characterizations of bounded semilinear languages by one-way and two-way deterministic machines

Characterizations of Bounded Semilinear Languages by One-way and Two-way

Deterministic Machines

Oscar H. Ibarra1 and Shinnosuke Seki2

1. Department of Computer Science, University of California, Santa Barbara, USA2. Department of Systems Biosciences for Drug Discovery, Kyoto University, Japan

Conceptual diagram of a counter machine

2011/8/18Automata and Formal Languages (AFL 2011)

2

finite state control

a binput tape

p

c1 c2 ck

can be tested for zero

counters…

Conceptual diagram of a counter machine


3


a binput tape

p

c1 c2 ck

can be tested for zero

counters

𝛿 (𝑝 ,𝑎 ,1 ,1 ,…,0 )=(𝑞 ,𝑅 ,0 ,−1 ,…,+1)

q

…

Variants of counter machine


4


a binput tape

p

c1 c2 ck

counters…

• Pushdown counter machines• 2-way counter machines

𝛼

𝛼𝛽

stackend markers

Formal definition of pushdown counter machines


5

For a pushdown -counter machine is a 7-tuple , where a finite set of statesinput and stack alphabetsthe bottom stack symbolthe initial statea set of accepting states

and is a relation

Reversal-bounded counter machines


6

For a positive integer , a counter machine is -reversal if for each of its counters, the number of alternations between non-decreasing mode and non-increasing mode – called reversal – is upper-bounded by in any computation.

Definition [Baker and Book 1974, Ibarra 1978]For a positive integer , a counter machine is -reversal if for each of its counters, the number of alternations between non-decreasing mode and non-increasing mode – called reversal – is upper-bounded by in any accepting computation.

Definition (Equivalent*)

A finite automaton augmented with 2 counters is Turing-complete. [Minsky 1961]

* This equivalence needs a help of finite state control to remember how many reversals each counter has made.

-reversal counter and 1-reversal counter


7

One can simulate a counter that makes -reversals by counters that make reversal.

This lemma enables us to focus on the -reversal counter machines.

For any positive integer , an -reversal counter can be simulated by -reversal counters.

Lemma

Classes of counter machines


8

• the class of deterministic -counters machinesDCM (𝑘)

• The class of (non-deterministic) -counters machinesNCM (𝑘)

• The class of deterministic pushdown -counters machinesDPCM (𝑘)

• The class of pushdown -counters machinesNPCM (𝑘)

• The class of 2-way deterministic -counters machines2DCM (𝑘)

For a machine class , by , we denote the set of all languages accepted by a machine in .

Classes of counter machines (cont.)


9

is the class of deterministic counter machines.

Note that .

Semilinear sets


10

A subset is a linear set if there exist -dimensional integer vectors such that

A set is semilinear if it is a finite union of linear sets.

Bounded semilinear languages


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A language is bounded if there exist and such that .

If are letters, is especially said to be letter-bounded.

For a bounded language , let

.

If this set is semilinear, then we say that is bounded semilinear.

Definition

Finite-turn machines


12

For an integer , a 2-way machine is -turn if the machine can accept an input with at most “turns” of the input head. A 2-way machine is finite-turn if it is a -turn for some .

Finite-turn

For an integer , a 2-way machine is -crossing if every accepted input admits a computation by the machine during which the input head crosses the boundary between any two adjacent symbols no more than times. A 2-way machine is finite-crossing if it is a -turn for some .

Finite-crossing

Finite-turn Finite-crossing

Main theorem


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The following 5 statements are equivalent for every bounded language :1. is bounded semilinear; 2. can be accepted by a finite-crossing ; 3. can be accepted by a finite-turn whose stack is

reversal-bounded; 4. can be accepted by a finite-turn ; 5. can be accepted by a .

Main Theorem

Main theorem


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The following 5 statements are equivalent for every bounded language :1. is bounded semilinear; 2. can be accepted by a finite-crossing ; 3. can be accepted by a finite-turn whose stack is reversal-bounded; 4. can be accepted by a finite-turn ; 5. can be accepted by a .

Main Theorem

and trivially hold since • , and • Let us prove 1

2 5

3 4

Proof direction for


15

LanguageAccepted by

is semilinear is semilinear

result 1[Ibarra 78]

Use result 1 and non-deterministic guess.

orLinear Diophantine-

equations turn out to be solvable by these

machines

Letter-bounded to bounded


16

Let s.t. is semilinear. Given a word ,

.

Let , and construct a for it.

for 1. reads ; 2. non-deterministically decomposes it as ; 3. runs on , and returns ’s decision as its decision for and .

Proof direction for


17

LanguageAccepted by

is semilinear is semilinear

result 1[Ibarra 78]

Use result 1 and non-deterministic guess.

orLinear Diophantine-

equations turn out to be solvable by these

machines• How can a deterministic machine non-deterministic

decompose an input?• Our answer is “this decomposition does not require

non-determinism.”

Deterministic decomposition


18

Let be a sufficiently-large constant. This is efficiently computable from a given language s.t. is

semilinear. Let .

Based on this theorem, we can figure out that looking-ahead by some constant distance on an input tape settles the decomposition.

Let be primitive words. If and share their prefix of length , then .

Theorem [Fine and Wilf 1965]

Main theorem


19

The following 5 statements are equivalent for every bounded language :1. is bounded semilinear;

2. can be accepted by a finite-crossing ; 3. can be accepted by a finite-turn whose stack is


Main Theorem

Proof for


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If a letter-bounded language is accepted by a finite-crossing , then it is accepted by a finite-crossing .

Lemma

Proof idea• In a computation by a finite-crossing 2DPDA,

contents of ’s stack consist of constant # of blocks of the form (pumped structure).

• Each of these blocks correspond to a writing phase during which the stack is never popped.

• # of “long” writing phases is bounded by a constant .

• We build a , and let its counters to store , and let its finite state control to remember 𝑢1𝑣1

𝑖1𝑤1

𝑢2𝑣2𝑖2𝑤2

𝑢3𝑣3𝑖 3𝑤3

stack

⋮

Proof for


21

If a letter-bounded language is accepted by a finite-crossing , then it is accepted by a finite-crossing .

Lemma

The previous result is generalized for a broader class of finite-crossing .

Proof for


22

A bounded language that is accepted by a finite-crossing is effectively semilinear.

Theorem

Proof. Let be a finite-crossing for some s.t. .

1. It is easy to construct a finite-crossing for . 2. This language is letter-bounded, and hence, this machine

can be converted into an equivalent finite-crossing . 3. It is known that if a letter-bounded language is accepted by

a finite-crossing , then the language is bounded semilinear. 4. Thus, is semilinear.

Main theorem


23

The following 5 statements are equivalent for every bounded language :1. is bounded semilinear; 2. can be accepted by a finite-crossing ; 3. can be accepted by a finite-turn whose stack is


Main Theorem (Proved!!)

Is every bounded language accepted by a finite-crossing bounded semilinear.

Open

AcknowledgementsThis research was supported by

• National Science Foundation Grant• Natural Sciences and Engineering Research Council of

Canada Discovery Grant and Canada Research Chair Award in Biocomputing to Lila Kari

• Funding Program for Next Generation World-Leading Researchers (NEXT Program) to Yasushi Okuno

2011/8/1824 Automata and Formal Languages (AFL 2011)

References


25

[Baker and Book 1974] B. S. Baker and R. V. Book. Reversal-bounded multipushdown machines. Journal of Computer and System Sciences 8: 315-332, 1974.

[Fine and Wilf 1965]N. J. Fine and H. S. Wilf. Uniqueness theorem for periodic functions. Proc. Of the American Mathematical Society 16: 109-114, 1965.

[Ibarra 1978]O. H. Ibarra. Reversal-bounded multicounter machines and their decision problems. Journal of the ACM 25: 116-133, 1978.

References


26

[Minsky 1961]M. L. Minsky. Recursive unsolvability of Post’s problem of “tag” and other topics in theory of turing machines. Annals of Mathematics 74: 437-455, 1961.

characterizations of bounded semilinear languages by one-way and two-way deterministic machines

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