Integrating Boolean and Mathematical Solving: Foundations, Basic Algorithms, and Requirements

Audemard, Gilles; Bertoli, Piergiorgio; Cimatti, Alessandro; Korniłowicz, Artur; Sebastiani, Roberto

doi:10.1007/3-540-45470-5_22

Cited by 8 publications

(11 citation statements)

References 13 publications

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“…Future work could include investigating the use of further solvers, for instance one with integrated computations for specialized mathematical domains [2]. Besides classification wrt.…”

Section: Resultsmentioning

confidence: 99%

Applying SAT Solving in Classification of Finite Algebras

Meier

Sorge

2006

J Autom Reasoning

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The classification of mathematical structures plays an important role for research in pure mathematics. It is, however, a meticulous task that can be aided by using automated techniques. Many automated methods concentrate on the quantitative side of classification, like counting isomorphism classes for certain structures with given cardinality. In contrast, we have devised a bootstrapping algorithm that performs qualitative classification by producing classification theorems that describe unique distinguishing properties for isomorphism classes. In order to fully verify the classification it is essential to prove a range of problems, which can become quite challenging for classical automated theorem provers even in the case of relatively small algebraic structures. But since the problems are in a finite domain, employing Boolean satisfiability solving is possible. In this paper we present the application of satisfiability solvers to generate fully verified classification theorems in finite algebra. We explore diverse methods to efficiently encode the arising problems both for Boolean SAT solvers as well as for solvers with built-in equational theory. We give experimental evidence for their effectiveness, which leads to an improvement of the overall bootstrapping algorithm.

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“…Future work could include investigating the use of further solvers, for instance one with integrated computations for specialized mathematical domains [2]. Besides classification wrt.…”

Section: Resultsmentioning

confidence: 99%

Applying SAT Solving in Classification of Finite Algebras

Meier

Sorge

2006

J Autom Reasoning

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“…The verification problem for LAL is well known, and it has been devoted a lot of interest in the past. In particular, our approach builds upon and improves our previous work on MATHSAT [5,4,7,6,14], along the lines described in Sect. 3.…”

Section: Related Workmentioning

confidence: 97%

“…[5,17,24]. Given the stack-based nature of the boolean search, the MATHSOLVE can significantly exploit previous computations.…”

Section: The Top Level Algorithm: Boolean+theory Solvingmentioning

confidence: 99%

An Incremental and Layered Procedure for the Satisfiability of Linear Arithmetic Logic

Bozzano¹,

Bruttomesso²,

Cimatti³

et al. 2005

Tools and Algorithms for the Construction and Analysis of Systems

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Abstract. In this paper we present a new decision procedure for the satisfiability of Linear Arithmetic Logic (LAL), i.e. boolean combinations of propositional variables and linear constraints over numerical variables. Our approach is based on the well known integration of a propositional SAT procedure with theory deciders, enhanced in the following ways.First, our procedure relies on an incremental solver for linear arithmetic, that is able to exploit the fact that it is repeatedly called to analyze sequences of increasingly large sets of constraints. Reasoning in the theory of LA interacts with the boolean top level by means of a stack-based interface, that enables the top level to add constraints, set points of backtracking, and backjump, without restarting the procedure from scratch at every call. Sets of inconsistent constraints are found and used to drive backjumping and learning at the boolean level, and theory atoms that are consequences of the current partial assignment are inferred.Second, the solver is layered: a satisfying assignment is constructed by reasoning at different levels of abstractions (logic of equality, real values, and integer solutions). Cheaper, more abstract solvers are called first, and unsatisfiability at higher levels is used to prune the search. In addition, theory reasoning is partitioned in different clusters, and tightly integrated with boolean reasoning.We demonstrate the effectiveness of our approach by means of a thorough experimental evaluation: our approach is competitive with and often superior to several state-of-the-art decision procedures. Motivations and GoalsMany practical domains require a degree of expressiveness beyond propositional logic. For instance, timed and hybrid systems have a discrete component as well as a dynamic evolution of real variables; proof obligations arising in software verification are often boolean combinations of constraints over integer variables; circuits described at Register Transfer Level, even though expressible via booleanization, might be easier to analyze at a higher level of abstraction (see e.g. [15]). Many of the verification problems arising in such domains can be naturally modeled as satisfiability in Linear Arithmetic Logic (LAL), i.e., the boolean combination of propositional variables and linear constraints over numerical variables. For its practical relevance, LAL has been devoted a lot of interest, and several decision procedures exist that are able to deal with it (e.g., SVC [17], ICS [24,19], CVCLITE [17,10], UCLID [36,33], HDPLL [30]).In this paper, we propose a new decision procedure for the satisfiability of LAL, both for the real-valued and integer-valued case. We start from a well known approach, previously applied in MATHSAT [26,4] and in several other systems [24,19,17,10, 35,3,21]: a propositional SAT procedure, modified to enumerate propositional assignments for the propositional abstraction of the problem, is integrated with dedicated theory deciders, used to check consistency of propositional assignments wit...

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“…But since then, many solvers that operate directly on ground term algebra appear in the literature [Shankar and Ruess, 2002;Barrett et al, 2000;Groote and van de Pol, 2000;Badban et al, 2007;Badban and van de Pol, 2005]. Solvers such as MathSAT [Audemard et al, 2002] are further specialized for efficiently handling the propositional component of the language.…”

Section: Related Workmentioning

confidence: 99%

On the projection problem in active knowledge bases with incomplete information

Belle

2014

AI Matters

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The problem of projection has been identified as a fundamental reasoning concern in dynamical domains, where we are to determine whether or not some conditions will hold after a sequence of actions has been performed starting in some initial state. Solving the problem requires, at the very least, effectively reasoning about how actions transform the world, and inferring the logical consequences of the initial knowledge base Completing this Ph.D. has been an enriching journey. What follows is my humble attempt to thank those who have been a part of this ride, through its many potholes. Let me begin by expressing my sincere gratitude to Professor Gerhard Lakemeyer, for first encouraging me to find my own path, and then offering valuable suggestions to realize my goals. My previous background was in machine learning, and I imagine my initial intuitions about epistemic logic must have seemed peculiar. In spite of that, Gerhard patiently helped me fix my technical knowledge, allowed me to explore tangential ideas freely, and more importantly, shared his thoughts on what makes a problem interesting. He has overseen my academic growth for the past four years, and his views on mathematical logic and artificial intelligence have greatly influenced my own. For all of this, for showing me how to write scientifically, and for your continued guidance and support: thank you. A special thanks also goes to my external examiner Professor Hector Levesque for painstakingly reading this thesis, and bringing significant bugs and typographical errors to my attention. Carrying out my work at the Knowledge-based Systems Group has been a lot of fun. I thank its members for great corridor and coffee conversations. Participating in soccer matches, and doctoral seminar escapes to picturesque countrysides, with the remainder of Informatik 5 was very exciting, and I hope these traditions are here to stay. I also thank Jens Claßen for technical discussions, and Tim Niemüller for offering to translate the abstract of this thesis. A number of people visited the department over the years, and among them, I would especially like to thank Yongmei Liu for insightful conversations on research. She also had shared her intuitions about "forgetting" with me, which prompted me to explore that topic. I thank her for this. Financial support for the years 2008-2010 was gratefully received by the Deutsche Forschungsgemeinschaft (German Research Foundation), which funded the graduate school Software for mobile communication systems, of which I was a stipendiary. During my time at the graduate school, Prof. Otto Spaniol provided constant encouragement, and supported our participation in Dagstuhl seminars. For this, I am thankful. Further, Prof. Wolfgang Thomas encouraged me to present my results to the AlgoSyn graduate school, and for this, I thank Wolfgang. For much of the year 2011, financial support was received by the BIT research school, and for this I express my gratitude to Gerhard and Prof. Matthias Jarke. For the remainder of 2011, financial support w...

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Integrating Boolean and Mathematical Solving: Foundations, Basic Algorithms, and Requirements

Cited by 8 publications

References 13 publications

Applying SAT Solving in Classification of Finite Algebras

Applying SAT Solving in Classification of Finite Algebras

An Incremental and Layered Procedure for the Satisfiability of Linear Arithmetic Logic

On the projection problem in active knowledge bases with incomplete information

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