fixed points in the propositional nonmonotonic logic

ARTIFICIAL INTELLIGENCE 199

RESEARCH NOTE

Fixed Points in the Propositional Nonmonotonic Logic

Grigory F. Shvarts Laboratory for Knowledge Representation Problems, Program Systems Institute o f the USSR Academy o f Sciences, 152140 Pereslavl-Zalessky, USSR

ABSTRACT

We give an explicit description of fixed points for propositional theories in the nonmonotonic logic of McDermott and Doyle. Using this description we refute two claims from McDermott and Doyle's original paper.

Introduction

One of the systems formalizing nonmonotonic reasoning is the nonmonotonic logic introduced by McDermott and Doyle [1]. They add to the usual predicate (or propositional) logic an unary connective M which means the unprovability of the negation (i.e. Mp means ~ ~p) . Unlike the provability logic (see e.g. Boolos [2]) Mp means here the unprovability in the same system. This leads to some difficulties in formalizing nonmonotonic reasoning: the "nonmonotonic proof" cannot be considered as a sequence of formulae, in which each formula is obtained by means of an inference rule from the previous ones. McDermott and Doyle [1] introduced an operator NM A (Z is a given set of nonlogical axioms) defined on sets of formulae. They considered fixed points of this operator as candidates for the set of nonmonotonic consequences of A. They denoted the intersection of these fixed points as TH(A) and proved the decidability of TH(A) for any finite set A of propositional formulae.

We modify slightly the decision procedure from [1] to obtain the procedure which enables to describe precisely all fixed points of a finite nonmonotonic propositional theory. Using this procedure we refute an assertion of McDer-

Artificial Intelligence 38 (1989) 199-206 0004-3702/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)

200 G.F. SHVARTS

mott and Doyle [1, p. 52] that, if a nonmonotonic theory A has exactly one fixed point, then this fixed point is the limit of successive applications of the operator NM a to A. Nevertheless for a finite set A each fixed point of NM A is the limit of successive applications of NM A to Th(B) , where B is obtained from A by adding some set of subformulae of A.

1. Fixed Points of Nonmonotonic Theories

For the reader 's convenience we reproduce the basic definitions from McDer- mott and Doyle [1].

The language L has an infinite number of atomic propositional letters (or atoms). Formulae of L are either atoms or, for formulae p and q, expressions of the form ~ p , (p D q), ( p /x q), (p v q) and Mp.

In this paper the letters A, B, S, T and U (possibly with indices) denote arbitrary sets of formulae of L; p, q and r denote formulae of L. S ~ - p denotes, as usual, the deducibility of p from S in the classical propositional calculus (using only modus ponens as an inference rule). T h ( S ) = { p: S ~- p}. Following [1] we define

As(S) = {Mq: q ~ L a n d - n q f ~ S } ,

NMA(S ) = Th(A U As(S)) .

Let N M ° ( T ) be T h ( A U T) and NMA+I(T) be NMA(NMA(T)) for each nonnegative integer i.

Lemma 1.1. Let T j denote NM~(B) for j = O, 1 . . . . . Let us assume that B C T ~ and B C T 2. Then:

_ I

(i) for all nonnegative integers" i and j

T 2i C T 2i+2 T 2i+1 ~ T 2i+3 T 2i C T 2j+ 1 .

(ii) if U i T 2~= n~ T 2i+l = T, then T is the fixed point of NM, and T is the unique fixed point of NM a containing B.

Proof. (i) is easy to obtain by induction on i from the obvious fact that for all S and U the condition U C_ S implies N M ~ ( S ) C NMA(U ).

The first part of (ii) is obvious. Let S be a fixed point of NM A such that B C S. Therefore Th(A U B) C Th(A U S), i.e. T ° C S. From NMA(S ) = S by induction on i we obtain T 2i C S , T 2i+1 ~ S, hence T = S. []

Let M(A) be the set of all subformulae of formulae from A of the form Mq. Let us call B admissible for A, if B C_ M(A) , and for each Mp E B, A U B ~- ~ p , and for each Mp E M ( A ) \ B , A U B ~ --qp. Note, that the admissibility of a subset of M(A) is decidable if A is finite.

FIXED POINTS IN NONMONOTONIC LOGIC 201

The modal complexity m(p ) of a formula p is defined as follows:

m ( p ) = 0 for a t o m i c p ;

m(--Tp) = m(p ) ;

m ( p /x q) = m ( p v q) = m ( p D q) = m a x ( m ( p ) , m( q)) ;

m(Mp) = m ( p ) + 1.

By an immediate occurrence of a formula into another formula we mean an occurrence, which does not lie in the scope of any M. Consider, for example, the formula M ( p & M q ) & r . The only occurrence of M ( p & M q ) in this formula in an immediate one, and the occurrence of Mq is not. By an immediate occurrence of a formula into a list of formulae (in particular, into a derivation) we mean an immediate occurrence of the formula into a m e m b e r of the list.

Formulae of the form Mp are called M-formulae.

Lemma 1.2. Let B C_ M( A ) be admissible for A. Then for each i

NMA(B ) N M(A) = T h ( B U A) n M ( A ) . (1)

Proof. Let us prove (1) by the induction on i. For i = 0 (1) follows immediately from the definition of N M ° ( B ) . Let (1) hold. We have to prove (1) for i + 1 instead of i. From (1) and the

definition of NM we obtain

NMA(B ) = Th(A U B U N ) , (2)

where N is some set of formulae beginning with M, which are not subformulae of any formula of A (empty if i = 0 ) . If Mp ~ M(A) \B , then, since B is admissible, A U B [- -Tp. Together with (2) this gives

Mp ~ ' A s ( N M A ( B ) ) . (3)

Let Mp E B. We claim that Mp ~ As (NM~(B) ) . Suppose on the contrary that this does not hold. From (3), (2) and the definition of As we can conclude that there exists a derivation of --np from A U B U N in the propositional calculus. Since none of the formulae f rom N is a subformula of a formula from A U {~p} we can obtain a derivation of 7 p from A U B (replacing immediate occurrences of formulae from N in the derivation by some fixed tautology). This contradicts the admissibility of B for A. Thus for Mp E B,

Mp E As(NMA(B)) ,

and since for all Mp in M ( A ) \ B condition (3) holds, we have

As(NMA(B)) = B U N ,

202 G.F. SHVARTS

where N is some set of M-formulae that are not subformulae of any formula from A. Hence

NMA+'(B) = Th(A U B U N ) .

Therefore

NMA+I(B) 71M(A) = Th(B U A) 71 M(A)

(if an element of M(A) is derivable from B U A and some formulae of the form Mp that are not in M(A), then it is derivable from B U A). We have now proved (1) for each i. []

L e m m a 1.3. Let A and B be as in Lemma 1.2. Then for each p with m(p) = i and for each j > i

p @ N M A ( B ) if and only if p E N M ~ ( B ) . (4)

Proof . Induction on i. Induction basis. We have to prove that for p not containing M and for each

j > 0

p E T h ( A U B) if and only if p E N M ~ ( B ) . (5)

From Lemma 1.2 we obtain B C_NM~(B) for each j, and A C N M ~ ( B ) by definition, so we have proved the "only if" part in (5). Let us now prove the "if" part. Let

p E N M ~ ( B ) = Th(A O {Mq: ~ q ~ ' N M ~ ' ( B ) } ) .

From Lemma 1.2 we obtain p E T h ( A U B U N ) , where N is the set of M-formulae not contained in M(A). Take any derivation of p from A U B U N and replace in it all immediate occurrences of formulae from N by tautology. We obtain the derivation of p from A U B.

Induction step. Let (4) be true for all p with re (p) = i and for all j > i. Let m ( p ) = i + l a n d j > i + l . We have to prove

p e N M ~ + I ( B ) if and only if p e N M ~ ( B ) . (6)

Let us prove the "only if" part. Let

p E T h ( A U {Mq: ~ q ~ N M A ( B ) } ) .

Then by Lemma 1.2

p e T h ( A U B U {Mq: ~ q ~ ' N M A ( B ) and m(q) ~ i} U f ) ,

where N is some set of M-formulae of modal complexity greater than i + 1, which are not in M(A). As in the proof of the "only if" part of the induction basis, we replace in the derivation of p all immediate occurrences of the


members of N by tautology, and obtain

p ETh(A U B U {Mq: --7q ~E'NMA(B ) and m(q) <~ i}).

Since i < j - 1 and by the induction hypothesis this is equivalent to

p ~ Th( A U B U {Mq: -7q ~NMJA-I( B ) and m( q) <~ i} ) . (7)

By Lemma 1.2,

B C_ NM~(B) = Th(A U {Mq: ~ q ~ N M ~ ( B ) } ) ,

which together with (7) gives p • NM~(B). Let us prove the "if" part of (6). Assume p E NM~(B). Then, by Lemma

1.2, we have

p ETh(A U B U {Mq: 7q~NM~t- ' (B ) and m(q) ~< i} U f ) ,

where N is the set of M-formulae of modal complexity greater than i + 1, which are not in M(A). Again as in the proof of the "only if" part we obtain, using the induction hypothesis and Lemma 1.2,

p ETh(A U B U {Mq: 7qJE'NMA(B ) and m(q) <~ i}).

Therefore, by Lemma 1.2, p ~ NMA + l(B). []

Theorem 1.4. Let B C M(A) be admissible for A. Then the sequence {NMA(B)} i converges to the fixed point of NM m which is the unique fixed point of NM A containing B.

Proof. By Lemma 1.3, the sequence i {NMA(B)} i converges to some T. By Lemma 1.2, B C_ NMA(B ) for each i, thus, by Lemma 1.1, Tis the unique fixed point containing B. []

The notion of an admissible subset of M(A) is analogous to the notion of an admissible labeling introduced by McDermott and Doyle [1]. The existence of a fixed point containing the admissible subset of M(A) was in fact proved in 111.

Corollary 1.5. I f S is a fixed point of NMA, then B = As(S)N M(A) is admissible for A, and S is the unique fixed point containing B.

Proof. If Mp E As(S) N M(A), then As(S) U A ~ -Tp, hence B U A ~ -Tp. If Mp E M(A)\B, then A U As(S) F 7p , and since p c M(A), A U B ~- 7p. []

Corollary 1.6. I f A is finite, NM A has a finite number of fixed points.

From Lemma 1.3 and Theorem 1.4 we can extract the deciding procedure

204 G.F. SHVARTS

for each fixed point of NM A if A is finite. Let us briefly describe the algorithm for deciding, whether p belongs to the fixed point corresponding to the set B admissible for the finite set A.

By Lemma. 1.3, it is sufficient to decide for p with m(p)= i, whether p E NMA(B ). For i = 0 it is equivalent to p E Th(A U B), which is decidable since A U B is finite. Assume that for each j ~< i we have the algorithm for deciding for each q with m(q) = j whether q E NM~(B) . Assume re(q) = i + 1. Then

p ENMA+~(B) iff p E T h ( A U {Mq:- lqJE 'NMA(B)} ) .

Using Lemma 1.2, we see that this is equivalent to

p E Th(A O B

U {Mq E M(A U {p}): ~ q ~ 'NMA(B) and re(q) ~< i} O f ) ,

where N is the set of M-formulae, which are not in M(A U {p}) or have modal complexity greater than i + 1. This is equivalent to

p E T h ( A U B U {Mq • M(A U { p } ) : - n q ~ " N M ~ ( B ) and m(q) <~ i} ,

which is decidable, since by the induction hypothesis and Lemma 1.3 we can decide whether ~ q ~ ' N M ~ ( B ) for all q with m(q)<~ i, and M(A U {p}) is finite.

A fixed point of NM A is called accessible, if it is the limit of the sequence {NM~(Th(A))}i. McDermot t and Doyle [1, p. 52] asserted that if NM A has exactly one fixed point, then this point is accessible. We shall prove that this is not the case.

Corollary 1.7. There exists a finite theory A such that NMa has exactly one fixed point, which is inaccessible.

Proof. Let

A : {(Mp

where p and q are

A U {Mq}

A Mq) D ~ q , M q D ' T p } ,

different atoms. M(A) is equal to {Mp, Mq}. Since

[---np and A U { M q } ~ m q ,

{Mq} is admissible for A, and it is the unique set, admissible for A, since

A U {Mp} ~ ~q, A U {Mp, Mq} ~---qq.

From Theorem 1.4 and Corollary 1.5 we obtain that the sequence {NMA(A U {Mq})} converges to the fixed point of N M A , which is unique. On the other hand, it is easy to prove by induction on i that, for even i, Mp, Mq, ~ p and


-nq do not belong to NMA(A ) and, for odd i, Mp, Mq, ~ p and ~ q belong to i NMA(A ). Therefore {NMA(A)} i does not converge. []

Note that by Theorem 1.4 for finite A each fixed point of NMA(A ) is accessible when we start with some finite extension of A.

2. Foreseeability and Realizability

McDermot t and Doyle [1] called a formula p foreseeable in A, if there exists A' such that A' _~ A and there exists a fixed point of NM A, not containing ~ p . They called a formula p realizable in A, if there exists B _D A U {p} such that B has a consistent (i.e. not coinciding with L ) fixed point; safe in A, i f p belongs to each fixed point of NM 8 for each B D A; undeniable in A, if ~ p is not realizable in A.

It is obvious that all realizable formulae are foreseeable and that all safe formulae are undeniable. McDermot t and Doyle [1] asserted that a foreseeable formula may not be realizable (and by duality, an undeniable formula may not be safe). Indeed, they asserted [1, example T9] that for a theory A with a single axiom p D (q&(Mq D -nq)), where p and q are different atoms, p is foreseeable, but not realizable. But this is not the case. Indeed, let A' D A, and let T be a consistent fixed point of NMA,. Clearly, A C_ T. If ~ q E T, then we easily obtain ~ p E T, since T is closed under propositional calculus. Otherwise, if ~ q ~ T then Mq C T. Thus, by propositional calculus, we have (Mq D-nq) =-- --nq in T, so we obtain p D (q&-qq) from A, hence -qp E T. Thus, ~ p E T in each case, and p is not foreseeable in T.

Theorem 2.1. for each theory A and for each formula p, if p is foreseeable in A, then p is realizable in A.

Proof. Let A' _D A and let T be the fixed point of NM A, not containing -rip. Let P be the set of all subformulae of formulae of A' U {p} beginning with M, which are in T, and Q be the set of all such subformulae, which are not in T. Obviously for each Mq E Q, ~ q E T holds. Therefore

T h ( A ' U Pt0 {~q: Mq E Q } ) c T

and, since -qp ~ ' T and by the deduction theorem, the set A' U P U {~q: Mq E Q} u {p} is consistent in the usual sense. Denote this set by R. Let

B = A' U {p} U {-nq: Mq E Q} u {-nr: Mr E P and R [ - -nr} .

Let

P ' = P\{Mr: R [- -nr} .

Since B to P ' C T h ( R ) for all Mq from P', we have B to P ' ~ -nq . By the construction of B, for all elements Mq from M(B)\P' we have -nq E B.

206 G.F. SHVARTS

Therefore P ' is admissible for B. By T h e o r e m 1.4 NM R has a fixed poin t which is equal to S = l im{NM~(B U P ' ) ) i . This fixed point does not coincide with L since B U P ' is monoton ica l ly consis tent , and by L e m m a 1.3 the modal- f ree parts of S and T h ( B U P ' ) coincide. Since A U {p} ~ B, p is real izable in A. []

Corol lary 2.2. I f p is undeniable in A , then p is safe in A .

ACKNOWLEDGMENT

My interest in nonmonotonic reasoning systems is due to the exciting lectures held by Professors John McCarthy and Vladimir Lifschitz in August, 1987 in Moscow during the 8th International LMPS Congress. I would like to thank Sergey K. Lando for the help in writing this note in English. Sergey G. Vorobyov read the earlier drafts of the paper and made useful comments. I am indebted to the referees for the useful comments on the first version of the paper.

REFERENCES

1. McDermott, D. and Doyle, J., Non-monotonic logic I, Artificial Intelligence 13 (1980) 41-72. 2. Boolos, G., The Unprovability of Consistency: An Essay on Modal Logic (Cambridge

University Press, Cambridge, England, 1979).

Received April 1988; revised version received August 1988

fixed points in the propositional nonmonotonic logic

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