Monadic Decomposition in Integer Linear Arithmetic

Hague, Matthew; Lin, Anthony W.; Rümmer, Philipp; Zhang, Wu

doi:10.1007/978-3-030-51074-9_8

Cited by 7 publications

(28 citation statements)

References 22 publications

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“…In view of applying our learning algorithm to the monadic decomposition problem [15,29] for quantifier-free integer linear arithmetic formulas, we consider two extensions. Firstly, we allow infinite hypercubes.…”

Section: Contributionsmentioning

confidence: 99%

“…In this section, we describe an immediate application of our learning algorithms to monadic decomposition of quantifier-free Presburger formulas [15,29]. We then report on experimental comparisons between our algorithms and existing methods for the problem.…”

Section: Applications and Experimentsmentioning

confidence: 99%

“…Of course, not all formulas admit a monadic decomposition (e.g., x = y). It was shown in [15] that deciding if a formula in the theory be monadically decomposable is coNP-complete 2 . Veanes et al [29] provides a generic semidecision procedure for computing a monadic decomposition of a quantifier-free formula as an if-then-else formula that is applicable to pretty much all theories considered in SMT.…”

Section: Application To Monadic Decompositionmentioning

confidence: 99%

“…. , x n ) into an equivalent boolean combination of monadic predicates ψ(x i ) in some special form, i.e., typically in DNF [5,7,15,19], or by an if-then-else formula [29], which could sometimes be exponentially smaller than the DNF equivalent representation. Veanes et al [29] provided a generic semidecision procedure for performing this monadic decomposition as an if-then-else formula, which works regardless of the base theory.…”

Section: Introductionmentioning

confidence: 99%

“…Veanes et al [29] provided a generic semidecision procedure for performing this monadic decomposition as an if-then-else formula, which works regardless of the base theory. The restriction of the problem to the quantifier-free theory of integer linear arithmetic (with and without extra modulo constraints) was studied in [15], wherein the problem was shown to be coNP-complete and a monadic decomposition could be exponentially large in general. For the subcase without modulo constraints, a monadic decomposition in DNF corresponds precisely to a finite union of (possibly infinite) rectilinear hypercubes, which is the subject of this paper.…”

Section: Introductionmentioning

confidence: 99%

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Learning Union of Integer Hypercubes with Queries

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2021

Computer Aided Verification

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We study the problem of learning a finite union of integer (axis-aligned) hypercubes over the d-dimensional integer lattice, i.e., whose edges are parallel to the coordinate axes. This is a natural generalization of the classic problem in the computational learning theory of learning rectangles. We provide a learning algorithm with access to a minimally adequate teacher (i.e. membership and equivalence oracles) that solves this problem in polynomial-time, for any fixed dimension d. Over a non-fixed dimension, the problem subsumes the problem of learning DNF boolean formulas, a central open problem in the field. We have also provided extensions to handle infinite hypercubes in the union, as well as showing how subset queries could improve the performance of the learning algorithm in practice. Our problem has a natural application to the problem of monadic decomposition of quantifier-free integer linear arithmetic formulas, which has been actively studied in recent years. In particular, a finite union of integer hypercubes correspond to a finite disjunction of monadic predicates over integer linear arithmetic (without modulo constraints). Our experiments suggest that our learning algorithms substantially outperform the existing algorithms.

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Section: Contributionsmentioning

confidence: 99%

Section: Applications and Experimentsmentioning

confidence: 99%

Section: Application To Monadic Decompositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Learning Union of Integer Hypercubes with Queries

Markgraf

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2021

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Quantifier elimination for counting extensions of Presburger arithmetic

Chistikov

Haase

Mansutti

2022

Lecture Notes in Computer Science

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We give a new quantifier elimination procedure for Presburger arithmetic extended with a unary counting quantifier $$\exists ^{= x} y\, \mathrm {\Phi }$$ ∃ = x y Φ that binds to the variable $$x$$ x the number of different $$y$$ y satisfying $$\mathrm {\Phi }$$ Φ . While our procedure runs in non-elementary time in general, we show that it yields nearly optimal elementary complexity results for expressive counting extensions of Presburger arithmetic, such as the threshold counting quantifier $$\exists ^{\ge c} y\, \mathrm {\Phi }$$ ∃ ≥ c y Φ that requires that the number of different y satisfying $$\mathrm {\Phi }$$ Φ be at least $$c\in \mathbb {N}$$ c ∈ N , where c can succinctly be defined by a Presburger formula. Our results are cast in terms of what we call the monadically-guarded fragment of Presburger arithmetic with unary counting quantifiers, for which we develop a 2ExpSpace decision procedure.

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CertiStr: a certified string solver

Kan

Lin

Rümmer

et al. 2022

Proceedings of the 11th ACM SIGPLAN International Conference on Certified Programs and Proofs

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Theories over strings are among the most heavily researched logical theories in the SMT community in the past decade, owing to the error-prone nature of string manipulations, which often leads to security vulnerabilities (e.g. cross-site scripting and code injection). The majority of the existing decision procedures and solvers for these theories are themselves intricate; they are complicated algorithmically, and also have to deal with a very rich vocabulary of operations. This has led to a plethora of bugs in implementation, which have for instance been discovered through fuzzing.In this paper, we present CertiStr, a certified implementation of a string constraint solver for the theory of strings with concatenation and regular constraints. CertiStr aims to solve string constraints using a forward-propagation algorithm based on symbolic representations of regular constraints as symbolic automata, which returns three results: sat, unsat, and unknown, and is guaranteed to terminate for the string constraints whose concatenation dependencies are acyclic. The implementation has been developed and proven correct in Isabelle/HOL, through which an effective solver in OCaml was generated. We demonstrate the effectiveness and efficiency of CertiStr against the standard Kaluza benchmark, in which 80.4% tests are in the string constraint fragment of CertiStr. Of these 80.4% tests, CertiStr can solve 83.5% (i.e. CertiStr returns sat or unsat) within 60s.

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Monadic Decomposition in Integer Linear Arithmetic

Cited by 7 publications

References 22 publications

Learning Union of Integer Hypercubes with Queries

Learning Union of Integer Hypercubes with Queries

Quantifier elimination for counting extensions of Presburger arithmetic

CertiStr: a certified string solver

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