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A branch and bound algorithm for extracting smallest minimal unsatisfiable subformulas

A branch and bound algorithm for extracting smallest minimal unsatisfiable subformulas
A branch and bound algorithm for extracting smallest minimal unsatisfiable subformulas
Explaining the causes of infeasibility of Boolean formulas has practical applications in numerous fields, such as artificial intelligence (repairing inconsistent knowledge bases), formal verification (abstraction refinement and unbounded model checking), and electronic design (diagnosing and correcting infeasibility). Minimal unsatisfiable subformulas (MUSes) provide useful insights into the causes of infeasibility. An unsatisfiable formula often has many MUSes. Based on the application domain, however, MUSes with specific properties might be of interest. In this paper, we tackle the problem of finding a smallest-cardinality MUS (SMUS) of a given formula. An SMUS provides a succinct explanation of infeasibility and is valuable for applications that are heavily affected by the size of the explanation. We present (1) a baseline algorithm for finding an SMUS, founded on earlier work for finding all MUSes, and (2) a new branch-and-bound algorithm called Digger that computes a strong lower bound on the size of an SMUS and splits the problem into more tractable subformulas in a recursive search tree. Using two benchmark suites, we experimentally compare Digger to the baseline algorithm and to an existing incomplete genetic algorithm approach. Digger is shown to be faster in nearly all cases. It is also able to solve far more instances within a given runtime limit than either of the other approaches.
Liffiton, Mark
13a68b0b-5c7e-470d-bb66-77887430e8b1
Mneimneh, Maher
68af350f-d215-4508-81e9-649b5ff22d91
Lynce, Ines
2325eb92-000d-4de6-8e5f-1a35f91e7c45
Andraus, Zaher
371b28c8-4993-4496-8166-3a330396ff13
Marques-Silva, Joao
f992f61f-cedd-4897-9f73-1a3ac7ebb35c
Sakallah, Karem
defb7e2c-080d-4c47-8330-d5c6d50ae78a
Liffiton, Mark
13a68b0b-5c7e-470d-bb66-77887430e8b1
Mneimneh, Maher
68af350f-d215-4508-81e9-649b5ff22d91
Lynce, Ines
2325eb92-000d-4de6-8e5f-1a35f91e7c45
Andraus, Zaher
371b28c8-4993-4496-8166-3a330396ff13
Marques-Silva, Joao
f992f61f-cedd-4897-9f73-1a3ac7ebb35c
Sakallah, Karem
defb7e2c-080d-4c47-8330-d5c6d50ae78a

Liffiton, Mark, Mneimneh, Maher, Lynce, Ines, Andraus, Zaher, Marques-Silva, Joao and Sakallah, Karem (2009) A branch and bound algorithm for extracting smallest minimal unsatisfiable subformulas. Constraints, 14. (doi:10.1007/s10601-008-9058-8).

Record type: Article

Abstract

Explaining the causes of infeasibility of Boolean formulas has practical applications in numerous fields, such as artificial intelligence (repairing inconsistent knowledge bases), formal verification (abstraction refinement and unbounded model checking), and electronic design (diagnosing and correcting infeasibility). Minimal unsatisfiable subformulas (MUSes) provide useful insights into the causes of infeasibility. An unsatisfiable formula often has many MUSes. Based on the application domain, however, MUSes with specific properties might be of interest. In this paper, we tackle the problem of finding a smallest-cardinality MUS (SMUS) of a given formula. An SMUS provides a succinct explanation of infeasibility and is valuable for applications that are heavily affected by the size of the explanation. We present (1) a baseline algorithm for finding an SMUS, founded on earlier work for finding all MUSes, and (2) a new branch-and-bound algorithm called Digger that computes a strong lower bound on the size of an SMUS and splits the problem into more tractable subformulas in a recursive search tree. Using two benchmark suites, we experimentally compare Digger to the baseline algorithm and to an existing incomplete genetic algorithm approach. Digger is shown to be faster in nearly all cases. It is also able to solve far more instances within a given runtime limit than either of the other approaches.

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More information

e-pub ahead of print date: 5 August 2008
Published date: December 2009
Organisations: Electronics & Computer Science

Identifiers

Local EPrints ID: 266149
URI: https://eprints.soton.ac.uk/id/eprint/266149
PURE UUID: e7302c4b-257e-42cc-b3ff-979188783c29

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Date deposited: 17 Jul 2008 12:51
Last modified: 06 Nov 2018 17:31

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