## Towards efficient satisfiability checking for boolean algebra with presburger arithmetic (2007)

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Venue: | In CADE-21 |

Citations: | 28 - 17 self |

### BibTeX

@INPROCEEDINGS{Kuncak07towardsefficient,

author = {Viktor Kuncak},

title = {Towards efficient satisfiability checking for boolean algebra with presburger arithmetic},

booktitle = {In CADE-21},

year = {2007}

}

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### Abstract

1 Introduction This paper considers the satisfiability problem for a logic that allows reason-ing about sets and their cardinalities. We call this logic quantifier-free Boolean Algebra with Presburger Arithmetic and denote it QFBAPA. Our motivationfor QFBAPA is proving the validity of formulas arising from program verifica-tion [12,13,14], but

### Citations

657 | The description logic handbook: theory, implementation, and applications - Baader - 2003 |

267 |
The classical decision problem
- Borger, Gradel, et al.
- 1997
(Show Context)
Citation Context ...∈X and λ1(x) ∈ {0, 1} for all x ∈ X, then λ2 = λ1. 3) For {x1, . . . , xn} = X (for x1, . . . , xn distinct), the system of d equations expressed in vector form as x∈X λ(x1)x1 + . . . + λ(xn)xn = � X =-=(5)-=- has (λ(x1), . . . , λ(xn)) = (1, . . . , 1) as the unique solution in Z n ≥0 . 4) NICG(X). Proof. 1) → 2): This is Lemma 1. 2) → 3): Assume 2) and let λ1(xi) = 1 for 1 ≤ i ≤ n. For any solution λ2 we... |

207 |
CVC Lite: A new implementation of the cooperating validity checker
- Barrett, Berezin
(Show Context)
Citation Context ...orresponding to the overall running time of the algorithm. We ran the experiments on 3GHz, 1MB cache, 2GB RAM workstation. As a decision procedure for quantifier-free PA we used CVC3 version 20070217 =-=[3]-=-. 3 The examples are available from http://lara.epfl.ch/ ∼ kuncak/cade07exampless14 VC# (∗=invalid) BAPA QFBAPA PA size(nodes) total time(s) PA size(nodes) iteration of N total time(s) 1 39 < 0.1 190 ... |

190 |
PP is as hard as the polynomial-time hierarchy
- Toda
- 1991
(Show Context)
Citation Context ...blems [2], because our partition cardinality variables are not restricted to {0, 1}.sRelationship to counting SAT. Although similarly looking, QFBAPA satisfiability is different from the #SAT problem =-=[26]-=-. Solving QFBAPA formula differs from counting the number of satisfying assignments of propositional formulas because set partitions may possibly be empty. An immediate consequence of our results is t... |

126 | Theory of Computation - Kozen - 2006 |

82 |
On the complexity of integer programming
- Papadimitriou
- 1981
(Show Context)
Citation Context ... ki+1 = ki + ki can efficiently encode large constants. Fundamentally, the reason we are interested in large set cardinalities is because they arise from small model theorem for Presburger arithmetic =-=[19]-=-; supporting them is necessary for verifying symbolic cardinality bounds and constraints such as |x ∩ y| = |z|. Abstraction using sizes of partitions. An alternative to examining set interpretations u... |

79 |
Set partitioning: a survey
- Balas, Padberg
- 1976
(Show Context)
Citation Context ...s X with NICG(X) do not form a matroid, and maximal NICG(X) need not be a Hilbert basis. The equations generated from QFBAPA problems are more difficult than set packing and set partitioning problems =-=[2]-=-, because our partition cardinality variables are not restricted to {0, 1}.sRelationship to counting SAT. Although similarly looking, QFBAPA satisfiability is different from the #SAT problem [26]. Sol... |

67 |
The first order properties of products of algebraic systems
- Feferman, Vaught
- 1959
(Show Context)
Citation Context ... this example by narrowing down the search using the observations on minimal counterexamples in the proof of Lemma 6.) = 3 λ1 + λ2 + λ3 λ1 λ1 + λ2 λ2 + λ3 + λ4 = 3 + λ3 + λ4 + λ5 = 4 + λ4 + λ5 = 4 11 =-=(8)-=-s12 It is easy to see that the system has (1, 1, 1, 1, 1) as the only solution in the space of non-negative integers. Note that all variables are non-zero in this solution. The five columns of the sys... |

44 | Complexity of the Two-Variable Fragment with Counting Quantifiers
- Pratt-Hartmann
- 2005
(Show Context)
Citation Context ...ing the validity of formulas arising from program verification [11,12,13], but QFBAPA constraints also occur in mechanized set theory [7], constraint data bases [23,24], as a fragment of other logics =-=[18,20,1]-=- and in the semantic analysis of natural language [15]. Figure 1 shows the syntax of QFBAPA. The logic contains 1) arbitrary boolean algebra (BA) expressions denoting sets, supporting operations such ... |

40 | Model theoretic methods in combined constraint satisfiability
- Ghilardi
- 2004
(Show Context)
Citation Context ...ers as BAPA formulas, so they essentially have the same expressive power. Quantifier elimination also makes BAPA interesting as a potential shared language for combining multiple reasoning procedures =-=[9]-=-.sVC# verification condition property being checked 1 x /∈ content ∧ size = card content −→ (size = 0 ↔ content = ∅) 2 x /∈ content ∧ size = card content −→ size + 1 = card({x} ∪ content) 3 size = car... |

39 |
The Complexity of Logical Theories
- Berman
- 1980
(Show Context)
Citation Context ...show that BAPA has the same complexity as the complexity of Presburger arithmetic (PA), namely alternating doubly exponential time with a linear number of alternations, denoted STA(∗, 22nO(1) , n) in =-=[4]-=-, [10, Lecture 24]. BAPA admits quantifier elimination, which implies that QFBAPA formulas define the same class of relations on sets and integers as BAPA formulas, so they essentially have the same e... |

39 | The SMT-LIB Standard: Version 1.2
- Ranise, Tinelli
- 2006
(Show Context)
Citation Context ...iminated by flattening the formula to contain no nested terms and then replacing t = ite(p, t1, t2) with the formula (p → t = t1) ∧ (¬p → t = t2). (It is also directly available in the SMT-LIB format =-=[22]-=-.) Using ite, we can express cijlj as ite(cij, lj, 0). Then (3) becomes �p �N i=0 j=1 ite(�bi�βj , lj, 0) = ki. Note that we can substitute the values ki back into the original PA formula G, so there ... |

37 |
Hilbert bases, Caratheodory’s Theorem and combinatorial optimization
- Sebö
- 1990
(Show Context)
Citation Context ... sets of different cardinality. Nevertheless, Lemma 2 and Lemma 3 show that some of the properties of independent sets do hold for vectors in X where NICG(X). Note also that X5 is not a Hilbert basis =-=[25]-=-. Namely, we have that (1, 1, 1, 1) ∈ cone(X5) \ int cone(X5) because (1, 1, 1, 1) = 1/3((1, 0, 1, 1) + (1, 1, 0, 1) + (1, 1, 1, 0) + (0, 1, 1, 1)). This illustrates why previous results on Hilbert ba... |

36 | Modular Data Structure Verification
- Kuncak
- 2007
(Show Context)
Citation Context ...es. We call this logic quantifier-free Boolean Algebra with Presburger Arithmetic and denote it QFBAPA. Our motivation for QFBAPA is proving the validity of formulas arising from program verification =-=[11,12,13]-=-, but QFBAPA constraints also occur in mechanized set theory [7], constraint data bases [23,24], as a fragment of other logics [18,20,1] and in the semantic analysis of natural language [15]. Figure 1... |

31 | Deciding Boolean Algebra with Presburger Arithmetic
- Kuncak, Nguyen, et al.
(Show Context)
Citation Context ...es. We call this logic quantifier-free Boolean Algebra with Presburger Arithmetic and denote it QFBAPA. Our motivation for QFBAPA is proving the validity of formulas arising from program verification =-=[11,12,13]-=-, but QFBAPA constraints also occur in mechanized set theory [7], constraint data bases [23,24], as a fragment of other logics [18,20,1] and in the semantic analysis of natural language [15]. Figure 1... |

26 | An algorithm for deciding BAPA: Boolean Algebra with Presburger Arithmetic
- Kuncak, Nguyen, et al.
- 2005
(Show Context)
Citation Context ...es. We call this logic quantifier-free Boolean Algebra with Presburger Arithmetic and denote it QFBAPA. Our motivation for QFBAPA is proving the validity of formulas arising from program verification =-=[11,12,13]-=-, but QFBAPA constraints also occur in mechanized set theory [7], constraint data bases [23,24], as a fragment of other logics [18,20,1] and in the semantic analysis of natural language [15]. Figure 1... |

19 | Decision procedures for set-valued fields
- Kuncak, Rinard
- 2005
(Show Context)
Citation Context ...formula of size polynomial in n that would express the property “all 2 n partitions of n sets are non-empty”. Reasoning about sets. The quantifier-free fragment of BA is shown NPcomplete in [17]; see =-=[14]-=- for a generalization of this result using the parameterized complexity of the Bernays-Schönfinkel-Ramsey class of first-order logic [5, Page 258]. The decision procedure for quantifier-free fragment ... |

19 | Systems of negative Boolean constraints
- Marriott, Odersky
- 1992
(Show Context)
Citation Context ...no QFBAPA formula of size polynomial in n that would express the property “all 2 n partitions of n sets are non-empty”. Reasoning about sets. The quantifier-free fragment of BA is shown NPcomplete in =-=[17]-=-; see [14] for a generalization of this result using the parameterized complexity of the Bernays-Schönfinkel-Ramsey class of first-order logic [5, Page 258]. The decision procedure for quantifier-free... |

18 |
Set Theory for Computing
- Cantone, Omodeo, et al.
- 2001
(Show Context)
Citation Context ...Arithmetic and denote it QFBAPA. Our motivation for QFBAPA is proving the validity of formulas arising from program verification [11,12,13], but QFBAPA constraints also occur in mechanized set theory =-=[7]-=-, constraint data bases [23,24], as a fragment of other logics [18,20,1] and in the semantic analysis of natural language [15]. Figure 1 shows the syntax of QFBAPA. The logic contains 1) arbitrary boo... |

10 | On algorithms and complexity for sets with cardinality constraints
- Marnette, Kuncak, et al.
- 2005
(Show Context)
Citation Context ...ed results. We suggested the possibility of the existence of sparse solutions in the final version of [13], where we also established the complexity of quantified BAPA. In a previous technical report =-=[16]-=- we identified a PSPACE algorithm for QFBAPA, but the techniques used there are different and not needed for the results of this paper. A preliminary version of the current result is described in [11,... |

10 |
On the abstract properties of linear independence
- Whitney
- 1935
(Show Context)
Citation Context ... As we have remarked before, the set of d unit vectors is a maximal NICG for every d, which means that, unlike linearly independent sets of vectors over a field or other independent sets in a matroid =-=[27]-=-, there are maximal NICG sets of different cardinality. Nevertheless, Lemma 2 and Lemma 3 show that some of the properties of independent sets do hold for vectors in X where NICG(X). Note also that X5... |

9 |
How to extend a formal system with a Boolean Algebra component
- Ohlbach, Koehler
- 1998
(Show Context)
Citation Context ...ing the validity of formulas arising from program verification [11,12,13], but QFBAPA constraints also occur in mechanized set theory [7], constraint data bases [23,24], as a fragment of other logics =-=[18,20,1]-=- and in the semantic analysis of natural language [15]. Figure 1 shows the syntax of QFBAPA. The logic contains 1) arbitrary boolean algebra (BA) expressions denoting sets, supporting operations such ... |

9 | Quantifier-elimination for the first-order theory of boolean algebras with linear cardinality constraints
- Revesz
- 2004
(Show Context)
Citation Context ...BAPA. Our motivation for QFBAPA is proving the validity of formulas arising from program verification [11,12,13], but QFBAPA constraints also occur in mechanized set theory [7], constraint data bases =-=[23,24]-=-, as a fragment of other logics [18,20,1] and in the semantic analysis of natural language [15]. Figure 1 shows the syntax of QFBAPA. The logic contains 1) arbitrary boolean algebra (BA) expressions d... |

8 | Combining sets with cardinals
- Zarba
(Show Context)
Citation Context ..., as we explain below. Unfortunately, its direct use introduces a number of integer variables that is exponential in the number of sets. This approach is the essence of previous algorithms for QFBAPA =-=[28,23,18]-=- and appears as a special case of our algorithm for quantified BAPA [12, 13]. All these algorithms would yield exponentially large certificates for satisfiability of QFBAPA, specifying the values of e... |

7 |
Using first-order theorem provers in a data structure verification system
- Bouillaguet, Kuncak, et al.
- 2007
(Show Context)
Citation Context ...hey contain some unnecessary assumptions that would normally appear in an automatically generated verification condition. Syntactically determining which assumptions are useful is a difficult problem =-=[6]-=-, so it is reasonable to leave this task to the the decision procedure. Formulas 1 − 6, 2a, 3a, 6a are all valid. Formulas 2b, 3b, 4b, 5b, 6b, 6c are obtained from 2a, 3a, 4, 5, 6a, 6a, respectively, ... |

5 |
The expressivity of constraint query languages with boolean algebra linear cardinality constraints
- Revesz
- 2005
(Show Context)
Citation Context ...BAPA. Our motivation for QFBAPA is proving the validity of formulas arising from program verification [11,12,13], but QFBAPA constraints also occur in mechanized set theory [7], constraint data bases =-=[23,24]-=-, as a fragment of other logics [18,20,1] and in the semantic analysis of natural language [15]. Figure 1 shows the syntax of QFBAPA. The logic contains 1) arbitrary boolean algebra (BA) expressions d... |

5 |
and Gennady Shmonin. Carathéodory bounds for integer cones
- Eisenbrand
- 2006
(Show Context)
Citation Context ...hose indices p i 1, . . . , p i 10 identify Venn regions that belong to b. 2. The computation of a sufficient polynomial value for N is the second contribution of this paper. We start with the result =-=[8]-=- that if an element is in an integer cone generated by a set of vectors X of dimension d, then it is also insan integer cone generated by a “small” subset of X of size N(d). This result implies that a... |

1 | Precise understanding of natural language. Stanford Univeristy PhD dissertation draft
- Lev
- 2007
(Show Context)
Citation Context ...ion [11,12,13], but QFBAPA constraints also occur in mechanized set theory [7], constraint data bases [23,24], as a fragment of other logics [18,20,1] and in the semantic analysis of natural language =-=[15]-=-. Figure 1 shows the syntax of QFBAPA. The logic contains 1) arbitrary boolean algebra (BA) expressions denoting sets, supporting operations such as union, intersection and complement, 2) arbitrary qu... |