## On Linear Arithmetic with Stars

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Citations: | 8 - 6 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

### Citations

445 | G.: Lazy abstraction
- Henzinger, Jhala, et al.
- 2002
(Show Context)
Citation Context ...metic formulas. 1 Introduction Decision procedures [5, 1, 15, 10, 7] are among key techniques that enable automated verification of infinite state systems, as, for example, in software model checkers =-=[2, 6, 12]-=-. These techniques are also increasingly used to raise the level of automation in interactive theorem provers [24, 8, 17]. We believe that an important step towards making such theorem provers even mo... |

393 | S.K.: Automatic predicate abstraction of c programs
- Ball, Majumdar, et al.
- 2001
(Show Context)
Citation Context ... Decision procedures and automated theorem provers [7, 3, 19, 12, 9] are among key techniques that enable automated verification of infinite state systems, as, for example, in software model checkers =-=[4, 14, 18]-=-. These techniques are also increasingly used to raise the level of automation in interactive theorem provers [27,10,8,20,2]. We believe that an important step towards making such theorem provers even... |

204 |
CVC Lite: A new implementation of the cooperating validity checker
- Barrett, Berezin
- 2004
(Show Context)
Citation Context ...bility problem for a class of “homogeneous” transition systems whose transitions are defined using integer linear arithmetic formulas. 1 Introduction Decision procedures and automated theorem provers =-=[7, 3, 19, 12, 9]-=- are among key techniques that enable automated verification of infinite state systems, as, for example, in software model checkers [4, 14, 18]. These techniques are also increasingly used to raise th... |

160 | Métayer: Programming by Multiset Transformation
- Banâtre, Le
- 1993
(Show Context)
Citation Context ...QFPA ∗ . The satisfiability of the example constraint (3) is equivalent to the satisfiability of the QFPA ∗ constraint us = 1 ∧ uy ̸= uL + 1 ∧ (us, uy, uL) ∈ {(s ′ , y ′ , L ′ ) | y ′ = L ′ + s ′ } ∗ =-=(4)-=- We prove equisatisfiability of (3) and (4); see [21, Theorem 2] for the analogous proof for an arbitrary formulas on multisets. Suppose that (3) has a solution E = {e1, . . . , eN}. Because functions... |

93 |
Ranjit Jhala, Rupak Majumdar, and Grégoire Sutre. Lazy abstraction
- Henzinger
- 2002
(Show Context)
Citation Context ... Decision procedures and automated theorem provers [7, 3, 19, 12, 9] are among key techniques that enable automated verification of infinite state systems, as, for example, in software model checkers =-=[4, 14, 18]-=-. These techniques are also increasingly used to raise the level of automation in interactive theorem provers [27,10,8,20,2]. We believe that an important step towards making such theorem provers even... |

83 |
On the complexity of integer programming
- Papadimitriou
- 1981
(Show Context)
Citation Context ... left the exact complexity open. This paper settles the question of the exact complexity by presenting a polynomial-time satisfiability-preserving reduction of (1) to QFPA, which is known to be in NP =-=[23]-=-. Moreover, in this paper we identify another application of our constraints. We consider infinite-state transitions systems whose state consists of finite control and a finite number of integer count... |

64 |
Presburger formulas and languages
- Semigroups
- 1966
(Show Context)
Citation Context ...ity is equivalent to (1), and then tests the satisfiability of φ. The idea behind the construction of φ is the following. Because F is a QFPA formula, its solution set {�v | F (�v)} is semilinear set =-=[13]-=-. This implies the existence of finitely many generating vectors �ai, � bij whose linear combination is {�v | F (�v)}. However, in worst case the number of generating vectors is exponential, so we avo... |

49 | Formal system development with KIV
- Balser, Reif, et al.
- 2000
(Show Context)
Citation Context ... 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 notions of multi... |

47 |
L.: Zapato: Automatic Theorem Proving for Predicate Abstraction Refinement
- Ball, Cook, et al.
- 2004
(Show Context)
Citation Context ...bility problem for a class of “homogeneous” transition systems whose transitions are defined using integer linear arithmetic formulas. 1 Introduction Decision procedures and automated theorem provers =-=[7, 3, 19, 12, 9]-=- are among key techniques that enable automated verification of infinite state systems, as, for example, in software model checkers [4, 14, 18]. These techniques are also increasingly used to raise th... |

45 | Indexed predicate discovery for unbounded system verification
- Lahiri, Bryant
- 2004
(Show Context)
Citation Context ... Decision procedures and automated theorem provers [7, 3, 19, 12, 9] are among key techniques that enable automated verification of infinite state systems, as, for example, in software model checkers =-=[4, 14, 18]-=-. These techniques are also increasingly used to raise the level of automation in interactive theorem provers [27,10,8,20,2]. We believe that an important step towards making such theorem provers even... |

39 | The PROSPER Toolkit
- Dennis, Norrish, et al.
- 2000
(Show Context)
Citation Context ...ication of infinite state systems, as, for example, in software model checkers [4, 14, 18]. These techniques are also increasingly used to raise the level of automation in interactive theorem provers =-=[27,10,8,20,2]-=-. We believe that an important step towards making such theorem provers even more effective is the development of decision procedures for new classes of formulas that go beyond the traditionally consi... |

39 | The UCLID Decision Procedure
- Lahiri, Seshia
- 2004
(Show Context)
Citation Context ...bility problem for a class of “homogeneous” transition systems whose transitions are defined using integer linear arithmetic formulas. 1 Introduction Decision procedures and automated theorem provers =-=[7, 3, 19, 12, 9]-=- are among key techniques that enable automated verification of infinite state systems, as, for example, in software model checkers [4, 14, 18]. These techniques are also increasingly used to raise th... |

36 | Verifying a file system implementation
- Arkoudas, Zee, et al.
- 2004
(Show Context)
Citation Context ...ication of infinite state systems, as, for example, in software model checkers [4, 14, 18]. These techniques are also increasingly used to raise the level of automation in interactive theorem provers =-=[27,10,8,20,2]-=-. We believe that an important step towards making such theorem provers even more effective is the development of decision procedures for new classes of formulas that go beyond the traditionally consi... |

36 | Modular Data Structure Verification
- Kuncak
- 2007
(Show Context)
Citation Context ... languages such as Java, data abstraction can be used to show that data structures satisfy set specifications, and then techniques 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 ... |

35 | N.: Efficient E-Matching for SMT Solvers
- Moura, Bjørner
- 2007
(Show Context)
Citation Context ...econd is the reachability problem for a class of transition systems whose transitions increment the state vector by solutions of integer linear arithmetic formulas. 1 Introduction Decision procedures =-=[5, 1, 15, 10, 7]-=- are among key techniques that enable automated verification of infinite state systems, as, for example, in software model checkers [2, 6, 12]. These techniques are also increasingly used to raise the... |

33 |
Minimal solutions of linear Diophantine systems: bounds and algorithms
- Pottier
- 1991
(Show Context)
Citation Context ...rating vectors �ai, � bij whose linear combination is {�v | F (�v)}. However, in worst case the number of generating vectors is exponential, so we avoid explicitly constructing them. We instead apply =-=[25]-=- to compute an upper bound on the size of generating vectors. This gives us bounds on coefficients in an exponentially large QFPA formula equisatisfiable with (1). We then combine two constructions to... |

29 | Solving quantified verification conditions using satisfiability modulo theories
- Ge, Barrett, et al.
(Show Context)
Citation Context |

28 | Towards efficient satisfiability checking for boolean algebra with presburger arithmetic
- Kuncak, Rinard
- 2007
(Show Context)
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... |

27 | Modular Pluggable Analyses for Data Structure Consistency
- Kuncak, Lam, et al.
(Show Context)
Citation Context ... languages such as Java, data abstraction can be used to show that data structures satisfy set specifications, and then techniques 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 ... |

23 | Combining WS1S and HOL
- Basin, Friedrich
- 2000
(Show Context)
Citation Context ...ication of infinite state systems, as, for example, in software model checkers [4, 14, 18]. These techniques are also increasingly used to raise the level of automation in interactive theorem provers =-=[27,10,8,20,2]-=-. We believe that an important step towards making such theorem provers even more effective is the development of decision procedures for new classes of formulas that go beyond the traditionally consi... |

22 | Cooperating theorem provers: A case study combining HOL-Light and CVC Lite
- McLaughlin, Barrett, et al.
(Show Context)
Citation Context |

20 |
Automated verification of shape, size and bag properties
- Chin, David, et al.
- 2007
(Show Context)
Citation Context ... languages such as Java, data abstraction can be used to show that data structures satisfy set specifications, and then techniques 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 ... |

18 |
On programming, an interim report on the setl project
- Schwartz
- 1973
(Show Context)
Citation Context ...f multiset constraints. Many modern programs perform extensive manipulation of collections of objects implemented either as built in data structures or as collection libraries. Languages such as SETL =-=[26]-=- directly support sets as primitives, whereas the Gamma parallel programming paradigm [6, Page 103] is based on multiset transformations. Sets and multisets would directly arise in verification condit... |

14 | Banâtre and Daniel Le Métayer. Programming by Multiset Transformation - Jean-Pierre - 1993 |

11 | query language for RDF. http://www.w3.org/TR/rdf-sparql-query - SPARQL - 2008 |

11 | Decision procedures for multisets with cardinality constraints
- Piskac, Kuncak
- 2008
(Show Context)
Citation Context ...cause semilinear set representation can be exponentially large. First algorithm for QFPA ∗ satisfiability that avoids explicit construction of semilinear set representation is the PSPACE algorithm in =-=[21]-=-. The present paper is the first to establish the exact complexity of QFPA ∗ satisfiability, namely NP-completeness. To show this result, we will use bounds on solutions of integer linear programming ... |

11 | Combining multisets with integers
- Zarba
- 2002
(Show Context)
Citation Context ...ns. Our decision procedure enables reasoning about collections of objects (sets and multisets) and their cardinalities, which was our original motivation for introducing it in [21]. Previously, Zarba =-=[25]-=- considered decision procedures for multiset constraints but without the cardinality operator, presenting a direct reduction to QFPA. The cardinality operator makes the reduction in [25] inapplicable.... |

10 |
de Moura and Nikolaj Bjørner. Efficient E-matching for smt solvers
- Leonardo
- 2007
(Show Context)
Citation Context |

8 | Using decision procedures with a higher-order logic
- Shankar
(Show Context)
Citation Context |

5 |
and Gennady Shmonin. Carathéodory bounds for integer cones
- Eisenbrand
- 2006
(Show Context)
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... |

2 |
Multiset theory version 1.30 (Isabelle distribution). http://isabelle.in.tum.de/dist/ library/HOL/Library/Multiset.html
- Nipkow, Wenzel, et al.
- 2005
(Show Context)
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... |

2 | Carathéodory bounds for integer cones
- Eisenbrand, Shmonin
- 2006
(Show Context)
Citation Context ...t, we will use bounds on solutions of integer linear programming problems with exponentially many variables [20], bounds on seminilinear set generators [22], and Carathéodory bounds for integer cones =-=[9]-=-. Our proof builds on some of the ideas previously introduced in [14, 21]. Application to reasoning about collections. Our decision procedure enables reasoning about collections of objects (sets and m... |

1 | Automatic predicate abstraction of C programs - Balser, Reif, et al. - 2001 |

1 | Automated verification of shape, size and bag properties via separation logic - Wenzel, Voelker |

1 |
Multiset theory version
- Nipkow, Wenzel, et al.
(Show Context)
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... |