## On Linear Arithmetic with Stars

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Citations: | 10 - 7 self |

### BibTeX

@MISC{Piskac_onlinear,

author = {Ruzica Piskac and Viktor Kuncak and Lara I and C Epfl},

title = {On Linear Arithmetic with Stars},

year = {}

}

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### Abstract

Abstract. We consider an extension of integer linear arithmetic with a star operator that takes closure under vector addition of the set of solutions of linear arithmetic subformula. We show that the satisfiability problem for this language is in NP (and therefore NP-complete). Our proof uses a generalization of a recent result on sparse solutions of integer linear programming problems. We present two consequences of our result. The first one is an optimal decision procedure for a logic of sets, multisets, and cardinalities that has applications in verification, interactive theorem proving, and description logics. The second is NP-completeness of the reachability problem for a class of “homogeneous ” transition systems whose transitions are defined using integer linear arithmetic formulas. 1

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Citation Context ...et X ⊆ Z d be a finite set of integer vectors and let � b ∈ X ∗ . Then there exists a subset ˜ X such that � b ∈ ˜ X ∗ and | ˜ X| ≤ 2d log(4dM), where M = maxx∈X ||x||∞. Theorem 1 has been applied in =-=[17]-=- in order to establishing membership in NP for quantifier-free Boolean algebra with Presburger arithmetic. However, in the case of linear arithmetic with stars, it is not sufficient to apply the theor... |

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Citation Context ...ly many atomic formulas, we obtain a polynomial bound on the number of bits needed for �u in the smallest solution of (1). 2. We apply twice a theorem on the size of minimal generator of integer cone =-=[11]-=- to prove that only polynomially many generating vectors are sufficient. Finally, we show that we can group linear combinations of generating vectors into linear combination of polynomially many varia... |

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Citation Context ...based on sets become applicable for verifying data structure clients [16,15,21]. Multisets and sets also arise in other domains, as witnessed by their use in libraries of interactive provers Isabelle =-=[22]-=- and KIV [5], and their presence in the Sparql query language forssemantic web [1]. Decision procedures for sets and multisets are therefore useful for automated reasoning about such domains. Our noti... |

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Citation Context ...set specifications, and then techniques based on sets become applicable for verifying data structure clients [13, 18]. Multisets and sets are also present in libraries of interactive provers Isabelle =-=[19]-=- and KIV [3]. Our results yield decision procedures that can increase the automation within such systems. As a simple running example, consider a verification condition for insertion of an element rep... |