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# On the Automata Size for Presburger Arithmetic

LICS, pp.110-119, (2004)

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摘要

Automata provide an effective mechanization of decision procedures for Presburger arithmetic. However, only crude lower and upper bounds are known on the sizes of the automata produced by this approach. In this paper, we prove that the number of states of the minimal deterministic automaton for a Presburger arithmetic formula is triple ex...更多

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简介

- Presburger arithmetic (PA) is the first-order theory with addition and the ordering relation over the integers.
- These results are not needed for the upper bound on the minimal DWA for a PA formula.
- From Lemma 7, it follows that the minimal DWA representing Ø ℄ has at least Ø · Ø · states.

重点内容

- Presburger arithmetic (PA) is the first-order theory with addition and the ordering relation over the integers
- PA contrasts with the upper bound on the automata size for the monadic second-order logic WS1S, or even WS1S with the ordering relation “ ” as a primitive but without quantification over monadic second-order variables
- In order to establish the upper bound on the automata size for PA, we give a detailed analysis of the deterministic word automaton (WA) for formulas by comparing the constructed WAs with the quantifier-free formulas produced by the quantifier elimination method in [25], which is an improvement of Cooper’s quantifier elimination method [10]
- We analyzed the automata-theoretic approach for deciding Presburger arithmetic and established a tight upper bound on the automata size
- The triple exponential upper bound on the automata size carries over when using the Ô’s complement representation
- The main technique to prove the triple exponential upper bound on the automata size was to relate deterministic word automata (DWA) with the formulas constructed by a qe method

结果

- The authors give an upper bound of the size of the minimal DWA for a formula Ø · , where 3⁄4, ØÜ1⁄2
- The authors give an upper bound on the size of the minimal DWA for a quantifier-free formula.
- This upper bound depends on the maximal absolute value of the constants occurring in theequations of the formula, the homogeneous terms, and the divisibility relations.
- The size of the resultant DWA using the product construction is in the worst case the product of the number of states of the two DWAs. Note that the authors allow the con-
- The authors give an upper bound on the size of the minimal DWA for an arbitrary formula.
- The authors use Reddy and Loveland’s qe method since the produced formulas are “small” with respect to the following parameters on which the upper bound of the minimal DWA in Theorem 10 depends.
- Using the qe method naively to eliminate all quantifiers in a formula is insufficient to prove the upper bounds in Theorem 14.
- The authors prove the main result of the paper: The size of the minimal DWA for a formula is at most triple exponential in the length of the formula.
- Note that Ò ¿ and that the numbers of quantifiers, atomic formulas, and the maximal absolute integer in 3 are at most Ò.
- The size of a DWA representing a PA definable set can be exponentially smaller by interpreting the first letter as the least significant bit.
- Using the formulas defined in [14], which are used to describe Turing machines in PA, and using lower bounds on the BDD size for Ñ-bit multiplication [8], it is straightforward to show a similar lower bound for the least significant bit encoding as in Theorem 17.

结论

- The main technique to prove the triple exponential upper bound on the automata size was to relate DWAs with the formulas constructed by a qe method.
- This technique can be used to prove upper bounds on the sizes of minimal automata for other logics that admit qe, and where the structures are automata representable [3, 21], i.
- The effect on the automata size of switching from a Ô’s complement representation to a Õ’s complement representation is open

基金

- This work has been partly supported by the German Research Council (DFG) as part of the Transregional Collaborative Research Center “Automatic Verification and Analysis of Complex Systems” (SFB/TR 14 AVACS)

引用论文

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