motivation -- --* -

only) Section 5

Extension of finite step idea to cope with, e.g., tree automata

As the diagram shows, the paper in its totality is very much a sequel

to [2].

Certainly, it is clear that the ideas of this paper unify a number of

apparently distinct ideas scattered in the literature. Unification is not

all, however: by proving results about linear systems in a category theory

rather than vector space setting, one strips avay features of the vector

space setting which may obscure possible applications or extensions of the

result in other settings.

Aclmowledgement:

The research reported in this volume was supported in part by

National Science Foundation grant number GJ 35759, which also supported

Dr. Anderson's visit to the University of Massachusetts at Amherst for the

period September 1973 through February 1974. This work was completed in

February 1974.

TABLE OF CONTENTS

1 . A General Setting for Discrete Action Nonlinear Systems . . . . 1

. . . . . . . . . . . . . . . . . . 2 . Nerode Equivalence Approach 27

3 . Tree Automata: Finite Successes and Infinite Failures . . . . . 43

. . . . . . . . . . . . . . . . . . . . . 4 . Finite Step Conditions 63

. . . . . . . . . . . . . . . . . . . . . . 5 Augmenting the Process 81

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . conciusions 89

. . . . . . . . . . . . . . . . . . . . . . . . . . . . References 92

1. A General Setting for Discrete Action Nonlinear Systems.

In the paper "Foundations of System Theory: Decomposable Systems", [2],

to which this is a sequel, we shoved how the techniques of category theory

allowed us to give a general theory of reachability, observability, reali-

zation and duality for decomposable systems: a class erbracing both linear

(discrete-time) systems and group machines. In '%chines in a Category:

An Expository Introduction", 111, we gave a more general categorical system

theory which handles nonlinear systems as well as decomposable systems.

In fact, as we shall see in Section 3, it also handles the tree automata

of formal language theory and computer semantics, so long as certain finite-

ness conditions are met.

In this section, we ass- the reader familiar with the notions of [ 2 ]

--such as category, power and copower, dyn-rphism of decomposable dynamics,

etc.--and provide the extra background from category theory required to

set forth the general framework of [IT.

The time-invariant system

= f(q.u) y = B(9) (1)

when discretized in time (into a difference equation), initialized

(coordinates are chosen such that f(0,O) = 0, B(0) = O), and linearized,

gives rise to the linear system

q(t + 1) = Fq(t) + Gu(t) y(t) = Hq(t) (2)

described by the linear mappings F: Q -4 Q, G: U 4 Q, H: Q 4 Y

of finite-dimensional real vector spaces. Similarly, the time-varying

system

4 = f(q,u,t) Y = 0(q,t) (3)

commutes, namely that satisfying the equation r - i n . = $G. I

0

& SFINITION: Let M - (Q,F,I,G,Y,B) be a system in the category &,

with countable powers and copowers. Then M has r m E r: I --rQ defined by r s in . = $G

J

and p b s e r v a b i l i t ~

o: Q - Y5 defined by nk'a = EIFk.

The ,re@sus of M is then the composition

A 5 f = a- r : I --r Y5.

The of M IS simply

A 0 f = To.f : I 4 Y.

A s id the case of l i nea r systems, r, o and f A (but not E) a r e a l l dyna-

morpbisms, a s we see by reading off the squares of diagram(6) and (7),

and noting that t he composite of dynmorphisms is again a dynamorphism.

MACHINES I N A CATEGORY

Po r e l a t e these concepts t o the general framework of [ I ] , we need

the concepts of functor and of adjoint from category theory. (For fur ther

background on standard category theory see rhe textbooks of Arbib and Manes

[121, Herrlich and Strecker [ l l ] and Mac Lane [lo].) 'Phe def ini t ions of

l e f t and r ight adjoint w i l l seem rather indigest ible a t f i r s t , but should

become clearer when we rework Lemas 1.2 and 1.3 in to the f o r m 1.12 and

1.11 below:

~BFINITION,: Let and i(. be categories. Then a H: -+

comprises a map A k- AH which sends objects A of t o objects AH of

L, together with, for each pa i r A,B of objects of $4, a map

$KA,B) - L(AB,BH): f w f~

which sends X-morphisms f : A - B to L-morphisms £E: AE - BE i n

such a way tha t

( i ) id> = i d AH for every object A of &

( i i ) (f.g)H - fE.gH for every pair f , g of composahle %-morphisms.

( i ) For any category X , the assignment A ++ A, f n f clearly

defines a functor--l.5(i) and ( i i ) are t r i v a l l y satisfied--called

the i den t i t y &&& id* of %.

( i i ) 1 n the category we may f i x on any s e t Xo t o form the assignment

A H AxXO; (f: A -C B ) I- (fxXO: AxXO -L BxXO: (8.x) Y (f(a),x)).

This defines a functor s ince

i d xX = idAxX. : (a,x) ++ (a,x) A 0 0

and (f-g)xXo = (fxXo).(gXfo): (a+) (f(g(a)) ,x).

We c a l l t h i s functor -xXo: %-+a. This notion of functor allows u s t o give one of the key definitions

of [I]. Below, we s h a l l r e l a t e it to the j u s t reviewed ideas of [2].

(i) A X i n a category .% is a functor X: & --+ %.

( i i ) An X-dvnamics is a pair (Q.6) with Q an object (the s t a t e object)

of k and 6: QX -+ Q a mrphism of X . t .

( i i i ) An a is a pair ( 1 , ~ ) with I an object (the input object)

'This ra ther neutral terminology is due t o the f ac t t ha t fo r decomposable machines we in te rpre t (1,~) as specifying the Input map, while fo r non- l inear systems, we think of it as giving the I n i t i a l s t a t e .

(iv) An W u t is a pair (Y,B) with Y an object (the output

object) and 6: Q -+ Y a morphism

(v) An X-svstem i s a 7-tuple (X,Q,&,I,r,Y,B) such tha t X is a

Process, (9.6) an X-dynamics, (1,~) an 1-frame, and (Y,B) an

output map.

(i) Let fi = &g, the category of r e a l vector spaces and l inear

maps. Let the functor X: W--C &g& be the ident i ty functor, QX = 4,

fX = f . Then (2) is captured a s the X-dynamics (Q,F) with I-frame t he

input map (U,G) and output map (Y,H). These a r e the decomposable systems

of above.

( i i ) Let & he the category whose objects are sequences (Qt I t € 8

of r e a l vector spaces and l e t the X-morphisms (Qt) 4 (R ) be sequences t

f f t I t c where each f t : Qt 4 Rt is linear. Iden t i t i es and composi-

t ion a r e the ordinary ones "t-wise". Let the functor X: % -+ % be

defined by (%)X = (Ot) with Qt = QtP1 and, for (ft): (Qt) -+ (Rt),

(ft)X = (Tt) with Et: 0 4 ii t t - ft-l. Then ( 4 ) is captured by the

X-dynamics ((Qt), (Ft)), with I-frame ( ( I ) . (Gt)) and output map

((Y),(Xt)). Here, (1) is the constant sequence (It) with It I, and

(Y) similarly.

(iii) Let & be the category &of s e t s and functions. If Xo is a

fixed s e t define X: &&--*.a by QX = QxXo and, fo r f : Q & R ,

fX = fxXo: QxXo - RxXo, (q,x) -+ (f (q) ,XI.

Then (5) i s described by the X-dynamics (Q,6), with I-frame the i n i t i a l

s t a t e (I,qo) (where I is a one-element' s e t ) , and output map (Y,B).

[Here, the following t r i ck is being used: one can identify

one par t icular element c of an arbi t rary s e t C with one par t icular

mapping y: I -+ C where I is a one-element s e t , since y is uniquely

specified by its image.] The sextuple (Xo,Q,6,qO,Y,8) is the well-known

sequential machine [ 7 1.

Other examples w i l l be given i n the course of the paper.

Next, following 111, we introduce the concepts that allow X-systems

t o "run", i.e. which allow us to set up morphisms that i n some way para l le l

the reachabili ty mrphism of decomposable machines, which, f o r l inear

systems, re la tes past input sequences (as opposed t o single inputs) t o the

present s ta te .

-: Let X: fi X be a process i n $$. The $ & e m &

- X, writ ten Dyn(X) is defined t o be the category whose objects

a r e X-dynamics (Q,6) and whose marphisms a r e X-&&a,r~~ombisms

$: (Q,6) -+ (Q' ,6 ') , being K-morphisms 9: Q 4 Q' sa t isfying

Notice that i d (Q.6) -+ (Q,6) i s a dynamorphism and (9,s) + (Q",6") Q:

is a dynamorphism i f 01: !Q,6) -+ ( Q ' . S ' ) and q2: (9' 16') + (Q1',6")

are; t h i s ensures tha t Dyn(X)is indeed a category.

OBSERVATION: "Forgetting the dynamics" which sends an X-dynamics

(Q,6) t o the s t a t e object Q, and which views a dynamorphism

9: (9,s) -+ (Q1,6') simply as a X-morphism +: Q -+ Q ' defines a functor

U: Dyn(X) -+ % --we c a l l i t the foreef fub &&&&.

With this , we my, r e ca l l from I l l :

pEFWTION: X is an *t Drocess i f the forgetful functor

Dyn(X) 4 X: (Q,S) ++ Q has a l e f c a ( l e f t adjoint is a standard

category theory notion) i.e., i f for each object I of there ex i s t s an

object IX@ with two associated morphisms, ri: I -+ IX@ and a a @ &uen&a uo: (IX@)X --r I X ; which a r e such that, for any X-dynamics

(Q,b) and any K-morphism h: I -+ Q, there exis ts a unique dynamorphism

# h : (1x@,u0) 4 (Q,6) such tha t h#.n = h :

# We c a l l h the @amorohic e x t e n s i q of h.

Several remarks should imedia te ly be made.

( i ) The above def ini t ion may be employed i n a s i tua t ion

where (1,h) is thought of as an I-frame g but t h i s is not always so.

(ii) To make contact with other viewpoints of l e f t adjointness which

the reader m y have, observe tha t the above definition associates with

any h e A [I,$(.[I,Q]] (for any I and (Q,S)), a unique ha i n

[ ( I x @ , ~ ~ ) , (9,611. It is a l so immediate from the diagram that any

B hB E Dyn(X) [(IX@,U~), (9.6) 1 uniquely determines an h by h n - h.

( i i i ) Observe that the f r e e dynamics is uniquely determined t o within

isomorphism by I. For i f Co: IUt + 8. {: I -+ R were another madel

@ of a f ree dynamics then one could form $ by the freeness of IX and

b j the freeness of R such t ha t

# #-# Now observe that h = n I? s a t i s f i e s h'I? = n and i s a dynamorphism.

# The same i s t rue of idIX@. Noting the uniqueness of h i n the def ini t ion

# # 0 8 above, i t follows that ri Pi = id. Similarly, one can show F n = id .

Hence (R.co) and 1 X @ , u 0 are isomorphic. (This r e su l t , when viewed

as a theorem about adjoints, is standard.)

To see that t h i s generalizes u, we may irmnediately note:

If each object of X has countable copowers, then idx is an

input process.

Proof: If one inserts i n (9)

I§ fo r LK @

ino: I -+ I§ for n: I - IX @

5 @ zI: I + I§ for uo: (IX )x - IX @

the f a c t that F and G determine a unique r i n (7) becomes equivalent t o

the f a c t that h (= G ) and 6 (- F) determine a unique ht (= r) i n (9). 0

"Reversing the arrows" i n 1.11, we may r eca l l the following def ini t ion

'from [31. (The m t i o n of r igh t adjoint is standard category theory, and

w i l l be taken up again a t below.)

DEFINITION: X is an p u t ~ u t urocess i f the forgetful functor

Dyn(X) -+ %: ( q , S ) - + Q has a r igh t adjoint; i . e . i f f o r each Y €3 there ex i s t s an object YX with two associated morphisms, A: PX@ A Y @

and a cofree dynamics L: (YX@)X + YX@; with the property tha t , given

any X-dynamics (Q ,6 ) and any X-rnorphisrn h: Q - Y , there exists a unique

dynamorphism hi,: (Q,6) - UX ,L) such tha t A-hg = h : @

For the moment, the choice of object names a s Q and Y has no special

significance.

Jus t as f o r f ree dynamics, so can the cofree dynamics be proved

unique up t o isomorphism.

We see now that a yields

U F I f each object of & has countable powers, then id% is an

output process, with

YX = Yg for each X-object Y @ A 7,

while Y X @ - Q - Y g 2 y

and L =Y ( Y X @ ) X + Y x = Yg + Y 5 . @ 0

&& DEPINITION: X: -3 is a &&$-behavior process i f i t is both

an input process and an output process.

The decomposable machines of 123, including of course l inear systems,

have, by and -processes which are state-behavior. The theory

also embraces tha t of sequential machines:

1 16 PACT The process -xX: i s state-behavior, A --. 0 -

Proof: Given a s e t Xo, l e t X: he the s e t bf a l l f i n i t e s t r ings . .

(xl,. . . ,X >' of elements of X --we write h fo r the empty s t r i ng . ( ) , and n o * distinguish ati element x of Xo from the corresponding s t r ing (x) of Xo...

We must f i r s t check that X = -xXo is an input process. We claim

tha t the fo l lw ing definitions do the t r i c E *

IX@ = IxXo for each s e t I

n: I - IX@ = I+ I ~ x * . i u (is*) 0 ' * *

Po : (IX@)X -+ IX@ - ( I ~ X ~ ) * X ~ - 1xx0: ((i,w),x) t+ ( i s m )

* Here, wx = (5 ,..., x ,x) for w = (xl,. . ..X ) E Xo 8 . d X E Xo. n n

What we must check is that , i n terms of the nbtation of &&, an

if a rbi t rary h: 1 4 Q gives r i s e t o a unique h with (from the l e f t

diagram)

h i , h for each i G I

and (from the r ight diagram)

a * hif(i,wx) = 6(h (i,w),x) fo r each i e I, w c Xo, x E X . if In f a c t , these uniquely define h by the rule

h'(i,w) = 6 *(h(i),w)

where 6* i $ the standard extension of 6 from QxXo * Q to

* * i * QxXo --c Q v ia the induction t u l e 6 (q,A) = q and 6 (q,wx) = 8(6 (q,w) .x) .

We next check that X - -xXo is an output process with the formulae

x* * YX@ - Y 0 the set of a l l maps from X t o Y * A: YX - Y = 3 ( 0 - Y : f - f t n )

@ * * L: (YX@)X 4 Yx, = Poxx + 20: (f ,x) M fLx

0

where fLx denotes composition of f with "the l e f t juxtaposition of x"

* * map, Lx: X -+Xo: wI+-xw. Thus fLx(w) - f(rm). The diagrams of

0

require that , given an arb i t ra ry h: Q - + Y , there be hg with

* [hl(q)] (A) = h(q) fo r each qcQ ( reca l l that hg(q) is a map Xo --+ Y)

and

hr(6(q,x)) = ht(q)Lx fo r each qrQ, xeX0

i.e. [hn(6 (Q,x)) 1 (w) = lha(q)Lxl (w) = Ihg(q) I (xw)

* fo r each q E Q, x E Xo and w E Xo. A routine calculation then ver i f i es

tha t hi(q) is uniqueLy determined t o be

* * hp(q): Xo + Y: w C I . h[6 (q.w)l . D

& pgELKUB4: Let M = @,Q, 6,I,r,Y,B) be an X-system. If X i s

an input process, the p& @ b i l i t v r: I X -+ Q of M i s the unique

dynamorphism r : (IX@,P,) 3 (Q.6) which s a t i s f i e s

If X is an output process, the o b s e r v a b i l i t e m o: q -+PX of M @ is the unique dynamorphism a: (Q,6) -+ (=@,I) which s a t i s f i e s

@ If X is a stare-behavior process, the t o t a l response fA: I X 4 YX@

A of M i s then the composite dynamorphism f = 0.r: ( I X @ , ~ ~ ~ ) -+ (YX ,L). @

The mnse (or behavior) of M i s simply

. . A @ f = A - f = 6-r: IX -+ Y

and is not adynamorphism.

Note that, by 1.12 and L&, these definitions subsume those far

decomposable machines in u. They also include many other situations, for example, the sequential machine case.

IE: An X-system (X,Q,G,I,r,Y,B) becomes a sequential k h i n e l . l a E X A M P

(Xo,Q,6,qo,Y,6) in the sense of example (iii), if we take % to be a, X to be the functor -xX . Q to be a set; 6 a map QxX' -+ Q; I to he a

0' 0

one element set so that r: I -+ Q becomes an element qo c Q and * *

IxX " X ; Y to be a set and 8: Q + Y . Using the formulae obtained in 0 0

1.16, - we see immediately that such a machine M has * *

reachability map r: Xo + Q : w* 6 (qo,w)

sending w to the state M after starting in its

initial state qo and receiving input string w; *

observability map 0:. Q 4 $0 : q u Mq

* * where Mq: Xo 2 Y : w n 8[6 (q,w)] tells us the

output of M started in state q which will be observed

after feeding it any input string w; *

total response f': X: --r $O : w ++ M * 6 (qo.w); * *

system behavior f: Xo --t Y : w H 6 [6 (qo,w) 1.

This does indeed match the 'classical' formulations of sequential machine

theory.

As we noted in Section 3 of [I], it has become standard category theory

to generalize the notion of one-to-one maps and onto maps between sets to

obtain classes 5 and (standing for generalized epi and mono respectively)

of morphisms in a category 3. Although in any category there are epi maps

and mono maps, the maps in & or may form proper subcategories of these

classes of maps. Just as in '& f: A -+ B always canbe thought of as

thecomposition of an onto map followed by a one-to-one map, so one can

postulate that in an arbitrary category X , any morphism f has f = gh

for h E E , g 7q. With several additional postulates, R is said to

have an image factorization system.

PEPINITION: An X-system (X,Q,6,1,r,Y,$) is &a& if X is input

@ and r: IX 4 Q is in .Z ("all states can be reached by proper choice of

inputs"). It is observable if X is output and a: Q + YX@ is in 712 ("different states can, in theory, be distinguished by input tests").

Sometimes a class E'oE epimorphisms is defined without an q ; in this case we can speak of "reachable" without an observability theory, but the

crucial definition of "minimal realization" (see =below) is still

possible.

Generalizing Definition && of 123 we have:

DEFINITION: Fixing I and Y, but letting the state-space Q vary,

consider the category whose oblects are systems M = (X,Q,G,I,r,Y,B); and

whose morphisms are simulations Ji: M --t M' (we say M simulates M I ) , i.e.

dynamorphisms Ji: (Q,6) --r (Q1,6') which comute with the input and output:

It is an inmediate consequence of the definitions that a simulation

commutes with (if X is input) the reachability and (if X is output) obser-

vability maps, i.e. if X is state-behavior,the dynamorphism Ji is a simulation

and satisfies the diagram

Of course, if the dynamorphism $ satisfies this diagram, it is certainly

a simulation. In particular, then, the existence of a simulation guarantees

A that the two machines have the same (total) response: f = or = o'qr =

= f'b.

1.21 DEFINITION: Let I, Y be fixed, let X: he an inpput (state- =

@ behavior) process and let f: IX -Y be an arbitrary X-morphism (with

A dynamorphic coextension f = f ' IX @

# . -+ EX@) . We say a system M is a A . realization of the (total) response f (fA) if f (f ) 1 s the (total) response

of M. We say M is a minimal realization if it is a reachable realization

of f with the property that, given any other reachable realization M' of f,

there exists a unique simulation I): M' -+M. In other words, M is a

terminal object in the category whose objects are reachable realizations of

f, and whose morphisms are simulations (composition and identities being at

the level of z). Minimal means "all states are used" and "states with the same effect in any simulation (M') are merged ($) to a single state (of M)".

Minimal usually coincides with "reachable and observable" if X is state-

behavior 131.

In the case of decomposable machines and sequential machines, "minimal"

certainly coincides with the standard meaning.

ADJOINT PROCESSES

In this subsection we introduce an important class of state-behavior

processes--the adjoint processes. This will require the presentation of

further (standard) category theory results through 1.26) which we

include here for completeness.

DFLNITION: We say that a collection (ini: Ai -+ A I i E J) of

$L,.-morphisms with common codomain A is a ~op= for the collection

(Ai / I E J) of objects of X if it has the property that, given any

other collection (fi: Ai -+ A' I L E J), there exists a unique morphism

f: A-+A' such that

ini Ai . A

A'

conmutes. [All coproducts of a given family Ai are isomorphic. If the

index set i s the set3 of nonnegative integers, and the Ai are equal, then their

coproduct is just the countable copower, familiar from 121.1

We say X has finite coproducts if each finite collection (AL, ..., An) has a coproduct, the object of which we may denote %Qn Ai, or

A1+ ...+A ,. Similarly, has co~mt.&&.e cooroducts if every countable

collection (%,A2, ..., A ,... ) has a coproduct. n

We say a functor X preserves coDroducts if, whenever (ini: Ai -+ A 1 i E J) is a coproduct of the Ai, then (iniX: AiX -I AX I i E J) is a

coproduct of the AiX.

Just as arrow reversal turns copowers into powers 121, so may We

reverse the arrows in 1.22 to define products (xi: A -f Ai I i € J) by

a diagram n i

A-

DEFINITIQN: A functor F: d +@ has a &&€ &&&& if there

exists a functor F': 8 -+d (the rigkt adjoint of F) such that to each

B in ,$ there corresponds a 6 -morphism BFF' -% B such that to each

b-morphism g: AF --+ B there corresponds a unique A-morphism 4: A - BF' such that

commutes.

Notice that given a $: A + BF', one immediately has a g: AF --+ B

by g = E.$F. Indeed, adjunctions can be thought of as a bijection between

sets of morphisms:

The reader for whom this concept is new could profitably see how

specializes this definition.

L& -: Let &&= 'Vector Spaces and Linear Maps,,. and let

F: -+ be the identity functor (Q w.Q; f f). Then F is its

own right adjoint--setting F - F' = id , E =; idg, we have that (i7) is

satisfied with 4 - g :

id^ B-B B

L : Let && = <Sets and Maps>, and let X - -xX : -+ & U E i L B Z J 0 - be the functor Q b QXXo; f H fXXo where

fxxo: gxxo ~ ' x x ~ : (q,x) ++ (f(q),x) . Then X = -xX has rlght adjoint which sends Q to the set gX0

0

of all maps from Xo to Q. E: B'OXX~ -+ B is the evaluation (f ,x) c f ( x ) , and we have that (17) is satisfied on taking Q (a) : X -+ B : x ++ g(a,x) :

0

The following is a standard result connecting the ideas of coproduct

and right adjoint:

bL2h -- LEB&: If F : 4 -8 has a right adjoint, then F preserves coproducrs.

ini Proof: Let (Ai A I i E I) be a coproduct. To show that

in.F (AiF A AF I i 6 I) is also a coproduct, we must show how to uniquely

complete the diagram

B

when gi: AiF 4 B, except for the specification of domain and codomain, is

an arbitrary collection of d-morphisms. Let us define Oi, for each i E I,

by the following variant of (17):

B BF-F BF'

? ! +i

A> *i

we may then use the fact that ( A ~ 3 A) is a coproduct to define

6: A -' BF' by the rule

ini Ai - A

C BF'

Now the g: A F 4 B corresponding to 4: A+ BF' in the style of (17) is

just g = E.c$F, and it is a straightforward exercise to check that this g

is the unique solution of (18). 0

,CONSTRUCTION: In preparation for delineating a class of input pra- *

cesses, we note the following construction of a functor X from a functor

x: -+ x . He

Let ( A + ] a E I) be a collection of functors; and assume that

(8 has I-indexed coproducts. Then we can form the functor

(uniquely up to isomorphism) defined on objects by

and on morphisms by

ing A 1 AHa

a I 5 If - Ifll

J. ing 14 2 ll A"$ a

BH6

[Identify fi: Ai -+ A' in (15) with in6.fHg: AH6 4 U A7Ha here.] a

It is then straightforward to check that He is a functor by considering

a its action on identities and composition of morphisms.

Now, given any functor X: & -+ a,. define X" for any n 2 0 by the mles

0 X = id . .

x*' = X".X for n t o

and then set

And now we have a statement on the existence of input processes:

&& -: Let X have, and let X: X -+ X preserve, countable co-

@ * products. Then X is an input process and X = X .

* * * Proof: If lX@ is to be IX we must define the morphisms uo: IX X -+ IX and

inn * 11: I -+ JX@ described in u. Since X preserves the coproduct IXn IX , we have that

in X * rxn+l = 11 Ix x

is also a coproduct. Thus we may define uo by the obvious rule

* We define q: I - IX to be simply ino Let us check that this works,

i.e. that the diagrams

="+l

i n n y uop in x

I ". Ix* b IX*

I] ://x

QX 6

# * define a unique h : IX 4 Q. But the left-hand diagramsays

hi.ino = h

while the right-liand diagram asser t s that

h#-inn+l = 6-h'x-innx = 6 . (hi'.inn)x n r 0

8 . Evidently, ha-ino, h -=nl, ... can be defined successively, so that these

# equations define the unique h which s a t i s f i e s the diagrams. 0

A special case of t h i s r e su l t (where X is of the form -exo for a

sui table "tensor product" @ and object Xo) l i e s a t the heart of the theories

of Goguen [13] and Wrig-Kreowski [14].

Using u, we have

L22 B: I f has countable coproducts and X: -+ X has a

r ight adjoint, then X is an input process, and X@ = x*. 0

The functor X = -XXo: &&-+a is cer ta inly included within the

ambit of the above discussions. We could have obtained the def ini t ions of

@ IX and so on i n the early part of once onechecks preservation of

coproduce, o r reacies the observation of tha t the r ight adjoint of

-*x, i s (-)'o.

The question a r i ses of when X is also an output process, i . e . when

X is s t a t e behavior. We have the following resu l t from [3]:

L.32 THKQEZ& Let $. have countable coproducts and products, and l e t X

have a r igh t adjoint. Then X is state-behavior.

Proof: I n outline, t h i s can be argued by observing that the r ight adjoint

X' of X defines a functor zop -t -&OP which i t s e l f has a r ight adjoint. It

then is an input process. Taking duals leads t o the conclusion t ha t X is

an output process. One has the important resu l t a lso that

For convenience, and because of the importance of the concept, we

formalize t h i s c lass of processes:

U -N: Let X: X, + X be a functor. Then X is an*

~rocess i f fi has countable coproducts and products, and X has a r ight

* adjoint.

IWXPLBS:

( i ) I n case X = -xXo: Set+ we have X is an adjoint process. * * n As computed ear l i e r , IX@ = IxXo " U IX and YX = 30 = y%n .

n20 @ n

( i i ) In case X = idx and 3C has countable coproducts and products,

X is an adjoint process. As computed ear l i e r , IX@ = 1' and YX = Y @ 5 -

( i i i ) Let X be a s i n the time-varying l inear system example of 1.8.

Then X again is adjoint. For products a r e formed t-wise i n w, and copro-

ducts a r e "weak di rec t sums". Define (Qt)X' = (Qt+l) and, fo r -

(ft): (Qt) + (Rt) , (ft)X. = ( f t ) : (Qt+l) -+ (Rt+l) . Then X' can be

shown to be a functor and the r igh t adjoint of X, using 1.23.

As discussed i n [3], adjoint processes a r e r i ch enough t o i nc lude the

processes appearing i n seq'uentiaL machines, nondeterministic machines, Boolean

machines, metric machines and topological machines.

* Actually, i f X is state-behavior, has a r igh t adjoint and X is simply

assumed tg have coproducts, a nontr ivial argument shows that YX has t o be @ Y(X') . So i n t h i s case, the specif ic assumption that X has products is n-0 not necessary.

MORE ON DPNAMOWHISMS

In this final subsection of Section 1, we present some useful properties

of dynamorphisms. We start by examining approximants to dynamorphic extensions:

1 . 3 1 L@E?&: Given any morphism a: A 4 Q, and a dynamics 6 : QX -+ Q,

@ let a': AX -+ Q be the dynamorphic extension of a

and let a be .the apuroximant to a # n

axn , (n) a n =~~"-----+QX~-Q

where 6(n) is the n-fold iteration of the dynamics 6 defined inductively by

, (n) 6(O)= idQ: Q -+ Q while = Q?" Q X ~ - Q.

S n+l& QX ----+ . By au easy induction,we have also 6(*') = QX

Clearly, then, a = 6.anX . As a special case, let lln be the -inclusion n+l

?J (n)

nn: Axn * AX@e L Ax @

where (") is defined from ilo just as 6(n) was defined from 6. Again, "0

= ".qnX. Then for all n 2 0 we have the commutativity of

n Proof: The case n - 0 is just a = a .q. Far the induction step, we

assume the commutativity of (20) for a given n, and show it also holds when

n is replaced by n+l. We have:

The triangle is obtained from (20) and the fact that X is a functor; while

B the square says that a is a dynamorphism. Thus

In case X is an adloint process, we can show that

This is easy to see by an inductive argument. As a basis, observe that

n1 = vo'nX = .in X = inl, using the last diagram in u. Suppose that uo 0

nn = in,. Then nnC1 = u -n X = vo'innX = in o n n+l' again using u. In the next lemma, we shall label v0 and n with the object A which

generates the free dynamics with which they are associated, so that we have

@ An: A -+ AX@; Avo: AX X 4 AX@. Just as we may consider a pair (q,x) G

* QXXo for a sequential machine as a pair (q,(x)) E QXXo by idenrifying

the input x with the input string ( x ) ; so may we, given any input process

X: X -+& and any object A of %, define (by J&J

Again, given eny morphism f: A 4 B we can define a morpbimr

@ fX : AX@ PX@ BX@ by the diagram

[It is a straightforward exercise to show that, with this action on mor-

@ phisms, X becomes a funetor 4 3C. I Then we have the following lema:

f U Lk2!&&: For all f: A 4 B, we have the comutativity of

Proof: We simply use the definition of Anl and Bnl to expand (22) to

the diagram

@

and note that the left-hand square commutes by the fact that (21) commutes

and that X is a functor; while the right-hand square just says that

@ fX = (Bn-f)' is a dynamorphism. 0

L35 DEFINITION: The g& 6@: QX@ 4 Q of a dynamics (Q,6) is

the dynamorphi~ extension

Lemma 132 then imediately yields:

' m e category theorist will recognize that (21) and (22) say that n and n1 are natural transformations, but this general notion need not detain us here.

Finally we observe that "a dyxamorphisn, since ic commutes with one-step

transitions, must commute with all transitions":

m: Given any dynamorphism 31: (Q,6) + (Q',6'), we have the

commutativity of @

QX@ 6 Q

@ Proof: $ a 6 =+.id = + @ # Q

since S = (idQ) . @ @ (6') -+X .Qn = (6')@.~'~1.31 by (21) with f = +: Q -+ Q'

= idq, ' $ = + since (6')@ = (idQ,) R

@ @ Thus *.6@ = 31' = (6') .@X , and SO our diagram commutes. 0

2. Nerode Muivalence Approach

In Section 3 of [Z] we presented the notion of an image factorization

system; and then showed in Section 4 how this yielded a simple minimal

realization theory for decomposable systems. In [3] we extended this

approach to X-systems for any state-behavior process X (yielding a theory

akin to that of Bainbridge [211).

However, in [I], we built upon the Nerode equivalence approach to

minimal realization to yield a theory applicable to input processes, even

if they are not state-behavior. Our aim in this section is to give an

internal Nerode approach which is both more elegant and more powerful than

the internal approach of [I].

In what follows, let X: 3 -+ X be a fixed input process, I, Y

fixed objects of k.

@ DEBINITION: Given a response morphism f: IX 4 Y, we say that a

@ # I pair af morphisms a,y: E --r IX abstractle if fa = fy

I\ . @ where a# (resp. y ) IS the unique dynamorphic extension EX --t IX@ of

a (resp. y).

We then say that a,y: Ef 4 IX@ is abstract Kezzk eauivalenre # f if fa = fya holds and if, wherever f (a')' = f(y')# also holds

for aT,y': E' -+ IX@ there exists a unique $: E' -+ Ef such that

commutes, i.e. a$ = a' and yiy = y'.

When X = -XXo: &43&$=, t h i s reduces [ I ] t o the familiar condition

* tha t two s t r ings wl and w2 of Xo are equivalent i f f f(wlw) = f (w2w) for

* a l l w i n Xo.

It is clear t ha t the abstract Nerode equivalence is a t e d n a l object

i n a category ( r eca l l 12, u) with morphisms of the kind depicted i n (1).

and so is unique up t o isomorphism (by Lemma && Of [?.I).

Por our work on t r e e automata, a minor variant on t h i s def ini t ion v i l l

be required. A pair t,u: A - B is termed reflexivq i f there ex i s t s a

v: B 4 A making the following diagram commutative:

t A-B

I n & t h i s simply means tha t every b 6 B is such tha t fo r some a E A,

t (a ) = u(a) = b. The abstract Nerode equivalence of t , when i t exis ts , is

a reflexive pair , because by the Nerode property, there e x i s t s $ making the

following diagram commutative:

a We say tha t a,y: E --+ IX@ is the reflexive Nerode ewi~aLenCc gf

f-7- # f i f fa' - fy' and i f , whenever f ( o r l ) = ffy ') ' fo r a ref lexive

a ' ,y7: E' -A IX@ there ex i s t s a unique 4 making (I) commutative.

Evidently, any abstract Nerode equivalence is a ref lexive Nerode equiv-

alence, hut not necessarily conversely. (Indeed, w e sha l l exhibit i n

Section 3 a ref lexive Nerode equivalence which is not an abstract Nerode

equivalence.)

An external version [XI of results by iterating X:

@ DEFINITION: Given a response f: IX - Y, a pair of morphisms

a,y: E IX@ are externally if

f.up)-u~n = f.lrf).ye (all n E 3 (2)

@ where (cf. && u:): IX xn -+ IX@ is the n-fold iteration of the free

dynamics vo. Note that =also shows that (2) can be rephrased as

# £-an = f.yn, where a is the n-step approximant to a . a,y: Ef + M @

n

is the e a d ~ a l a a ~ . Q£ f if a,y are externally f-equiva-

lent and if whenever uT,y': B' 4 IX@ are externally £-equivalent there

exists a unique $: E' 4 Ef as in (1).

The Nerode equivalence approach to minimal realization is: ."the state-

@ space Q of the minimal realization is defined by Qf = IX /EfU. The f

categorical problem is one of capturing "how to divide out by the equivalence

relation EfU. This is what coequalizers are for:

DEFINITION: We say a K-morphism A h B is e coeoualizer iff

there exists a pair p1,p2: R - + A of morphisms such that hapl - h'pZ, and such that whenever A B' satisfies hl-p = h'-p2, there is a 1

unique K-morphism B B such that $.h = h'

In this situation, we call h the ~oeaualizer & pl& p2, and write

h = coeq(pl,p2). Standard category theory arg-ts establish the

uniqueness up to isomorphism of coeq(pl,p2) if it exists.

An immediate consequence of the definition is the following standard

property.

LLc Every coequalizer is an epimorphism.

Proof: Suppose h = coeq(pl,p2) and that kl.h = k2*h for some kl and k2.

We must prove that kl = k2. But if we take h' = kl-h in Q-- which we

' may since hf'p1 - 5-(hap1) = kl.(h'p2) = hr-p2 --we see that there is

a unique $ such that $ah = h'. But h' = kl-h = k2-h . by hypothesis, and so we must have that k = k2. Thus h is an epimorphism. 1 0

2.5 EZQEUi: In && coeq(pl,p2) can be found in the following way.

Say that two elements al,a2 E A are strictly equivalent if there exists

some r E R for which pl(r) = al, p2(r) = a2, and say that two elkmencs

al.a2 E A are equivalent if al = a2 or if there exists for some bo,bl, .. .,bn E A,

adjacent pairs of the sequence (al,bo,bl, ..., b ,a ) which are strictly n 2

equivalent. This defines an equivalence relation R, which is generated

by the image of R, ( p ) p 2 1 r E R in AXA. Define h as the

canonical projection A 4 A/E.

Suppose that h'-p = h"p2 For a e AIR, define $(a) = hl(al) 1

where al is any member of the equivalence class AIK. If al,a2 are in the

same equivalence class, set up the sequence (a ,b b . . . ,bn,a2) With 1 0' 1'

adjacent pairs strictly equivalent. Then h' (al) = hpl(r) .= hp2(r) =hl(bo).

- - ... = h1(a2). This shows that $ is well defined; $h = h' is immediate,

and $ is obviously unique.

In coeq(pl,p2) exists and is obtained as A/kar(p -p ). 1 2

In both & and %, coequalizers coincide with the epimaps.

The Nerode equivalence theorem of [l] generalizes easily into the

framework of Definition &. The four postulates have been well motivated

in 111. We shall see below that they are satisfied in a wide range of

circumstances:

2.& llljQUX: Let the response morphism f: IX@ + Y satisfy the following

four postulates*:

@ Postulare 1: f has an abstract Nerode equivalence Ef 5 IX . @ Y

Postulate 2: rf = coeq(a,y): IX 4 Qf exists.

Postulate 3: There exists a dynamics (Qf,6£) with rf: (IX@,~,) --r

(Qf, Gf) a dynamorphism.

Postulate 4: X is such that if an X-dynamics (Q,S) has reachability map

@ @ . r: IX --t Q with r a coequalizer, then either rX or rX 1s

an epimorphism.

Then f has a minimal coequalizer-reachable (i-e., E is the class of

coequalizers) realization.

If in Postulate 1 the assumption of abstract Nerode equivalence is

replaced by reflexive Nerode equivalence, then f has a minimal reflexive-

coequalizer-reachable realization (where a reflexive coequalizer is one

coequalizing a reflexive pair).

i/ ii Proof: By definition of a', we have a q = a. By u, we have faB = fy . nus

# 5.a - f.a'.q = fay = f.y . Then we may use the fact that r = coeq(a,y): f

* The postulates are identical with those of 111, save that the abstract

rather than external Nerode equivalence is used, and Postulate 4 is mildly loosened, by allowing +x@, rather than rX, to be an epimorphism.

By Corollary the perimeter of (A)(C) is the statement that r is

a dynamorphism. 0

At first &ht, it might seem that verification of postulates 1 to 4

would be a major bar to application of this theorem in many cases, and the

postulates look very much as if they are tailored to guarantee the desired

results, rather than reflecting commonly encountered properties. However,

we shall see below that the postulates can be satisfied if certain common

conditions are fulfilled. Purther, we shall see later that in a wide

variety of cases, all coequalizers are reflexive coequalizers, so that the

# reflexive assumption is not a major one. Recalling Lemma (a . q = an

for any a: A + Q), we can verify that the external Nerode theory [I] is

a special case of the abstract approach. [Actually, this claim is open to

a charge that it is subjectively based. As the following theorem shows, it

is not true that if the external Nerode equivalence holds, then the abstract

equivalence holds. Rather, we need side conditions associated with the

external Nerode equivalence for the abstract equivalence to hold. To the

extent that these side conditions are minor, then, the abstract generalizes

the external.]

@ 2.7 THECREW Let f : IX --c Y satisfy

Postulate 1': f has an external Nerode equivalence a,y: E --c IX @ f

aatisfyingpostulates 2 and3 of2.6.

Then Ef is the abstract Nerode equivalence of f. -

/t # Proof: As in the proof of u, we note that , (rf-a) = Ifa - Thus

To show E is indeed a Nerode equivalence in the sense of a. we just f

note that if T -& IX@ satisfy fats = f.u# then, by U

B {I (n) . Uxn (n).tl(n=f.t =tf'Inn=uf.Ilin=f.u =£.,lo f -vO n n

for all n c g, yielding the desired $: T -+ Ef by postulate 1'. O

Taking into account Theorem &&, and a corresponding theorem of 111

using the external equivalence, it is evident that if an f : IX@ -+ Y is

guaranteed to have a minimal coequalizer-reachable realization via the

external theory of [I], then it is guafanteed to have such a realization

via the abstract equivalence theory of 2.6. In this sense, the abstract

equivalence is a true generalization of the external equivalence.

In one situation of some importance, the abstract and external Nerode

equivalences become the same thing. For adjoint machines, recall from =that

# a and a are related by n

# and to say that fa = fy' is the same thing as to say f .an = fDyn

for all n E g. @ In &&, we shall give an example of an f: IX -+ Y for which

the reflexive Nerode equivalence exists but for which the external Nerode

equivalenca fails.

Our task now is to see what happens to the Nerode equivalence relation

when X is a state-behavior process. The key is the commutative diagram:

From this we read off two diagrams

A Since f and a' are dynamorphisms, so is their composite, and we deduce

that

Hence, it is immediate from a t h a t :

@ Q -: Given E: IX + Y for X a state-behavior process,

the pair of morphisms a , y : E -+ IX@ are abstractly f-equivalent iff

A A f . a = f . y . ( 8 ) 0

In order to find conditions under which there can be morphisms a,y asso-

A ciated with a prescribed f and satisfying ( a ) , we recall from category

theory the definition:

a PEPINITIOEI: Given a morphism g: A 4 B, the pair p,q: E -+ A

is called the M e 1 Dair of g if it satisfies the equality sap = g.q,

and if, whenever the pair pl,q': E' - + A satisfies g-p' - g-q', there

exists a unique $: E' -+ E with p.$ = p' and .q.$ = q' :

k has kernel oairs if every %-mrphism g has a kernel pair.

Set3 Vect and most familiar categories have kernel pairs. Given

g: A - B, the usual construction is to set E = ( (al,a2) 1 g(al) - g(a2) > c AxA, with p,q: E--+ A the projections p(a ,a ) = al, q(a a ) = a 1 2 1'2 2'

Comparing 2.1 and Qwe imnediately deduce from 2.8 that

2.10 Let X be a state-behavior process, and let

A A f : IX@ + YX@ be a total response map. Then if f has a kernel pair

@ a,y: Ef-f IX , it is the abstract Nerode equivalence of f, and conversely.

0

The desired theorem asserts that the Nerode realization theorem a applies to a state-behavior process in any category equipped to "divide out

by equivalence relations".

Let X be such that every morphism has a kernel pair which,

in turn, has a caequalizer, and let X: X 4% be state-behavior. Then

@ every response morphism f: M + Y has a coequalizer-minimal realization

by the Nerode realization construction of &; indeed, the minimal

"state-space" object Q is the coequalizer of the kernel pair E of f f

A @ f : IX h Y X @ and E is the abstract Nerode equivalence of f. f

Proof: As a result of we see that we have only to establish postulates

@ 3 and 4 of 2.6. The fundamental observation is that the functor X has

X as a right adjoint: @

Therefore, by a var iant on t h e argument of Lemma u, (see [lo, p. 114,

@ dual of Theorem I ] ) , X preserves coequalizer diagrams, i .e . i f

@ @ c = coeq(a,h) then CX@ - coeq(aX ,hX ). P ~ s t u l a t e 4 is now immediate,

noting 2. For t h e proof of pos tu la t e 3, argue a s follows:

Let a,y: E + IX@ be t h e kernel p a i r o f f A , and l e t rf - coeq(a,y). f

The f a c t t h a t fA.a = fA-y impl ies t h a t there e x i s t s t sa t i s fy ing

Now by de f in i t ion of the f r e e run map II: we have @ @ u0.IX n = id=@ . Again, by (21) of Section 1, we have the a and y squares of

each commute. Thus we have t h a t

@ @ @ @ poeaX 'E q = II .IX 11-a = a . f 0

@ Since f A , uo and ax@ a r e a l l dynamorphisms, it then follows t h a t

# fA.l l ,@-a~@ = (fA.a) . A But fi.a = f -y and s o

A Then, s ince (a,y) is t h e kernel pa i r of f , t he re e x i s t s u completing

mus rf.ll:.ax@ = rf.a.u = r -y.u since r = coeq(ci,~) f £

@ @ = r .p -yX . f o

@ @ Since, by our clalm on x@, r X@ = coeq(aX ,yX ) there then ex is t s such

f

that v.rfx@ = r f . p @ o

NOW by Lemma we have the c o m t a t i v i t y of

Noting that , by m, we have u : - ~ ~ @ n l = u0 we may sp l ice (10) atop

(9) t o obtain

IX@X r f X r QX

which asser t s that rf i s a dynaaorphism (IX@,~~) 4 (Q,Qnl-v). Thus

postulate 3 i s sa t i s f ied , wfth 6 £ = v'Qnl : QX -+ Q the requis i t e dyna-

mics of our minimal realization. 0

% tie this back to the image factorizations of [ZJ, let's see how we

might attempt to construct a coequalizer-mono factorizationt of f: A + B

in X (suggested by the "first isomorphism theorem" of set theory): Let

p: A + G r coeq(t t ) where (ti,t2) is the kemel pair of f. Since 1' 2

ftl = ft2, there ezists a unique i: Q 4 B with f - p-i; call this the canonical factorization of f. While it is not always possible to prove that

i is mono, we do have the following result which says this is the only way

to construct the coequalizer-mono factorization if there is one.

2.12 -- EKC: Assume that the coequalizer of the kernel pair of f: A + B

exists. Then if E has a ~~equalizer-mono factorization it must coincide

with the can&ical factorization.

Proof: Let f = psi be a coequalizer-mono factorization of f, and let

(tl,t2) be the kernel pair of f. By hypothesis, we can write p = coeq(ul u ) ' 2

for some pair (u1,u2).

It suffices to show that p coeq(tl,t2). Suppose r-tl = rat2. As

f.u - f . ~ ~ rhere exists unique r with rSti - ui 6ince (tl,t2) is the 1

kernel pair of f; and it follows that r.ul - r.u2 inducing a unique

JI : Q 4 R with $.p '- r as degired to yield our conclusion that . P = coeq(tl,t2). 0

t ~ t is straightforward to prove that, if every morphism in .% may be factored as f - i.p with p a coequalizer and i a monomorphism, then E = {coequalizers~, @ = Imonomorphisms) defines an image factorization system for &,.

At the risk of minor redundancy, let us explain how =integrates

with the preceding material. For state-behavior machines, we have two

distinct approaches to realization:

i) Given existence of kernel pairs and coequalizers, we set up the

A kernel pair of f (which we can show is the abstract Nerade equi-

valence m, obtain its coequalizer, and note that it is a dyuamorphism u. A minimal realization results.

ii) Given existence of caequalizer-mono factorizations, we factor f A

and note [31 that the coequalizer and mono morphisms are both

dynamorphisms. A minimal realization results.

What Fact &g says is that if there exist kernel pairs and coequalizer-

mono factorizations, both approaches to realization yield the same thing.

Actually, even more is true, and we can connect realization theories

using reflexive and abstract Nerode equivalences.

PROPOSITIW Let X be a category possessing coequalizers and kernel

pairs. Then every coequalizer is a reflexive coequalizer.

Proof: Let r: A+ B = coeq(pl,p2) for pi: C 4 A. Let tl,t2: D 4 A

be the kelnel pair of r. As shown in 2.12, r = coeq(t,, t2).

It remains to show that kernel pairs ere reflexive.

BY the kernel pair property, there exists $: A 4 D making the above

diagram commute. 0

Now refer back. to the statement of Theorem a. Evidently,. the

abstract or reflexiveNsrode equivalences together with Postulates 2

through 4, with a kernel pair existence assmaption in the case of the

reflexive equivalence, yield a-minimal coequalizer-reachable realization

Of f by essentially identical construction procedures.

3. Gee Automata: &ire Successes and Infinite Failures.

In this section we apply the realization the~ry of Section 2 to a

class of systems in which have proved very important in computer science:

the tree automata. Since the study of tree automata plays no role in the

study of reachability and observability conditions in Section 4 (and only

a partial role in Section 5 ) , many control theorists may wish to omit this

section unless they have an interest in theoj of computation. (An over-

view of applications is given in Chapter 4 of 171).

Consider the arithmetic tree:

which we may regard as a representation of the arithmetic expression

To evaluate it we need two functions (where 3 is the set of all integers)

which we may combine into a single map (m-n if w = -

when we use the coproduct (disjoint union) diagrams with in-(m,n) = (m,n,-),

in,(m,n) = (m,n,x) in in.

[Note that the meaning of the symbols + and x in * Zf + X is not

addition and multiplication, butdisjointmion and cartesian product.]

We cell {-,XI the label set and we say that - and ,: are both 2-ary labels

since each labels an operator which acts on 2 arguments. With this back-

ground, the reader can appreciate that the automaton which processes whole

arithmetic trees such as that shovn above is a special case of the tree

automata which we arrive at in the next.few paragraphs:

a DEFINITIOB: A label set,is a set 82 together with a map v which assigns

to each w in Cl a cardinalv(w). [For much of this section, the reader may

think of v (w) as simply being an integer, v (o) E &. When v(o) is in 2 for

all o in Q, we call Q finitare.] We call v(w) the of w, and set Q n

to be the set of w in a with arity n. Note that v(o) may equal zero,

i.e. n may be nonempty. 0

We shall now show how to associate an input process with each label

set 0.

DEFINITION: Given a label set Q, we define a functor Xn : &&+&

by the object mapping

while the action of Xn on morpbisms is given by

n where f (ql,. . . ,q,,) = (f (ql), . . . .f (qn)).

When n = 0, we view Qn as a one-element set (11, and so the only

candidate for fO: Q0 - (Qt)O sends 1 to 1.

In our motivating example, R2 = {-,XI; On = 0, the empty set,

for n + 2. Thus

and f% : 11 9' + fi (Q')~ sends (q1.q2,w) to (f (ql) ,f (q2) .u). {-,x> {-,XI

Now just as our arithmetic tree processing required maps 6- and Ax, so do

we obtain the more general concept:

3.3 DEFINITION: An ,--is simply an %-dynamics

6: QXQ 4 Q

which is the same thing as a collection of maps 6 : QV(') -+ Q, one for 0

each w in Q:

Note that when v(w) = 0, 6 has domain a one-element set, and so can be W

thought of as a particular single element of Q.

An ,-algebra hmammUm h: (Q,6) A (Q1,6') is then an X,-dyna-

morphism, i.e. a map h: Q + Q' for which

Equivalently, for all o in ,,with n = v(w),

6'-hXn.in = h-6.i~ n,u n.0'

By && this is equivalent to

6'ain -hn = h.6.inn n,w .o

and by the remark above to

6'o-hn - h.6 W

i.e.

6'w(h(q1),....h(Q) = h(60(~l....,~)) ( 5 )

for all (ql,. ..,a) in Q ~ , and w 6 whatever the choice of the n'

cardinal n. In case n = 0, let 6 pick out qo E Q and 6Iw pick out W

<>

, 9,' E Q'; then (5) says that qo' = h(qO).

We say an Q-algebra is finitarv just in case Q is finitary (On # 0 only if

n finite) whether or not g is a finite set. The theory of finitary

n-algebras is studied in 1151 and [16 I.

3.4 Dm=: A automaton or -a &!smmn is an X -system, R

i.e. a 7-tuple (Xn,Q,6,I,r,Y,6) where XQ, Q and 6 have the interpretations

given above, T: I -+ Q is an arbitrary morphism of a, and 5: Q + Y is

an arbitrary morphism of s. Justification for the adjective "tree" has yet to be presented: note that I, Q and Y are not trees.

We shall now work towards establishing that XQ is an input process.

Q DEFINITION: Given any set I, define the set 3 inductively by 1,n

Basis Step: For each i E I, or w %, put (i) or w in 3 1,n Induction Step: For each o e R and each n-tuple (tl,. . . , t ) already

n n

in 35,n, put (tl, ...,t in 31,n.

At this point, (tl, ..., tn)w is a formal symbol, not the evaluation of

a function; in &below, however, we reinterpret (.)w as an n-ary

function on gISn.

We call elements of 3 (R,I)-trees, or Sl-trees over I generators. I,n

The reason for this terminology is best presented by a build-up of the tree

that introduced this section in the following steps:

Set R2 = I-,%>, all other R being empty. Take I = &. The basis n

step provides us with the one-node trees: . . * (2) = 2 ; (3) = 3 ; (4) = 4 ; and (7) = 7 .

(Of course, there are other one-node trees than these.) A single application

of the induction step yields two typical two-node trees:

and it requires two further levels of induction to yield our tree as

((((3),(4))-,((2).(7))-)~*(4))-.

As an example in which no gl 0, suppose Ro = Cn), R2 = C+,x) and

otherwise R = 8. Take I I & n Q . = & + m~ Typical one-node trees are . ( 2 ) = ; ; ( 3 ) s ; ; ( 4 ) = : ; +'zS+'z

A single application of the induction step yields for example

+ + ((3),(4))+=/\ ; ( ( 2 ) , f i ) + = / \

3 4 7. a More generally, we can argue inductively; each ieI and o e R0 deter-

mines a one-node tree:

(b "I

If tl,.. . ,tn are all trees and w E On, then (tl, ..., t )w can be thought n

of as the tree

where is ihorthand f.1 a picture of a whole tree. ~.

Now as the p i c tu r ea t the foot of p. 47 suggests, we may use the t rees

themselves t o form an 0-algebra:

with the simple composition ru le tha t for w c 0 11'

which i s cer ta inly a val id map by the induction step i n the def ini t ion of

31,Q. Let us check that t h i s yields the f ree XQ-dynamics i n the sense that

31,'n corresponds to I(x*)@ and 61 corresponds t o the morphism

@ @ @ 11,: I(Xn) Xn+- I(X ) . With s l igh t abuse of notation, wri te IXn fo r n

&& 2iiEBEZ: Let R be a l abe l s e t . Then for any set I, the XR-dynamics

11,: (I.", )Xn -C IX@n @ with IX, = J,,, and II = 6' is free over I. In other words, given any

0

n-algebra (Q.6). and my map T: I + Q, there ex i s t s a unique homomorphism

@ $: (IX , ,110) -C (9,s) such tha t

where the "inclusion of generators" In : I -+ LX@n views i B I as the

@ "one-node tree" n ( i ) - ( i ) a J1,* = I X n . Proof: We simply use the inductive def ini t ion of 3 t o specify $. I , n The t r iangle says

$ ( t ) - ~ ( i ) i f ' t = ( i ) for i B I. (7)

while the square says t h a t l i f $( t l ) , ..., $(tn) a r e already defined, then

for any w i n On we must have, by the equivalence of (4) and ( 5 ) ,

$((tl.. . . , t n ) d = 6W($(tl)... . .$(tn)). ( 8 )

In t h i s way, $ is both well-defined and uniquely defined, with (6)

holding. 0

Now we can see the reason for the n a m e s automaton: any element of

@ I X n is a t r ee , and $ describes how t h i s t r e e is processed t o yie ld an

element of Q. More precisely, we use r t o re label t h e i n i t i a l nodes ( i ) with

r ( i ) 6 Q and o with 60 when v(w) - 0 t o re labe l the i n i t i a l nodes with elements

of Q-for =ample

Then one "runs" (Q,6) on t h e t ree , passing from "leaves" t o "root" and

applying t h e appropriate 6 . For example w

m o t \

. /-\ yields /-\ i\ i\ / x \

2 + m o wCi1 3 m o 2 a o

leaves

and then / - \ and then -4 + a (-1).

2 + f l ( - l ) 6 6 0 . .

The mapping $ associa tes with 3 element of 3 an element q E Q.

It is interest ing t o note that $ can be bu i l t up i n another way.

@ @ @ @ Let 7 X n : I X , + QXn be the mapping taking a t r e e i n I X n and r ~ l a b e l l i n g

the leaves by changing i e I t o ~ ( i ) c Q (note tha t it is only a t [not

necessarily a l l of] the leaves tha t elements of I appear--at che other nodes,

@ operator labels appear). In t h i s way then, r~@n yields a t r e e i n QXn . Also, l e t A*: QX@~ -+ Q be the unique dymamorphic extension of id (obtained Q by taking s = i d : Q 4 Q i n (6)). Then we have, by an easy induction Q argument, that

* @ @ @ JI = 6 'TXn : IXn + QX, 4 Q

* wh&h generalizes the familiar formula ( q , ) = 6 ( ( q w of sequential

machine theory.

3.7 D E F m O N : A subset R of Q x Q is a soneruencfi on t he $2-algebra

(Q,6) i f it is an equivalence re la t ion which is also a subaleebra &

(QxQ.6~6); i.e.

wenn. and (qi,p;)eR for i en -=%' (6w(q1.. . . .9) ,6w(qi3. . ,<))<R. (9

Note tha t (9) is automatic fo r n = 0 since i t asser t s (60,6w)~R i f

oe Qo.

&& DSERVATIOX: Let QIR be the s e t of equivalence classes of Q with

respect t o a congruence R. Then the assignment

F : (Q/R)X, +

defined by

xu: ( Q / R ~ ~ ( ~ ) - (Q/R) : ([qll ,..-, [%I) * 16w(q1... .,¶,,)I

provides the unique n-algebra s t ructure on Q/R such tha t

r : Q -* Q/R : q * [ql

is an $2-homomorphism:

The proof is an immediate consequence of the definition. 0

u -: Let @: (Q, 6) + (Q1,6') be a dynamorphism. Then

$(Q) = (.$(q)I 9 E 91 is a subalgebra of (Q1,6'), i . e . @ven w E On and

(qT1.. . . .¶In) 6 ( J I ( Q ) ) ~ , 6',,,(si,-. .,<) E $(Q). Also,

E = ( (q1,4i) 1 $(ql) = JI(qi) 1 is a congruence of (Q,6).

Proof: Recalling ( 4 ) and (51, we have 6A(q; ,..., q') = 6;($(ql) ,..., $(qn))

= J1(6w(q1),...,6 (4 )), proving the f i r s t claim. u n

To prove the second claim, observe that E is clear ly an equivalence

relation. Suppose that (qi,ql) E E fo r i = 1,2,. ..,n. For a rb i t ra ry

o an. Ji(6w(41).-...6w(Q) = 601(JI(q1), .... $(B)) = 6w'($(4;) .-.., $(q;))

= JI(6w(qi), . . . , 6w(<)) and t h i s proves the subalgebra, and hence con-

gruency, property. 0

320 I?&PIBDIa Let (Q,6) Dyn(XQ). I f ja;i = I1,2 ,...., n l and i f

c Q are given for a l l i E ii with i + j define (q

Thus, i f n = 4 and j = 3, (qi;j,q) = (q1,q2,q,q4). A" elementarv translation

df (Q,6) is a function +: Q - + Q f o r which there ex i s t n > 0, o 6 On,

j E n and fixed q E Q for a l l i e n, i # j such t ha t i

+(a,) = 6w(qi;j,q) . (10)

[For example, i f a vector space Q is regarded as an Q-dynamics with (among

other things) + i n Q2, q rr q + qo is an elementary t ranslat ion for each

fixed qo . I

BUcadatioq - of (Q,6) is an element of the submonoid of Q~ generated

by the elementary translations; i.e., r: Q- Q is a translation if

rPid or if r is a composition r ....orl of elementary translations. Q n

For each set I the Chm, M (I), of I is the monoid of translations R

i3 of (IXn .llo).

U&@gJ&: Let nl = Xo (i.e. each x 6 Xo is the label of a unary

operator), Rn = 0 for n f 1. Then QXQ = Q x Xo. i.e. sequential

machine theory is a particular case of tree automata theory. An element *

(i.xl.. .xn) of IX@ = I x X is the tree 0

Every elementary translation r: M@ -+ M@ has the form 6 for X

some x E X with gx(i,w) =(i,wx). The general translation is then 0

* (i,w) * (I,ww1). In case I - I, M(1) = {Rw, I W' E Xol, where

* * Rw: Xo -+ X is the "right translation" w rr ww'.

0

We recalled, at the top of p. 28, that the classical Nerode equiva-

lence for -xXo: &&-+Slit has

* (w1,w2) E Ef ++ f (wlw) = f(w2w) for all w E Xo

which may now be rewritten as

* (wl,wZ) 6 E f ~ f R w w l = f R w w 2 farall w6Xo.

* Thus the classical Nerode equivalence of the response f: Xo 4 Y is seen

to be

Ef = I(wl,w2) I fml - fm2 for all T 6 M(1)).

But this construction works for finitary Xn:

@ &I2 I!WBEd: Let X = XQ for a finitary label set Q, let f: IXn -+ Y

be a function. Define

E = {(x,y) 6 IX@ x IX@ I ~ T X = f ~ y for all r e ~~(1)). f n R

@ Then Ef is a congruence of (IXn,Ipo) and

@ a,y: E + %, where a(x,y) = x and y(x,y) = y f

is the reflexive Nerode equivalence of f.

Postulates 2, 3, 4 of theorem =are satisfied and so (recalling &&)

the coequalizer-minimal realization Mf of f exists with the canonical

@ @ projection rf: Ign -+ (IXn /Ef) = Qf as the reachability map.

Proof: E is obviously an equivalence relation. We must show that if f

(x1,y1), . .. , (xn,y,) 6 Ef and if o 6 n n then ((xl.. .. ,x,)w. (y1.. . ..Y~)u) F. Ef. This is clear if n = 0 since it is true that (wo,wo) E Ef. .If

n > 0, the result follows (and here we make crucial use of the fact that

n is finite!). from the following chain of equalities in which T 6 %(I) is

arbitrary:

To see why the first equality holds, observe that

T': a- (a,x 2,.... xn)w

is an elementary translation. Hence T T ' is a translation, with

T T ' (x1) = T ( ( X ~ , X ~ , . . . ,xn)w) and T T ' (yl) = T((yl,x 2... .,xn)w). Because

( X ~ , Y ~ ) E Ef , ~ T T ' (xl) = f i r ' (yl), i .e. the f i r s t equality holds. The

other equal i t ies a r e obtained similarly.

Now that Ef is a congruence we may use && t o construct (Qf,Sf) =

@ IX /Ef with canonical projection sf R

# # First, we sha l l prove f a = fy . Since rf = coeq(a,y) i n && and

f a = fy (i .e. consider T = i d i n the def ini t ion of Ef), there ex i s t s

unique B with # # #

f Bfrf = f . Therefore, f a = B r a# = E ( r -a) [ i . e . r f a , f f f f

# (rfa) are both dynamorphisms equal to r f a when preceded by Efn and hence

# a r e equal by = Bf(rfey)' = fy . Now, t o show that E is t he . reflexive

f

Nerode equivalence, suppose t ,u: T + IX@~ is another reflexive p a i r

# B sat isfying f t = fu . To induce the desired (d w i l l be explained below)

a s shown above it is necessary and suf f ic ien t to show that f ( t (a) ,u(a)) I @ @ #

a E TI is a subset of EE. (Of course, E c I X x I X .) Since t = t n, f sr a # U u = u n, one has Im t c Im t8 and Emu c Im u ; it then suff ices t o show

# @ that T1 = i ( t (b),u'(b) I b E T X ~ } is a subset of Ef, i.e. tha t

# # @ f s t (b) = ~ T U (b) for a l l b € TXR, T E' M(1).

To this end we pause to observe that T1 is reflexive, i.e. (x,x) e T1 @ for all x e IXo; for, there exists d with td = id = ud and hence

a a if c @ t (nd) - id = u Old). Also, Imt and Imu by are subalgebras of IXQ,

a a @ @ @ so that TI = Imt x Im u is a subalgebra of IX x IX . Hence, if b E TXn

and T is the elementary translation x* (~~,...,x~_~,x,x. ..., x ) w then, l+l' n a %

because (xl.xl), . - -, ( ~ ~ - ~ , x ~ - ~ ) (t (h) ,U (b)), (x~+~,x~+~), . . . , (xn.xn) E T1

# # (~(t (~)),T(u (b)) E T1 by the subalgebra property. By iteration,

# # (r(t (b)),r(u (b)) E TI for any T E N(I).

Now let (x,y) E TI; then (TX,T~) 6 T1 by what we have proved for all

# # @ T M(1). Thus rx = t (b), ~y = u (b) for some b e TXn But by

a # assmption, ft = fu , so that frx = f ~ y and then (x,y) e Ef, i.e.

T1 c Ef. This completes the proof that a,y: Bf -+ IX@n is the reflexive

Nerode equivalence of f.

The rest is easy. Postulates 2 and 3 of && have already been estab-

lished as r and 6 above and postulate 4 holds for any X: &&-+&for f f

the following reason. Observe that for any epi a: A-+ B in &, there

is a readily constructed 5: B -+A with a-B = i$. Then aX-BX = idgX,

so that UX is an epi. 0

In previous literature on realization theory for tree automata 1171

it has been the practice to consider f: 4 Y only for I = 0 . A

justification for this practice was that if I # k? we may define a new

label set n(I) by

(n(r))o = no + I,

(n(I))= = On for n > 0

and then observe that there is a canonical bijection

Since each element of OX: is a "Pary derived operation" [IS], it follows

@ that elementarytranslations--hencealltraoslations--on OX are derived

unary operations. This observation is the basis of the Nerode equivalence

BE = I(~,~)~BX\X~X~ I ~ T X = f ~ y for all derived unary operations

@ T on an}

found (in different notation) as [17, w. Theorem 3.12 above shows that in general the relevant T's are translations, nor derived operations. The R(I) approach is not always natural. For example, in a study of machine

interconnection--where the output set of one machine is the input set of

another--it is natural to fix Xn and let I vary.

Lll&Xk&FX% E as in need not be the abstract Nerode equivalence: f

let X = % where n2 = [w), en = 8 for n f 2. Let I = {a,b},

Y = (0,11 and define f: IX@ - Y by @ Define Ef as in and define t,u: I + 1% with

by T = f(a,(aa)o)), t(a, (aa)~) = a, u(a,(aa)w) = (ae)w. It is clear by

# # . @ induction (as in 3.5) that the images of t . u 1n IX contain no trees in

# # which b appears as a leaf and, in particular, ft = fu . TO show that r does not exist it is necessary and sufficient to find

T E XI) such that fr(a) f fr((aa)o). Talte T defined by xrt (xb)~. Then

fr(a) = f ((ab)o) = 1, while fr((aa)w)) = f(((aa)wb)w) = 0. Of course,

@ t,u: T -+ IXn is not reflexive.

We now give compelling evidence as to why the reflexive Nerode equiva-

lence is of wider applicability than the external Nerode equivalence of &&:

w: Let n be finitary and contain at least one operation of arity 5 2. Let Y have 2 elements. Then for any I # 0 there exists

@ f: IX + Y such that postulate 3 of a-the existence of a minimal

dynamics--with respect to the external Nerode equivalence does not hold

(although it does hold for the reflexive Nerode equivalence of u b y 3.12).

Proof: Recall that u(~): lx@*xnn -+ IX@n is defined by

U(O' - id (n) " Xn @ s3 @

11 Cn+l) - lx@xnx __, lxnxn - n n n IXn @ Set Dn - 1m(!J(")) c IX n. Suppose, for illustration, that there

exists w E Q2, a,b,c E I. Then abo E Dl, abwacow 6 D2, but abwco C Dn

for any n. Dl coincides with trees of the form , D2 with trees of

the form f i (i.e. trees obtained by replacing the leaves of a

D tree with a whole Dl tree); Dg coincides with trees obtained by replacing 1

the leaves of a D tree by whole D trees, and so on. Notice that the tree 1 2

for abocw is

which will not look like a Dn tree for any n. CO

Set D - and define

@ f: lx,-+n=xD.

(I

Set E 2 IX@ t b be tbe kernel pa i r of f . Y

We claim that E is the external Nerode equivalence of f. Por clearly

it s a t i s f i e s f a = fy , i .e . f - u ( o ) - a ~ o = f . ~ ( ~ ) - y X ~ . Far a l l n 5 1,

~ m ( u ( ~ ) . d ) , IIQ(IJ(~)'Yx~) c ~ m ( p ( ~ ) ) = Dn c D, s o that f . ! ~ ( ~ ) . a x = f .u(n).y~n a'

fo r a l l n 2 0. ?umber i f E' + is such tha t f . u ( n ) . ~ ' ~ - f . ~ ( ~ ) . y ' ~ ~

for a l l n 2 0 , taking n = 0 and using the kernel pair property yields a

unique $ completing

Now l e t io be any operation of a r i t y > 1 and s e t n = vfo). Let a E I.

Define

P1 = (a,.. .,a)w D 1

P2 - (P l,..., P1)W E D2

Then (P ,P ) E E and (P1,P2) E E but ((PI. ..., P1.P1)w, (PI, ....P1.P2)w) h 1 1

a s P2- (P1 ,..., P I P ) w ~ D cD, (PI ,..., P 1 P ) w 4 D . ' 1 2 ' 2 a @ @ r Thus E 2 I X , is not a congruence. It follows from Q t h a t I X ,-+ Q = Y

G O ~ ~ ( U , Y ) carr ies no 0-structure making r a homomorphism, so postulate 3 f a i l s

as asserted. 0

Consider the labe l s e t n 2 = {w), nn = O n t 2. Consider the inclusions

@ jn: -+ IX ,. The image of jn is the s e t of b g n e o u s formulae pf

n. [Thus (((a,b)w, (c,d)u)w) is homogeneous of degree 2.1 The downfall

of the external Nerode equivalence is tha t not a l l formulas are homogeneous.

This d i f f i cu l t y can he surmounted by extending t o ?i by adding a unary operation

.,:. , .. . ..: ,>A, called "!&.L&g&". il-formulas a r e now representable a s horaogeneous 3.. ;i; i.0 formulas. To i l l u s t r a t e , l i ... .: ., ... ((a,b)w, (c,d)w)w - ((a,b)w,(c,d)w)o

((a,b)o,c)o - (ta,b)o,(c)A)o

.'j((a,b)w.((c,d)w,e)o~o,(x,y)olo ---t [~(~a~A,<bM)o,((c,d~~,~e~A)o~w,(c~x~AW,<<y~6~A)lwl~

: Clearly QX; - QXn + Q. This suggests the idea of augmenting fbe input

:process , which we s h a l l take up i n Section 5.

We conclude t h i s section with some remarks about in f in i t a ry l abe l s e t s .

There a r e many examples of in f in i t a ry $7-algebras such as l a t t i c e s with varping

degrees of completeness, Boolean a-rings 44s w a d i n measure theory), commutative

* G -algebras and so on. See 118, khapter 1, sect ion 51. Even though the minimal

rea l i za t ion problem f o r in f in i t a ry t r e e automata has no h e d i a t e appl icat ion t o

problems of e i t h e r control theory o r computer science it is worth our while t o

show t h a t , i n t h e in f in i t a ry case, not every f has a minimal real izat ion; f o r

t h i s ,demonstrates tha t "existence of minimal realizations" is not a consequence

of completeness and cocompleteness (see [ l o , chap. 1111) of Dyn(X) and 3 , and

strongly suggests t h a t "existence of minimal realizations" is a "f ini teness

condition" on X.

Let R be a l a b e l s e t , X = XR and l e t I be a s e t . Then t o construct

@ a response f : I X 4 Y with no minimal rea l i za t ion it suff ices t o construct a

@ sequence (Rn) of congruences on IX such t h a t Rn c Rn+l f o r all n and such t h a t

@ @ R = uRn is not a subalgebra of I X x IX . '2 Proof: Since R c Rn+l, R is an equivalence re la t ion. Set Y = IX /R and let n

f: IX@ - Y be the canonical projection. Suppose t h a t f has a minimal rea l i za t ion

M as shown below but tha t R is not a subalgebra. We s h a l l deduce a contradiction. f

Define S = f(x,y) I rf(x) = r f (y)>. Then S is a congruence by u. Set

@ with canonical projection m and l e t 6n: S X - Qn be the unique Q,, - I X /Rn

dynamics (see such tha t m is a dynamorphism. Since Rn c R there exis ts

unique B with Bnrn = f . Since N is minimal there must exist a unique $, with n f

$nrn - r and Bf$n = Bn. Let (x,y) R. Then ( x , ~ ) E Rn for some n. Hence f

x - r n = r n = r y , i.e. ( x , ~ ) 6 S. Conversely, l e t ( x , ~ ) E S.

Then f (x) = Bfrf(x) = Bfrf(y) = f (y ) , so (x,y) E R. Since R = S and S i6 a

@ @ subalgebra of I X x I X , t h i s contradicts the assumption about R. 0

@ The following theorem builds on t h i s Lemma to exhibit an f : IX -+ Y with

no m i n i m a l realization.

&J& Let a be the countably i n f i n i t e cardinal, let Q be the in f in i ta ry

label set with Ra = Iw), On = 0 i f n * a, l e t X = )46 and l e t I = {0.1,2 ,... 1.

Then there exis ts Y and f : IX@ - Y such tha t f has no minimal real izat ion.

Proof: Define hr I -+ I by

@ ~h~~ hn(i) = 0 i f 0 5 i 5 n and hn(n + k) = k. Let $: I X -+ I X @ = hx@,

Specifically, r e ca l l that IX@ is inductively defined by

(basis step) i E EX@ ( i f i E I)

@ (inductive step) I f Pk E IX@ f o r a l l k c a, (Pk)w E I X .

(Bere (P ) is short hand far the sequence P1,P2, ... ). Thus $ is inductively k

defined by

B(i) = h(i)

*((Pk)w) = (*(Pk))O

Note that $n = (hX@)" - h%@ is inductively defined by

$n(i) = hn(i)

*"((%)w) = (lj"(pk))w

@ Define Rn = I(x,y) I = $"(y)). Then Rn is a congruence on IX by u. If qn(x) = &y) then +(gn(x)) = $($n(y)), 80 R c R*. By a, it suffices n

to show R = uKn is not a subalgebra; to do this we will define P,, E IX@ so

that (PO,Pn) € R but ((POPOPO.. .)w, (PoP1PZ,..)u) 4 R. Accordingly, let , .

Pn = (012...nOOO;..)o

Then *n(~n) =, (hn(0)hn(l). . .hn(n)h"(0)h"(O) . . .)w = (000 ... )o = Po = *n(p0)

So (PO,Pn) E Rn c R. If ( (POPOPO.. .)o, (POP1P2.. .)w) E R we must have

,. IIY(P~P~P~. . .)o) = $n((~O~l~Z.. .)o) (12) . for some n. But

. . *"((P~P~P~. . .lo) = ($n(~O)*n(~O). . .)w

= (POP, ... ) w

whereas

n n n *n((~o~,~2..:)o) = (* (Po)* (PI)* (P2) ... )w

If (12) holds then we m s t have

Po = *n(~$

for every k. But

2 (00...)o = Po

This exhibits the desired contradiction.

@ The question of when f : IX 4 Y has a minimal realization has

been considered by Adhek [221 and Trnkov6 123, 241. The most powerful

'nm-existence theorem' at this vriting is [24, proposition 31:

@ Suppose there exists an infinite cardinal a S card(= ) for

which there exists p in aX such that p is not in the image of fX

for any f : A--r a with card(A) < a. Then there exists a

@ subset of IX whose characteristic function has no minimal

realization.

The hypotheses of this theorem hold for the X of &&with

a = L/IO thereby shoving that the statement of && can be' considerably

strengthened. Tmlrov&'s construction, even for this X, is rather compli-

cated, and we have failed to construct an example vith I = 1, Y = 2

that compares in simplicity to &&,

4. F i n i t e Step Conditions.

Arbib and Zeiger [9] i n giving a unified (but "pre-categorical") view

of l i n e a r systems and sequential machines, observed t h a t the subspaces

SIC s 2 c s 3 c ... with S. being t h e set of s t a t e s reachable i n a t most k s teps from t h e i n i t i a l

J

s t a t e had the property t h a t i f ever Sk equalled SWl, then a l l the S.'s were 3

equal fo r j 2 k. They observed that whenever S was not equal t o SjC1, then j

S. exceeded S. by a t l e a s t 1 i n cardinal i ty for sequential machines, and 3+-1 3

by a t l e a s t 1 i n dimensionality fo r l inea r systems. They could then exhibi t

the commonality of t h e r e s u l t s t h a t for a sequential machine of n s t a t e s t h e

s e t of reachable staLes was S while f o r a linear system of dimension n n-la

t h e space of reachable s t a t e s was Sn. We now provide a general categor ical

s e t t i n g fo r these resu l t s , and the dual observabili ty resu l t s . In t h i s

section, we provide a theory applicable, in par t icular , t o adjoint processes

(which include our c l a s s i c examples of sequential maebines and l i n e a r systems);

while Section 5 gives a more general theoty applicable t o t r e e automata,

which a re not even state-behavior.

Unt i l fur ther not ice i n t h i s section we assume tha t our category % has

f i n i t e coproducts, and an image factor izat ion system (see Section 3 of [21,

[11] or [121). By the theory of and followin8 &&we have:

&L Given an input process X, and a system M = (X,Q,G,T,r ,Y,B) , t h e i n i t i a l s t a t e r : I -t C! has dynamorphic extension the reachabi l i ty

@ map r: IX + Q, and (by Section 1 (20) with a = T) t h i s s a t i s f i e s

k = 8(k).T~k : IX 4 p. 0

Given the linear system, G: I --t Q, P: Q -+ Q, H: Q 4 Y (% = && k X = id ) r sends u to F Gu; an input string (. ..O u 0...0) where u

1(. k

occurs at time -k takes the zero state to the state P Gu at time 0 . *

For X = - x X o : &&-+&, r'qk : I x ~ f - 4 9 sends (qo,w) to 6 (qo.w)

the state reachable from initial state qo by applying the input string w of k n-1

length k. Since the coproduct (ink: IX -+ I$) gives us the object j=o .

of "less-than-n-step" input applications, we see the usefulness of the

definition:

3 DEFINITION: The less-than-n-steps-reachabilitv mar, af a system with

reachability r is the map rn: $ d + p given by = I(~)&.

n-1 Define kOn-l : L$ - ii by

3 =o J =o

(The diagram should make clear the two different uses of the symbol ink.)

Then

n-1 commutes, since for 0 S k C n, rn+l'ino 'ink = L+l- ink - 6(k)-r~k = r -ink. n

Thus we. may apply diagonal f i l l i n (See Lennna 3.5 of [ Z ] ; o r [ I l l , [12])

t o obtain:

4Li Let each m have 5-?ti factor izat ion 5 = mn-en. Then

there e x i s t s a unique hn+* i n such that

n-1 i.,"" 1xj j-i 14

j=O j = O

(2)

comutes. 0

I n our c lass ic examples, t h i s says tha t is, up t o isomorphism, a

subset of Q@, and so captures the existence of t h e chain . . . Sn c S n+l c . . . noted above. The next r e s u l t then captures t h e idea tha t i f we can ever

reach t h e whole state-space ( r is onto i n our c l a s s i c examples; r E E n n

i n our general theory) then we can always reach it a l l thereafter:

4.~4 IXFQW&X: I f m 4, then r e 6 , and q@s Q for a l l k 2 n. n+l

Proof: Lemma 5.6 of [2 ] (or 1111) t e l l s us t h a t whenever gf E &, we must

have g E E. Then rn+l E 6 follows from ( I ) , and so rk 5 fo r a l l k 2 n.

But i f rk is i n 5 , then % i n its E-??l fac to r iza t ion must be an isomorphism, --

and Q@" Q. I n f a c t , w e can choose rk = ek, mk - idQ, t o force

4 0 - Q. 0

In the r e s t of t h i s sect ion we develop our theory fo r processes X which

preserve f i n i t e coproduct diagrams (hence, by 1.31 and 1.26, we include - adjo.int processes, and, with them, l i n e a r systems and sequential machines).

n-1 . We may thus apply X t o the coprod;ct diagram in.: d --c U I X ~ i n order

3 j=o t o define

from the coproduct diagram

For example, for X - x X : && tl,,l i s simply the map (w,x) w wx 0 -

sending a s t r ing w of length n, and a l e t t e r x 6 Xo, t o the s t r ing wx of

length n + l . [Incidentally, by u, i f X presehles countable coproducts,

and % has countable coproducts, then IX@ = d . ] k20

4.5 TBBOREM: Let % have f i n i t e coproducts, and an image factorization

system (5,q). Let the input process X preserve f i n i t e coproducts and

E-morphisms. Then the following diagram defines a unique kncl: Q@x + Q @.

(NOTE: The passage from 6 t o (kn+l) describes a functor from Dyn(X) t o the

category of time-varying X-dynamics is the sense of [19]).

Proof: To check comonrtativity of the outer part of the diagram, we use the n-1

fact that (IXj) X is a coproduct to reduce our checking to showing j=o

that

-in. X = 6.r X.in.X, for 0 2 j < n. rn+~'"n,l 3 n I

But r ,-(%,,,injX) a r n+lainj+l by the definition of v n,l

- - 6(i+l) .Txj+l by definition

- 6.6(j)X.TXjC1 by the definition of 6 (j+l)

- 6.(6(j).Txj)X since x is a functor

- 6-(\*in.)X .I

by 4.2. asah

= 6-r X-in n j

That kn+l exists follows by the diagonal fill-in lemma, since enX €5 and

mn+l ' 3. D

In our classical examples, kel is simply the map that acts on a state

reachable in less than n steps to provide a state reachable in less than

n+l steps. We now come to the main result which may be paraphrased "if one

step gets you no further then no finite number of steps can take you

further: when you stick, you're stuck!"

4& THECREM: Under the conditions of && and with the definitions of Q,

if hn is an ioomorphism then hn+l is an isomorphism, and hence h is an n+k

isomorphism for all k I 0.

Proof: We shall show that if h is an isomorphism, then we may define n

v ~ + ~ as shown in the diagram.

A s soon a s such a v ~ + ~ ex is t s , so tha t hn+l~n+l = e E G . i t follows n+l

that hn+l 6 & . But hn+l is i n nflZ by &&, and so h is an isomorphism. n+l

Assuming tha t hn i s an isomorphism, so tha t hn-' ex i s t s , we may define

the iorphism

where lie have applied X t o the diagram

.and followed (hi l*en)X With kn : - Q@. Letting ino be the

inject ion I -+% I X ~ we a l so define j =o

' en.ino : I -+ Q@.

Since we c lear ly have a coproduct diagram

these two morphisms combine uniquely t o yield a s our candidate f o r the

desired mrphism,

It only remains to check that (4) commutes. To do this we shall only

verify that

hn+l'Vn+l = en+l

since the other triangle (which is irrelevant to our theorem anyway)then

commutes as h n+I E T. As mn+l €772, it suffices to verify that

mn+~ ' hn+l ' Vn+~ = mn+l ei+l*

i.e. not only that

mn+l ' hn+l ' en ' ino = rn+l ino,

[which is immediate since mn+l'hn+l.en = m .e by (Z), and since n n

\'ino = rn+l'inol but also that

-1 mn+l - h,+l - kn ' hn X ' e X - r

n n+l'l1n,l' ( 5 )

But

-1 mn+l-hn+l-kn-h n X e e X = m n n -kneh;lx-enX by (2)

= 6.m n-1 X - h i l ~ - e n x by (3)

= 6 . (mn-l - hi1. en)x = S - rnX by (2)

= r n+l ' 'h.1 by (3)

which establishes ( 4 1 , and with it, our theorem. 0

Since E-n2 factorization is only defined up to isomorphism;

we may, in the above circumstances, set

5 = Q@ = $3 = .,, = s G 3 = . . . forcing each hn+k, k20, to be the identity id- It then follows from(2) 9'

- tha t m n+k = mn-l = m, say for a l l k 2 0.

Thus - m-kn+& = S . i i X far a l l L 2 0;

and since i is mono, i t follows tha t the k ' s are a l l equal, say t o 5. n+a

Then - - m - 6 = 6 . k . ( 6 )

By repeated application of u, we obtain a monomorphism

i - hn. . . . . hl . ho : Q@ 3 ii such that m - i = m yielding 0'

Now consider the system with dynamics : GX -+ 6 and I-frame - @ 7 = fi . e : 1 --+ 5. Lat its reachabili ty map be ; : IX -+ 6. We then

0

have the diagram .

(Recall ( 6 ) in checking the commutativity.)

Thus, since T has a unique dynamorphic extension, we have

- - r = m-r.

But from (3) ,

and since ii E?, it follows that ; = enfl € &. Hence we have shown:

4J lXUB& Let % have f i n i t e coproducts; and l e t the input process X

preserve f i n i t e coproducts and &-morphisms. I f hn as defined i n (2) for

the pair ( 6 : QX + Q, T: I --t Q) is an isomorphism then, se t t ing

- m=hn-l, "k and d - h h . . . . h o e e o , wehave tha t ( ; : Q x - + ~ , n' n - @ : I 4 6) is reachable, with reachability map r: IX - 6 i n & sat isfying

- - m r = r. That i s , (8,;) i s the "reachable part" of (6,~). 0

In particular, we reiterate that i f X is an adjoint process (so

t ha t X has countable coproducts and products, and X has a r igh t adjoint)

then X preserves coproducts. An adjoint process a lso preserves &-morphisms

for most popular choices of E . Thus && and apply to adjoint processes.

Let us now sketch t he observability theory for adjoint processes (we assume

the reader familiar with the (x.)OP duality of [31):

n-1 1 4.8 DEFINITION: The dual of m : U IX --t Q yields the at-most-n-steps - i = O

3 [ I t is clear tha t t h i s approximates 0 : Q -+ YX = Y(X') .] Then the @ jzo

dual of (2) f o r (x-)OP yields a unique i n & such that

commutes. For sequential machines, Qn is the set of equivalence classes

of states distinguishable at the output by an input string of length at

most n; for linear systems, Q is the set of equivalence classes of states n

distinguishable by studying output sequences of length n.

The dual of (3) for (x')OP yields a unique eel such that

and in place of u w e obtain

4.9 DEOREM: Let have finite products; and let X be an output process

that preserves products and morphisms. Then if sn as defined in (7) for

the pair (6: QX + Q, 8: Q + Y) i.9 an isomorphism then, setting - - - - Q = @ y U = U n-1' 6 = a o and B = to. so. ... . s n-1 we have that

- (6: CX -+ Q; 0 : T( - Y) is an observable pair with observability map

0: C-+YX, in? satisfying o . ; = a. 0

Let us consider the special case of decomposable systems, so that

X = id and

In linear system theory, the finite-dimensionality of a state-space can

reflect itself in the condition F"G = rna for some morphism a. To

generalize t h i s situation, we now introduce two new concepts, with the

speculation tha t they w i l l be useful i n other contexts.

&J,Q DERINITIOQ Let 'k have an image factorization system ( E ,v 1. We

say an object A & E-M e B i f there ex i s t s a chain

el e2 e A - A + e

1 A2- ... d B

of 9. E-morphisms none of which is an isomorphism while any such chain of

length e+l must contain an isomorphism. We say A has &-u 9. i f 9. is

the maximum g-height of A over any %-object B.

Dually, we define Q-&&& by considering chains of wmorphisms

4.11EXAMPLES

( i f I n the category & of vector spaces and l inear maps, the

&-height of srn over sn is defined i f f m 2 n, i n which case it is m-n.

The $-height of gm is m. %-height yields the same numbers, and thus both

heights correspond t o dimension.

( i i ) In the category a, the &-height of A over B is defined

i f f 2 I B I , where the cardinalities I A ~ and [ B I a re f i n i t e and i f we

do not have A # + hut B = #. It 1s then [ A J - J B I . I f r > 0 ,

then A has 5-height - 1. %-height yields the same numbers, and--apart

from a 1--both heights correspond t o cardinality.

Then && and yield:

4.12 TREORFM: Let have f i n i t e coproducts, and an image factorization --

system (E,?). Let the input process X preserve coproducts and E-morphisms.

Let M = ( X , Q , ~ , I , T , Y , B ) be a reachsble X-system. Then

where n is the %-height of Q over I. 1

Dually, let % have f i n i t e products, and l e t the output process

X presenre products and Nmorphisms. Let M = (X,Q,G,I,r,Y,B) be an

observable X-system. Then

. when n2 is the &-height of Q over Y.

4.13 gEFINITPON: An object A of a category % is termed proiective i f

given an arbi t rary 0: B- C E & and y: A 4 C, there exis ts

a making the fallowing diagram commute

In f ree modules are projective, as a r e re t rac t s of f r e e

objects i n B; similarly fo r a. In &&, all objects are projective.

We now generalize the l inear system resu l t s as follows:

: Let %. be a category with products, coproducts and 6-nZ

factorization, and suppose 6: I --t Q, F: Q -+ Q and 8: Q + Y define a

decomposable system with I projective. With r as defined i n a, hn+l a s n

i n (2), hn+l is an isomorphism i f and only i f fo r some a: I -+ nI,

e ~ = r a . n

Proof: Suppose F"G = r a. l e t zn+l : (nil11 4 Q@ be the coproduct n

mapping of e : n I -+ Q@ and ena : I - Q@. Then n

+ = = - = r . by (9) . n m e a n n

Further, E E & since en e E . So mn . 'Z defines an c-? factori- n+l n+l

zation of r . whence Q@ is isomorphic to @. [Notice that for this n+&

part of the proof, the projective property was not used.] Conversely,

suppose hn+l is an isomorphism. From (2), we have

a - - - - - - - - - - - - - - - - I I

Noting the morphism e n+lainn : I -+ @ and the '2-character of en: nI + Q@, rhe projective property of I yields a: I. + n1 with

euma = en+l-in . Then \ .a = m .e .a = m n re re n.en+l'inn = rn+l-in, = P"G. 0

Various consequences follow from this theorem:

i) One can take duals (using injective (i.e. co-projective) output

objects Y.

if) One can take Q to be Y g and r to be fA.

We then become interested in conditions on the so-called parameters

i Ai = HF G, such as

(The column denoting a coproduct morphism, the row a product morphism.)

iii) Conditions such as F ~ G = %-a can even Arise from

conditions like

Such a condltlon is fulfilled in -when F is finite-dimensional by the

Cayley Hamilton Theorem, or in a when 9 is finite (for then 'dN = id

for some N).

A11 the theory to this point of the section has used the notion

of f.-nl factorization. Let us now change the vier~point in order to study finite step conditions via the Nerode equivalence ideas.

m: For the rest of this section the category jG has countable

coproducts and products. The process X of the machine M = (X,Q,G,I,r,Y,6)

is an adjoint process.

Recall from that the abstract (see and external (see

Nerode equivalences are the same as X is adjoint. We shall study conditions

under which we can deal with only a finite number of the external equivalence

@ conditions £-an = fey where the Nerode equivalence is a,y: Ef + IX , n

# - and an = ?A:). axn is the n-step approximant to a . DEFINITION: A par of morphisma a, y : E~ + 12 are partiallv

f-equivalent to level N if feun = fayn, n = 0,1, ..., N where, as usual,

a N an denotes the n-step approximant to a . We say that a, y : Ef- M @

is a partial Nerode equivalence off to level N if they are partially

f-equivalent and if whenever a, 7 : E~ -+ IX@ are again partially

-N N f-equivalent, there exists a unique $: E -+ Ef such that

I f % h a s kernel pairs, then as might be expected, a,y i s a

A kernel pair, actually of a type of approximant t o f . We digress to

es tabl ish t h i s result .

DEFINITION: The N - W o A . roximant t o fA, denoted fN, is

obtained a s follows. Let [fmll0 (") 1. : I X ~ -+ Y ( x ' ) ~ for n20 i SO .

be the morphism obtained by the X,X0 adjunction from feu?) : (fl d ) x " -+ Y. j 20

A . Then f N is defined by the product diagram:

1 . U PROPOSITION: The kernel pa i r of f is the p a r t i a l Nerode equivalence N

of f t o 1evel.N in the sense that i f e i ther ex i s t s so does the other, and

they are equal.

Proof: The resu l t w i l l follow f r o m u and the kernel pa i r definition

-N @ i f f o r arbi t rary z,?: E 4 IX , the conditions +a = +y and

- - feun = feyn, OSnsN a r e equivalent. That t h i s is so follows from the

following equivalences:

OcnSN (use X,X' adjointness)

- f - a = f a y

£:.;= f i . y (use m. 0 :

\

Now we present the analogue of the morphisma established i n a. N 4.19 -0- Let aN,yN : Ef - IX@ and aW1,y*l : --t IX@ be,

p a r t i a l Nerode equivalences t o levels N and N+l fo r a prescribed

@ f : IX --t Y. Then there ex i s t s a unique Ef -+ E! and a unique

N+I . aN+$

W l Proof: f ' a n = f - y n w1 for 0 S n I N+1 implies the same f o r 0 i n S N.

Identifying a*' and yNC1 with a and y i n &,&, sN+l exis t s (by identi-

f icat ion with JI i n &&). For 0 I n i N, we have

= £-,I, (n+l) uWIx"+' (by def ini t ion of vo (n+l)

= fa,,o (n+l)- yN+IxNfl (F is a p a r t i a l equivalence)

= f.!l,(n) * (PO.Y"lx)x".

Then we identify iN,?N: 8 --t IX@ i n 4.16 with

N+lX P o , U, .yN+'x : E? + IX@ t o conclude the existence of 2 N+1' 0

The "when you s t ick, you're stuck" resu l t follows:

0 : With the def ini t ion and notation of -and u, sN+l 4L2P23uiF22 an isomorphism implies sNC2 is an isomorphism and thus s ~ + ~ is an isa-

morphism for a l l k > 2.

N+l N+2 Proof: F i r s t , l e t us show there exis ts #: Ef -+ Ef . By & it is

N+l N+1 suff ic ient t o show tha t f'an - f .yn , 0 5 n S N+2. [Think i f E~ a s

EY1. I Certainly t h i s is t rue for 0 < n S N+1. Now

(N+2). ,N+Y+2 f . JJ+1 = f ' uo N+2

(N+2)) (*+I). p1 (aN+$)$'' (by def ini t ion of uo "0

(N+1) . . aN+\)XN+l = f .!lo (110 - f . pocN+1'. (aN . I ? ~ + ~ ) X N + ~ (by &&l

= f . (aNf1 . (s N + l 1 -1 .%+l)~N+l

(N+l) aN+12+1) ( (sN+l)-l)XNfl. = (f - No

N + l (N+l). y*++p+l) . ( ( sN+l ) - l )p l . Then Similarly, f ' yN+2 - (f - pO

N+1 since f - uo

A t t h i s stage, we have

This diagram of i t s e l f does not show that sNi2 is an isomorphism. However,

by applying the pa r t i a l Nerode equivalence def ini t ion t o

N+2 N+2 a IX Ef - @, it follows tha t both i and $.sN+2 are candidates

N+2 Y

fo r t he unique morphism JI : EY = EY -+ EY of def i n i t i o s &&.

Thus id N12 = @ ' s ~ + ~ and likewise. sW2'$ - id.$+l , so that sN+2 is Ef f

an isomorphism. I3

N N N Finally, and a s one might expect, a ,y : E - IX@ can be f

taken as the external (and abstract , since X is adjoint) Nerode equivalence.

4.2L WOW: With the def ini t ion of 4.16, 4.19 and a and with sNCl an

isomorphism, then aN,yN: E: -+ IX@ is the Nerode equivalence of f .

N+l Proof: Use the isomorphism established in t o identify $, Ef ,

F, ... and a , a N W l N+2 aN+', ... and y . y , y , ... . Then

(") - aNxn = f .iio(n). yNX" for 0 < n S N by def ini t ion and for a l l n > N fa!J0

N (n) . ;X" = f.,,p). YX" by noting that an - a , $ - yN. Moreover, fapo

f o r a l l n implies t h i s resu l t fo r OSnSN, whence there ex i s t s i j i (by t he

N N par t ia lNerode equivalence property of a ,y ) such t ha t

But t h i s i s precisely what I s required t o conclude the (non-partial)

Nerode equivalence property of I$.

5. Augmenting the Process.

In section 4, we considered "application of l e s s than n inputs" n-1 k i n terms of t h e coproduct 1 IX , and then required X t o preserve coproducts. j=O

Unfortunately, t h i s requirement r u l e s out t r e e autamata--we know from 131

tha t t h e only X: &-+ && which preserves coproducts is t h e sequential

machine process X = -xXo. I n t h i s section, we show how t o achieve the goals

of Section 4 i n a less r e s t r i c t i v e se t t ing , by augmentingx t o obtain a new

functor 5

Q ? = Q X + Q

(where + is the coproduct) whose n-fold application d i rec t ly generates

"application of a t most n inputs". Consider the case X = -xX . It is 0

easy t o check that gjt" is then essen t i a l ly (Q X X ~ ) , where the 3 s n

"essentially" reminds ns t h a t QXX: is actually present a s (j) d i s j o i n t

copies. With t h i s motivation, we turn t o the general development:

DEFWITION: Let X: k +'j$ where has f i n i t e coproducts. Then we - define the augmentation of X t o be the process ?: $ -+& : Q w QX + Q.

Thus an it-dynamics is j u s t a p a i r

6 9~ --+ Q and Q 2 Q

so t h a t an i-dynamorphism h: (Q,S,F) -+ (Q',6',F1) must s a t i s f y

In applications, we sha l l be most interested i n the case F = i d Q' Where the theory of Section 4 only applied t o processes which

preserved f i n i t e coproducts, thus excluding the processes of t r e e automata,

the current theory i s not so rest r ic ted:

L2 BBX@k I f X i s an adjoint process o r an Q-algebra process in Set,

then ii is an input process.

Proof: I f X i s an adjoint process, then ? is an adjoint process

so tha t % has adjoint 2: R I-+ RX' x R and so is an input process with

If X corresponds t o the operator s e t St, then % corresponds t o the

operator s e t = St u (1) where 1 4 St and haa a r i t y 1. Thus is an input

-@ process with QX the f ree ;-algebra on Q generators. 0

We now retrace the stages of Section 4 in the augmentation set t ing.

The r e su l t s w i l l not only have appl icabi l i ty t o algebra automata, but w i l l

have somewhat simples proofs:

Given an X-dynamics 6 : QX -+ Q we define its auwentation t o be

the %-dynamics

Define the under-n-steps reachabili tv mao of (6: QX- Q, 7: I -+ Q ) t o be

of course, < = ;.iin, where is t he Z-reachability map of 8.

It is then clear (observe tha t I%n = l z n - l X + lzn-') that

commutes. Analogously t o &&we then use diagonal f i l l - i n t o obtain

W TXEOREM: Let each Gn have an 6-'t( factorization $ = g . Sn. Then

there ex i s t s a unique 6 in such tha t n+l

5.4 C&OI,IARY: '1f 1 E & , then ;a+l E & and 6 @ = Q for a l l kzO. II - We now turn t o the generalization of a. W e no longer require

t ha t X preserves f i n i t e coproducts (a r e s t r i c t i ve condition tha t in & forces

x = - x x ) but only require tha t f preserves c-morphisms (which i n && with 0

&= onto --or i n any category with 6 = s p l i t epimorphisms -is s a t i s f i ed by

t every functor 1:

A s p l i t epimrphism e: A -+ B is one such that there ex i s t s f : B + A with ef = idg. Note chat not every category has E-q factorization, and i f a category has E-n/factorizations, it is not necessarily t rue tha t &= s p l i t epimorphisms is allowed.

@ while IX is sent i n to E@ by the X-dynamorphism rl defined by

The square of (3) tells us tha t

so t h a t $ is a l so an X-dgnamorphism. Moreover, splicing the diagrams of ( 4 )

above those of ( 5 ) . we deduce that

- q .$ = idIx@.

[On the other hand, the square of (4) yields

- p0.in

This w i l l imply $. = idE@ only it it is the case tha t io= [ ,, '1 . but t h i s case i s too r e s t r i c t i ve t o detain us here.]

Now by the def ini t ion of 6, we may factor T through 6 , say - A

m T = I L i j - + Q .

Consider, then, the %-dyn-rphic extension o f T:

with t h e right-hand rectangles yielding

Splicing (4) atop t h i s we see that ; = i?$ is the X-reachability map of i,

and t h a t A

r = m a r .

...,- A A A

Since m.r.$ = m = m.e , we have ?';in = & E E , and so ; E 5 . n n

We saw above t h a t b F is the ?-dynamorphic extension of r. But so too

is r.T, i n view of the diagram

A ?. 0 0 Thus we have r-V - m a r , and so we may l e t m.e be an ( E,q)-factor izat ion

of r t o form t h e diagram:

Recalling t h a t $'JI = idm@. we see t h a t $ E & , and hence (F:).; is - 0

another &-))1 factor izat ion of ;. Thus the re is an isomorphism o: Q + Q

such that

- 0 - o - r = e.$ so that w.?.$= g,

so that ? - * = ~ l . 8 E G .

Thus we may use the 6- f l factorization

We have proved the following:

L Z Let X be an input process whose augmented process is an input

process % which preserves 5-morphisms. If in as defined in for the

.. pair (6: QX- Q, T: 1 -+ Q) is an isomorphism, then setting m = h n-1'

= and ; = h ....' ho.eo, n n

we have that (i: 6~ - 6, ;: I -+ 6) A A

is reachable, with reachability map ;: IX@ 4 6 in 5 satisfying m a r = r.

hat is, ( 8 , ) is the "reachable part" of (6,~). . .

0

Note that this does say that (8,;) is reachable in any finite-

number of steps using the unaugmented input process--compare the comments

f ollorring u. We shall not belabor the observability story in this section,

since the &algebra process does not have an observability theory [3], save

to note that the dual of the augmentation of an adjoint process is the

augmentation of its dual:

CONCLUSIONS

To sumarize the preceding material i n a short space, we claim

three major ideas:

1 ) The application of &-q factorizations i n a category t o the real izat ion problem;

2) The definition of a Nerode equivalence within a category, and its application t o the real izat ion problem;

3) The formilation of the finite-dimensionality idea of l inear systems theory and sequential machines i n a category theory framework, and its application t o machine realization.

In the remainder of these conclusions, we sha l l discuss some

possible criticisms of the theory. A t the outset, l e t i t be seen that we

do not claim tha t t he approach of t h i s paper should supplant other and more

t rad i t iona l approaches. Rather, we would claim the advantages noted in the

introduction, of unification and exposure of the essent ia ls of some results.

[Nor, indeed, would a category theoris t claim tha t the study of group theory

could be supplanted by the study of category theory, despite the f a c t tha t

there is a category of groups.1

In rebut ta l t o the positive claims concerning the material of the

paper, we could argue as follows:

1) The unification may be more i l lusory than rea l . To say tha t state

behavior machines unify adjoint machines is not that helpful, when we believe

tha t the r ight way t o study adjoint machines is a s adjoint machines, rather

than as state-behavior machines. What then is t h e value of state-behavior

machines, since we know of no examples (other than contrived ones) of such

machines which a r e not a lso adjoint machines. [As p a r t i a l defense against

t h i s attack, we can claim tha t decomposable machines, adjoint machines, and

input process machines each include more than one example of stature--linear

systems and group machines, sequential machines and Goguen's aff ine machines

1131, t r e e automata and non-deterministic automata 111, respectively.]

2) The unification comes only with the aid of concepts which many,

including engineers at leas t , f ind very hard t o master. Though the general

theory of re la t iv i ty may unify special r e l a t i v i t y and Newtonian mechanics,

engineers prefer t o leave the general theory of r e l a t i v i t y t o the physicists,

and s t i ck with the Newtonian mechanics. Likewise, they may wish t o leave the

category theory t o category theorists. [As par t i a l defense, we could claim

' tha t the ideas of category theory used here are the more elementary ideas of

the subject, and tha t the subject would i t s e l f be much easier t o learn i f a

textbook written ass+ng l e s s advanced knowledge of the reader had been

available; that such a defect has now been remedied is the contention of

the authors 112, 201. And we might a l so argue tha t the l inear algebra now

regaraed a s both commonplac~ in and essent ia l fo r the study of control theory

was once viewed by engineers a s being alarmingly d i f f icu l t . ]

3) As yet, the theory has done l i t t l e new for the control theor i s t

remaining within t he confine- of l inear systems, or finite-dimensional non-

l inear systems. Nor has it shown how t o even formulate i n a category theory

framework t ha t most fundamental of control theory notions-feedback. The

search for such a formulation is a major challenge for those who would bring

category theory t o the at tent ion of control theorists. On the other hand,

Goguen's aff ine machines [13] model a discretized system ((1) of Chapter 1 )

which is neither i n i t i a l i z ed nor f u l l y l inearized a s a bi l inear term in q

and u is retained. A s adjoint machines, the real izat ion theory i s c lear .

e Rowever, an element of the "object of inputs" IX is more complex than a

f i n i t e sequence of inputs. Here, the philosophy of category theory suggests

a new principle i n systemengineering: fo r nonlinear systems, the s t ruc ture

of "input strings" i s not dictated by a priori intuition. A suitable

algebraic theory of discretized nonlinear systems (which does not exist

at t h i s writing) may be due to the failure t o recognize the proper formu-

e lat ion of the response f: I X + Y of such systems.

- -

Vol. 59: J. A Hanson, Growlh in Open Ecanomier V, 128 pages- 1971.

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Vol. 1W: D. E Poyce. A Farhi. R. Weischedel. Optimal Subset Selection. Multiple Regression. Intetdependenc. and Optimal %wnn Algorifhmo. XIII, 187 1974.

Vol. 104: S. Fujino. A NewKeynesian Thecry of Inflation and Economic Growth. V. 96 pages. 1974.

Vol. 105: Optimal Control Theory and its Appliwtions Part I. Pro ceedingsl973. EdiM by R J.I(irby.VI, 425 pages. 1974.

Vol. 106: Oplimal CMLml Theory and ib ApplioaUons. Part I!. Pro d i n g s 1973. Edited by R J. Kirby.VI. 40.9 uaaea 1974. . - VoI. 107: Control Theary. Numerical Methods and Computer S y s t m MdeJing. Interns~nal Sympcslum, Rwquencourf June 17-21,1974. E d i i byA Eensouasan and J.L tiansV1II. 757 p a g e Ism. Vol. 108: F. Bavar el el.. Supercritical Wing SRtions 11. A Hend- book V. 296 pages 1915. Val. 109: R van Randow, introduction to UleTheoryof Matroids. IX.102 pages.1975.

Vd. 110: C Sbiebel, Op(imai Contml ot Discrele Time Slochsstio SystamP. Ill. 208 pages. 1973. Vol. 111: Vaiable Structure System with Applicationto Economies andBiolo~y.Pr~ngs1974.Ed'iedbyARubwtiandRRMohler.. W, 321 p a w 1975.

Vol. n2: 1. Wfihlem Objsolives and Multi-Objeolive Decision Making Under Uncertainb IV, 111 pages. 1975.

Val. 113: G A Aschiw81, StabiiitdtsaussagPn fiber Klassen von Mauizen mil veisrhwindenden Zeilenaummen. V. 102 Seiten. 1975.

Vo1.114: G Uebe, Produldionsthmria XVII. 301 Seiten. 1976.

VoI. 116: Anderson el al.. Foundations 01 System Theory: Fin* and lnfinilaryConditions.VII93 pages1976

dkonometrie und Unternehmensforschung Econometrics and Operations Research

Vol. I Nichtlineare Programmierung. Von H. P. Kiinzi und W. Krelle unter M~twirkung von W. Oettli. Vergr~ffen

Vol. II Lineare Programmierung und Erwe~terungen. Von G. 6. Dantzig. Ins Deutsche iibertragen udd bearbe~tet von A. Jaeger. - Mit 103 Ab- bildungen. XVI. 71 2 Seiten. 1966. Geb.

Vol. Ill Stochastic Processes. By M. Girault - With 35 figures. XII. 126 pages. 1966. Cloth

Vol. IV Methoden der Unternehmensforschung im Versicherungswesen. Von K. H. Wolff. - Mit 14 Diagrammen. VIII, 266 Seiten. 1966. Geb.

Vol. V The Theory of Max-Min and its Appl~cation to Weapons Allocation Problems. By John M. Danskin. -With 6 figures. X, 126 pages. 1967. Cloth

Vol. VI Entscheidungskriterien bei Risiko. Von H. Schneeweiss - Mit 35 Abbildungen. XII, 21 5 Seiten. 1967. Geb.

Vol. VII Boolean Methods in Operations Research and Related Areas. By P. L. Hammer (Iv&nescu) and S.Rudeanu.With a preface by R. Bellman. - With 25 figures. XVI, 329 pages. 1968. Cloth

Vol. VIll Strategy for R 8 D: Studies in the Microeconomics of Development By Th. Marschak Th K Glennan Jr., and R. Summers. - With 44 figures. XIV, 330 pages. 1967. Cloth

Voi. I X Dynamic Ppagramming of Economic Decisions. By M. J. Beckmann. - With 9 figures XII, 143 pages. 1968. Cloth

Vol. X Input-Output-Analyse. Von J. Schurnann. - Mit 12 Abbildungen. X, 31.1 Seiten. 1968. Geb.

Vol. XI Produktionstheorie. Von W. Wittmann. - Mit 54 Abbildungen. VIII, 177 Seiten. 1868. Geb.

Vol. XI1 ' Sensitivitiitsanalysen und pararnetrischeProgrammierung.Von W. Din- kelbach. - Mit 20 Abbildungen. XI. 190 Seiten. 1969. Geb.

Vol. Xlll Graphentheoretische Methoden und ihre Anwendungen. Von W. KnLidel. - Mit 24 Abbildungen. VIII, 11 1 Seiten. 1969. Geb.

Vol. XIV Praktische Studien zur Unternehmensforschung. Von E. Nievergelt 0. Miiller, F. E. Schlaepfer und W. H. Landis. - Mit 82 Abbildungen. XI!, 240 Seiten. 1970. Geb.

Vol. XV 0ptimaleReihenfolgen.Von H. Miiller-Merbach. - ~ i t 4 5 ~ b b i l d u n ~ e n . IX, 225 Seiten. 1970. Geb.

Vol. XVI Preispolitik der Mehrproduktenunternehmung in der statischen Theo- lie. Von R. Selten. - Mit 20 Abbildungen. VIII, 195 Seiten. 1970, Geb.

Vol. x\nl Information Theory for Systems Engineers. By L. P. ~~v;irinen. -With 42 figures. VIII, 197 pages. 1970. Cloth

Vol. XVlll Unternehmensforschung im Bergbau. Von F. L. Wilke. - Mit 29 Ab- bildungen. VIII, 150 Seiten. 1972. Geb.

Vol. XIX Anti-Aquilibrium, Von J. Kornai. - Mit 31 Abbildungen. XVI, 382 Seiten. 1975. Geb.

Vol. XX Mathematische Optimierung Von E, Blurn; W. Oeffli. IX 413 Seiten. (davon 75 Seiten Bibliographic). 1975.Geb.