## On the Automata Size for Presburger Arithmetic (2004)

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Venue: | In Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LICS 2004 |

Citations: | 10 - 1 self |

### BibTeX

@INPROCEEDINGS{Klaedtke04onthe,

author = {Felix Klaedtke},

title = {On the Automata Size for Presburger Arithmetic},

booktitle = {In Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LICS 2004},

year = {2004},

pages = {110--119},

publisher = {IEEE Computer Society Press}

}

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### Abstract

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 exponentially bounded in the length of the formula. This upper bound is established by comparing the automata for Presburger arithmetic formulas with the formulas produced by a quantifier elimination method. We also show that this triple exponential bound is tight (even for nondeterministic automata). Moreover, we provide optimal automata constructions for linear equations and inequations.

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Citation Context ...tter is interpreted 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 m-bit multiplication =-=[8]-=-, it is straightforward to show a similar lower bound for the least significant bit encoding as in Theorem 17. Note that we cannot use lower bounds on the sizes of BDDs (or more generally, branching p... |

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Citation Context ...xponential worst case complexity in deterministic time. Another approach for deciding PA or fragments of it that has recently become popular is to use automata; a point that was already made by Buchi =-=[9]-=-. The idea of the automata-theoretic approach is simple: Integers are represented as words, e. g., using the 2's complement representation, and the word automaton (WA) for a formula accepts precisely ... |

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Citation Context ... (parametric) systems of linear Diophantine equations, integer programming, and various problems in system verification. The decidability of PA was established around 1930 independently by Presburger =-=[23, 24, 34]-=- and Skolem [32, 33] using the method of quantifier elimination. Due to the applicability of PA in various domains, its complexity and the complexity of decision problems for fragments of it have been... |

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Citation Context ...ation. Due to the applicability of PA in various domains, its complexity and the complexity of decision problems for fragments of it have been investigated intensively. For example, Fischer and Rabin =-=[14, 15]-=- gave a lower bound on any decision procedure for PA, namely double exponential in nondeterministic time. Later, Berman [2] showed that the decision problem for PA is complete in the complexity class ... |

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Citation Context ...WS1S, or even WS1S with the ordering relation " over monadic second-order variables. There, the number of states of the minimal WA for a formula can be non-elementary larger than the formula's le=-=ngth [26, 35]-=-. In order to establish the upper bound on the automata size for PA, we give a detailed analysis of the deterministic WAs for formulas by comparing the constructed WAs with the quantifier-free formula... |

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Citation Context ...domain of integers. Gradel [19] and Schoning [29] investigated the complexity of decision problems of fragments of PA. Oppen [22] showed that Cooper's quantifier elimination decision procedure for PA =-=[10]-=- has a triple exponential worst case complexity in deterministic time. Another approach for deciding PA or fragments of it that has recently become popular is to use automata; a point that was already... |

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Citation Context ... constructed by a qe method. This technique can also 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. e., these structures are provided with automata for deciding equality on the domain and the atomic relations of the structure. Examples are the mixed first-order theory over the structure (R; Z;s... |

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Citation Context ...ely constructed from the formula, where automata constructions handle the logical connectives and quantifiers. Specific algorithms for constructing WAs for linear (in)equations have been developed in =-=[1, 4, 7, 18, 37]-=-. A crude complexity analysis of automata-based decision procedures leads to a non-elementary worst case complexity. Namely, for every quantifier alternation there is an exponential blow-up in the wor... |

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Citation Context ...ative Research Center "Automatic Verification and Analysis of Complex Systems" (SFB/TR 14 AVACS). number of alternations. The upper bound for PA is established with a result from Ferrante an=-=d Rackoff [13]-=- showing that quantified variables need only to range over a restricted domain of integers. Gradel [19] and Schoning [29] investigated the complexity of decision problems of fragments of PA. Oppen [22... |

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Citation Context ... constructed by a qe method. This technique can also 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. e., these structures are provided with automata for deciding equality on the domain and the atomic relations of the structure. Examples are the mixed first-order theory over the structure (R; Z;s... |

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Citation Context ...ely constructed from the formula, where automata constructions handle the logical connectives and quantifiers. Specific algorithms for constructing WAs for linear (in)equations have been developed in =-=[1, 4, 7, 18, 37]-=-. A crude complexity analysis of automata-based decision procedures leads to a non-elementary worst case complexity. Namely, for every quantifier alternation there is an exponential blow-up in the wor... |

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Citation Context ...13] showing that quantified variables need only to range over a restricted domain of integers. Gradel [19] and Schoning [29] investigated the complexity of decision problems of fragments of PA. Oppen =-=[22]-=- showed that Cooper's quantifier elimination decision procedure for PA [10] has a triple exponential worst case complexity in deterministic time. Another approach for deciding PA or fragments of it th... |

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Citation Context ...mber n is represented as a word b n 1 : : : b 0 2f0; 1g with n= P 0i b i sb i . The relationship between nonstandard numeration systems and formal language theory has been investigated, e. g., in [16=-=, 20, 30]-=-. It remains as future work to investigate the sizes of DWAs using different nonstandard numeration systems. The main technique to prove the triple exponential upper bound on the automata size was to ... |

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Citation Context ...f it have been investigated intensively. For example, Fischer and Rabin [14, 15] gave a lower bound on any decision procedure for PA, namely double exponential in nondeterministic time. Later, Berman =-=[2-=-] showed that the decision problem for PA is complete in the complexity class LATIME(2 2 O(n) ), i. e., the class of problems solvable by alternating Turing machines in time 2 2 O(n) with a linear Th... |

38 | On the construction of automata from linear arithmetic constraints
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Citation Context ...ely constructed from the formula, where automata constructions handle the logical connectives and quantifiers. Specific algorithms for constructing WAs for linear (in)equations have been developed in =-=[1, 4, 7, 18, 37]-=-. A crude complexity analysis of automata-based decision procedures leads to a non-elementary worst case complexity. Namely, for every quantifier alternation there is an exponential blow-up in the wor... |

35 | Numeration systems, linear recurrences, and regular sets
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Citation Context ...mber n is represented as a word b n 1 : : : b 0 2f0; 1g with n= P 0i b i sb i . The relationship between nonstandard numeration systems and formal language theory has been investigated, e. g., in [16=-=, 20, 30]-=-. It remains as future work to investigate the sizes of DWAs using different nonstandard numeration systems. The main technique to prove the triple exponential upper bound on the automata size was to ... |

29 |
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Citation Context ...mata size for PA, we give a detailed analysis of the deterministic WAs 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]. From this analysis, we obtain that the minimal deterministic WA for an arbitrary formula of length n has at most 2 2 2 O(n) st... |

28 | A comparison of Presburger engines for EFSM reachability
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Citation Context ...ased decision procedures leads to a non-elementary worst case complexity. Namely, for every quantifier alternation there is an exponential blow-up in the worst case. However, experimental comparisons =-=[1,18,31]-=- illustrate that automata-based decision procedures for PA often have good and competitive performance in comparison to other methods. In [7], the authors claimed that the minimal deterministic WA for... |

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25 |
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Citation Context ... (parametric) systems of linear Diophantine equations, integer programming, and various problems in system verification. The decidability of PA was established around 1930 independently by Presburger =-=[23, 24, 34]-=- and Skolem [32, 33] using the method of quantifier elimination. Due to the applicability of PA in various domains, its complexity and the complexity of decision problems for fragments of it have been... |

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Citation Context ... we define hhaii N , for a 2 N r , as the shortest word w 2 (f0; 1g r ) with a = hwiN . 3.2. Linear Equations and Inequations In this subsection, we first recall the automata constructions given in [=-=4, 6, 18, 37]-=- for linear (in)equations. Then, we improve these constructions such that they are optimal, i. e., the constructed DWAs are minimal. Assume that the (in)equation tsc is in normal form, where t(x 1 ; :... |

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Citation Context ...WS1S, or even WS1S with the ordering relation " over monadic second-order variables. There, the number of states of the minimal WA for a formula can be non-elementary larger than the formula's le=-=ngth [26, 35]-=-. In order to establish the upper bound on the automata size for PA, we give a detailed analysis of the deterministic WAs for formulas by comparing the constructed WAs with the quantifier-free formula... |

18 | Deciding presburger arithmetic by model checking and comparisons with other methods
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Citation Context |

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Citation Context ... (parametric) systems of linear Diophantine equations, integer programming, and various problems in system verification. The decidability of PA was established around 1930 independently by Presburger =-=[23, 24, 34]-=- and Skolem [32, 33] using the method of quantifier elimination. Due to the applicability of PA in various domains, its complexity and the complexity of decision problems for fragments of it have been... |

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11 |
Subclasses of Presburger arithmetic and the polynomial-time hierarchy
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Citation Context ...oefficients appearing in these atomic formulas, and the number of distinct coefficients and divisibility predicates that may appear. Similar analysis of improved versions of Cooper's qe method are in =-=[19, 25]-=-. We refine the analysis [25] of Reddy and Loveland's qe method. Such analyses are technical and subtle. For the sake of readability, we have put the proofs of this subsection in the appendix. For a s... |

10 |
Numeration systems
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Citation Context ...s) when using the most significant bit encoding since the reading of the next digits by a DWA involves a left bit shift. There are also nonstandard numeration systems for representing integers, e. g. =-=[1-=-7]. In a nonstandard numeration system, the base is an infinite sequence of integers that has certain properties. A standard example is the sequence of Fibonacci numbers (b i ) i0 : a natural number n... |

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Citation Context ...13] showing that quantified variables need only to range over a restricted domain of integers. Grädel [19] and Schöning [29] investigated the complexity of decision problems of fragments of PA. Oppen =-=[22]-=- showed that Cooper’s quantifier elimination decision procedure for PA [10] has a triple exponential worst case complexity in deterministic time. Another approach for deciding PA or fragments of it th... |

8 |
Proof of a conjecture by Erdös and Graham concerning the problem of Frobenius
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Citation Context ...equation c 1 x 1 + + cs xs= z has no solution in the natural numbers. Fors= 1, it trivially holds that G(c 1 ) = 1. Fors> 1, the upper bound G(c 1 ; : : : ; cs) c 2s1 was proved by Dixmier [11]. It is straightforward to show that for alls> 0, G(c 1 ; : : : ; cs)s2 c1 ++cs(c 1 + + cs) : (2) 5 The cases for r = 0 and r = 1 are trivial since S = ;. Assume that r 2. By Lemma 2, it su... |

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8 |
Greedy numeration systems and regularity
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Citation Context ...mber n is represented as a word b n 1 : : : b 0 2f0; 1g with n= P 0i b i sb i . The relationship between nonstandard numeration systems and formal language theory has been investigated, e. g., in [16=-=, 20, 30]-=-. It remains as future work to investigate the sizes of DWAs using different nonstandard numeration systems. The main technique to prove the triple exponential upper bound on the automata size was to ... |

8 |
Über einige Satzfunktionen in der Arithmetik, Skr. Norske Vidensk
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Citation Context ... linear Diophantine equations, integer programming, and various problems in system verification. The decidability of PA was established around 1930 independently by Presburger [23, 24, 34] and Skolem =-=[32, 33]-=- using the method of quantifier elimination. Due to the applicability of PA in various domains, its complexity and the complexity of decision problems for fragments of it have been investigated intens... |

7 |
Complexity of Presburger arithmetic with fixed quantifier dimension
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Citation Context ... The upper bound for PA is established with a result from Ferrante and Rackoff [13] showing that quantified variables need only to range over a restricted domain of integers. Gradel [19] and Schoning =-=[29]-=- investigated the complexity of decision problems of fragments of PA. Oppen [22] showed that Cooper's quantifier elimination decision procedure for PA [10] has a triple exponential worst case complexi... |

6 | Upper bounds for a theory of queues - Rybina, Voronkov - 2003 |

6 |
gewisse Satzfunktionen in der Arithmetik , Skr. Norske VidenskapsAkademie i Oslo 7
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Citation Context ... linear Diophantine equations, integer programming, and various problems in system verification. The decidability of PA was established around 1930 independently by Presburger [23, 24, 34] and Skolem =-=[32, 33]-=- using the method of quantifier elimination. Due to the applicability of PA in various domains, its complexity and the complexity of decision problems for fragments of it have been investigated intens... |

4 |
Greedy numeration systems and regularity. Theory Comput
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(Show Context)
Citation Context ...nce of Fibonacci & & $ numbers : a natural number is represented as a word #" $ & . The relationship between nonstandard numeration systems and formal language theory has been investigated, e. g., in =-=[16, 20, 30]-=-. It remains as future work to investigate the sizes of DWAs using different nonstandard numeration systems. 9 ( 7 with B $), & - &#@ The main technique to prove the triple exponential upper bound on ... |

3 |
Grädel,Automatic structures
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(Show Context)
Citation Context ... constructed by a qe method. This technique can also 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. e., these structures are provided with automata for deciding equality on the domain and the atomic relations of the structure. Examples are the mixed first-order theory over the ! structure [5, 3... |

3 |
uchi, Weak second-order arithmetic and finite automata
- B
- 1960
(Show Context)
Citation Context ...xponential worst case complexity in deterministic time. Another approach for deciding PA or fragments of it that has recently become popular is to use automata; a point that was already made by Büchi =-=[9]-=-. The idea of the automata-theoretic approach is simple: Integers are represented as words, e. g., using the ’s complement representation, and the word automaton (WA) for a formula accepts precisely t... |

1 |
Proof of a conjecture by Erd ös and Graham concerning the problem of Frobenius
- Dixmier
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(Show Context)
Citation Context ...A( . The Frobenius number $ O O"! @ 5@ @ l $ OU : ( ( A has no solution in the natural numbers. For , it trivially holds that . For O#! @S !V , the upper bound $ O O#! ) % ! " was proved ( by Dixmier =-=[11]-=-. It is straightforward to show that for all , 9 @ @ @ @ @ @ ! O _ (2) : !. % : !. % : ( 7 or T $% ; & ; & " : . $ O O ! '& %)(*(*( % % : O 5U _ 6 6 I . I . & & & & . _ & & ( < & 6 & & & & & 1 & & & ... |

1 |
ädel. Subclasses of Presburger arithmetic and the polynomial-time hierarchy
- Gr
- 1988
(Show Context)
Citation Context ...oefficients appearing in these atomic formulas, and the number of distinct coefficients and divisibility predicates that may appear. Similar analysis of improved versions of Cooper’s qe method are in =-=[19, 25]-=-. We refine the analysis [25] of Reddy and Loveland’s qe method. Such analyses are technical and subtle. For the sake of readability, we have put the proofs of this subsection in the appendix. For a s... |

1 |
öning. Complexity of Presburger arithmetic with fixed quantifier dimension
- Sch
- 1997
(Show Context)
Citation Context ... The upper bound for PA is established with a result from Ferrante and Rackoff [13] showing that quantified variables need only to range over a restricted domain of integers. Grädel [19] and Schöning =-=[29]-=- investigated the complexity of decision problems of fragments of PA. Oppen [22] showed that Cooper’s quantifier elimination decision procedure for PA [10] has a triple exponential worst case complexi... |