## Deciding Boolean Algebra with Presburger Arithmetic

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Venue: | J. of Automated Reasoning |

Citations: | 31 - 26 self |

### BibTeX

@ARTICLE{Kuncak_decidingboolean,

author = {Viktor Kuncak and Huu Hai Nguyen and Martin Rinard},

title = {Deciding Boolean Algebra with Presburger Arithmetic},

journal = {J. of Automated Reasoning},

year = {},

volume = {2006},

pages = {10--1007}

}

### Years of Citing Articles

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### Abstract

Abstract. We describe an algorithm for deciding the first-order multisorted theory BAPA, which combines 1) Boolean algebras of sets of uninterpreted elements (BA) and 2) Presburger arithmetic operations (PA). BAPA can express the relationship between integer variables and cardinalities of unbounded finite sets, and supports arbitrary quantification over sets and integers. Our original motivation for BAPA is deciding verification conditions that arise in the static analysis of data structure consistency properties. Data structures often use an integer variable to keep track of the number of elements they store; an invariant of such a data structure is that the value of the integer variable is equal to the number of elements stored in the data structure. When the data structure content is represented by a set, the resulting constraints can be captured in BAPA. BAPA formulas with quantifier alternations arise when verifying programs with annotations containing quantifiers, or when proving simulation relation conditions for refinement and equivalence of program fragments. Furthermore, BAPA constraints can be used for proving the termination of programs that manipulate data structures, as well as

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Citation Context ...many integer variables to reduce the problem to PA. Among the systems for interactively reasoning about richer theories of sets are Isabelle (Nipkow et al., 2002), HOL (Gordon and Melham, 1993), PVS (=-=Owre et al., 1992-=-). First-order frameworks such as Athena (Arkoudas et al., 2004) can use axiomatizations of sets along with calls to resolution-based theorem provers (Voronkov, 1995; Weidenbach, 2001) to reason about... |

500 | T.: Introduction to HOL: A Theorem Proving Environment for Higher Order Logic: Cambridge - Melham - 1993 |

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Citation Context ...m α in Figure 11 maps each BAPAsentence F0 into an equivalent PA-sentence α(F0). The validity of PA sentences is decidable (Presburger, 1929). In combination with a decision procedure for PA such as (=-=Pugh, 1991-=-; Boigelot et al., 2005; Henriksen et al., 1995), the algorithm α is a decision procedure for BAPA sentences. 4.1. Example Run of Algorithm α As an illustration, we show the result of runing the algor... |

264 |
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Citation Context ...); see (Kuncak and Rinard, 2005) for a generalization of this result using the parameterized complexity of the main.tex; 12/07/2006; 11:49; p.28sBernays-Schönfinkel-Ramsey class of first-order logic (=-=Börger et al., 1997-=-, Page 258). (Cantone et al., 2001) gives an overview of several fragments of set theory including theories with quantifiers but no cardinality constraints and theories with cardinality constraints bu... |

204 | CVC Lite: A new implementation of the cooperating validity checker - Barrett, Berezin - 2004 |

181 | Languages, automata and logic
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Citation Context ...rs. Therefore, α can decide an extension of BAPA where the constraints on cardinalities of sets are expressed using relations on integers definable in MSOL over strings; these relations go beyond PA (=-=Thomas, 1997-=-, Page 400), (Bruyére et al., 1994). 8.3. Quantifier-Free Fragment of BAPA The consequence of the quantifier elimination property of BAPA is that the formulas in the quantifier-free fragment of BAPA, ... |

161 |
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Citation Context ...of finite sets, so we obtain a generalization of Theorem 9 to subsets of a countable set. 8.2. BAPA and MSOL over Strings The weak monadic second-order logic (MSOL) over strings is a decidable logic (=-=Thatcher and Wright, 1968-=-; Henriksen et al., 1995) that can encode Presburger arithmetic by encoding addition using one successor symbol and quantification over sets of elements. There are two important differences between MS... |

147 |
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Citation Context ...nts of arithmetic that go beyond PA are described in (Bozga and Iosif, 2005; Bruyére et al., 1994). Reasoning about Sets. The first results on decidability of BA of sets are from (Loewenheim, 1915), (=-=Ackermann, 1954-=-, Chapter 4) and use quantifier elimination, from which one can derive small model property. (Kozen, 1980) gives the complexity of the satisfiability problem for arbitrary BA. (Martin and Nipkow, 1989... |

138 | An algorithm to evaluate quantified Boolean formulae and its experimental evaluation - Cadoli, Schaerf, et al. |

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Citation Context ...thm α for translating BAPA sentences to PA sentences THEOREM 3. The algorithm α in Figure 11 maps each BAPAsentence F0 into an equivalent PA-sentence α(F0). The validity of PA sentences is decidable (=-=Presburger, 1929-=-). In combination with a decision procedure for PA such as (Pugh, 1991; Boigelot et al., 2005; Henriksen et al., 1995), the algorithm α is a decision procedure for BAPA sentences. 4.1. Example Run of ... |

92 |
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Citation Context ...ur implementation uses quantifer-elimination based Omega test (Pugh, 1991). Among the decision procedures for full PA, (Chaieb and Nipkow, 2003) is the only proof-generating version, and is based on (=-=Cooper, 1972-=-). Decidable fragments of arithmetic that go beyond PA are described in (Bozga and Iosif, 2005; Bruyére et al., 1994). Reasoning about Sets. The first results on decidability of BA of sets are from (L... |

88 | Combining superposition, sorts and splitting - Weidenbach - 2001 |

83 |
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Citation Context ...ially larger quantifier-free PA formula, guessing the truth values of conjuncts and then using a non-deterministic polynomial time algorithm to solve the resulting integer linear programming problem (=-=Papadimitriou, 1981-=-). (Quantifier-free PA formulas can be solved using implementations such as CVC Lite (Barrett and Berezin, 2004) and UCLID (Lahiri and Seshia, 2004).) Therefore, the worst-case complexity of α is no w... |

70 |
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Citation Context ...tion of transition relations, with appropriate widenings, can be used to generalize static analyses such as (Lam et al., 2005) and model checking approaches such main.tex; 12/07/2006; 11:49; p.10sas (=-=Bultan et al., 1999-=-). The quantifier elimination process here ensures that the transition relation remains represented by a quantifier-free formula throughout the analysis. Quantifier elimination is also useful for quer... |

67 |
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Citation Context ... we can already define all operations of PA. This leads to the structure BAPA, which properly generalizes both BA and PA. As we explain in Section 2, a version of BAPA was shown decidable already in (=-=Feferman and Vaught, 1959-=-). Recently, a decision procedure for a fragment of BAPA without quantification over sets was presented in (Zarba, 2004c), cast as a multi-sorted theory. Starting from (Lam et al., 2005) as our motiva... |

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Citation Context ...e an extension of BAPA where the constraints on cardinalities of sets are expressed using relations on integers definable in MSOL over strings; these relations go beyond PA (Thomas, 1997, Page 400), (=-=Bruyére et al., 1994-=-). 8.3. Quantifier-Free Fragment of BAPA The consequence of the quantifier elimination property of BAPA is that the formulas in the quantifier-free fragment of BAPA, denoted QFBAPA, define the same cl... |

51 | Symbolically computing most-precise abstract operations for shape analysis - Yorsh, Reps, et al. - 2004 |

44 | Generalized typestate checking for data structure consistency - Lam, Kuncak, et al. - 2005 |

39 |
The anatomy of Vampire: implementing bottom-up procedures with code trees
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Citation Context ...(Gordon and Melham, 1993), PVS (Owre et al., 1992). First-order frameworks such as Athena (Arkoudas et al., 2004) can use axiomatizations of sets along with calls to resolution-based theorem provers (=-=Voronkov, 1995-=-; Weidenbach, 2001) to reason about sets. Combinations of Decidable Theories. The techniques for combining quantifier-free theories (Nelson and Oppen, 1979; Ruess and Shankar, 2001) and their generali... |

36 | Verifying a file system implementation - Arkoudas, Zee, et al. - 2004 |

35 |
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Citation Context ...on quantifier elimination. 3.1. Verifying Data Structure Consistency Figure 4 presents a procedure insert in a language that directly manipulates sets. Such languages can either be directly executed (=-=Dewar, 1979-=-) or can arise as abstractions of programs in standard languages (Lam et al., 2005). The program in Figure 4 manipulates a global set of objects content and an integer field size. The program maintain... |

32 | Combining nonstably infinite theories - Tinelli, Zarba |

28 |
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Citation Context ...1915), (Ackermann, 1954, Chapter 4) and use quantifier elimination, from which one can derive small model property. (Kozen, 1980) gives the complexity of the satisfiability problem for arbitrary BA. (=-=Martin and Nipkow, 1989-=-) study unification in Boolean rings. The quantifier-free fragment of BA is shown NP-complete in (Marriott and Odersky, 1994); see (Kuncak and Rinard, 2005) for a generalization of this result using t... |

27 | Automatically generating loop invariants using quantifier elimination - Kapur - 2005 |

27 |
Complexity of Boolean Algebras
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Citation Context ...out data structures as dynamically changing sets of objects. Here we are interested in BA of all subsets of some set; this theory was shown decidable already in (Loewenheim, 1915; Skolem, 1919), see (=-=Kozen, 1980-=-) for the decidability and the complexity of all models of BA axioms. The motivation for this paper is the fact that we often need to reason not only about the content of a data structure, but also ab... |

26 | An algorithm for deciding BAPA: Boolean Algebra with Presburger Arithmetic - Kuncak, Nguyen, et al. - 2005 |

25 | M.: Structural subtyping of non-recursive types is decidable - Kuncak, Rinard - 2003 |

25 | Quantitative Shape Analysis - Rugina - 2004 |

23 | Generalized typestate checking using set interfaces and pluggable analyses - Lam, Kuncak, et al. - 2004 |

23 |
Untersuchung über die Axiome des Klassenkalküls und über Produktations– und Summationsprobleme, welche gewisse Klassen von Aussagen betreffen. Videnskabsakademiet i Kristiania
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Citation Context ...ies and reasoning about data structures as dynamically changing sets of objects. Here we are interested in BA of all subsets of some set; this theory was shown decidable already in (Loewenheim, 1915; =-=Skolem, 1919-=-), see (Kozen, 1980) for the decidability and the complexity of all models of BA axioms. The motivation for this paper is the fact that we often need to reason not only about the content of a data str... |

19 | Decision procedures for set-valued fields - Kuncak, Rinard - 2005 |

19 | Systems of negative Boolean constraints
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Citation Context ...n, 1980) gives the complexity of the satisfiability problem for arbitrary BA. (Martin and Nipkow, 1989) study unification in Boolean rings. The quantifier-free fragment of BA is shown NP-complete in (=-=Marriott and Odersky, 1994-=-); see (Kuncak and Rinard, 2005) for a generalization of this result using the parameterized complexity of the main.tex; 12/07/2006; 11:49; p.28sBernays-Schönfinkel-Ramsey class of first-order logic (... |

18 | Set Theory for Computing - Cantone, Omodeo, et al. - 2001 |

14 |
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Citation Context ... typestate properties and reasoning about data structures as dynamically changing sets of objects. Here we are interested in BA of all subsets of some set; this theory was shown decidable already in (=-=Loewenheim, 1915-=-; Skolem, 1919), see (Kozen, 1980) for the decidability and the complexity of all models of BA axioms. The motivation for this paper is the fact that we often need to reason not only about the content... |

14 | Combining sets with elements - Zarba - 2004 |

13 | The first-order theory of sets with cardinality constraints is decidable - Kuncak, Rinard - 2004 |

13 | A quantifier elimination algorithm for a fragment of set theory involving the cardinality operator - Zarba - 2004 |

10 | On algorithms and complexity for sets with cardinality constraints - Marnette, Kuncak, et al. - 2005 |

9 |
How to extend a formal system with a Boolean Algebra component
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Citation Context ...formula. We next describe observations that, although not an improvement in the worst case, may be helpful for certain classes of formulas. main.tex; 12/07/2006; 11:49; p.22sFirst, as pointed out in (=-=Ohlbach and Koehler, 1998-=-), if the input formula entails any BA identities (which can be represented as |b| = 0), then the number of non-empty Venn regions decreases, which reduces the number of integer variables in the resul... |

9 | Quantifier-elimination for the first-order theory of boolean algebras with linear cardinality constraints - Revesz - 2004 |

8 | Decision procedures in automated deduction - Tiwari - 2000 |

8 | Combining sets with cardinals - Zarba |

7 | On decidability within the arithmetic of addition and divisibility - Bozga, Iosif - 2005 |

5 |
The Complexity of Logical Theories. Theoretical Computer Science (TCS
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Citation Context ...procedure for the full first-order theory of BAPA. Furthermore, we analyze our decision procedure and show that it yields optimal computational complexity for BAPA, identical to the complexity of PA (=-=Berman, 1980-=-). This solves the question of the computational complexity of BAPA. 1 We have also implemented our decision procedure; we report on our initial experience in using the decision procedure in a system ... |

5 | Rybalchenko A (2005) Transition predicate abstraction and fair termination - Podelski |

5 | The Combination Problem in Automated Reasoning - Zarba - 2004 |