Constrained Pseudo-Propositional Logic

Abstract

Propositional logic, with the aid of SAT solvers, has become capable of solving a range of important and complicated problems. Expanding this range, to contain additional varieties of problems, is subject to the complexity resulting from encoding counting constraints in conjunctive normal form (CNF). Due to the limitation of the expressive power of propositional logic, generally, such an encoding increases the numbers of variables and clauses excessively. This work eliminates the indicated drawback by interpolating constraint symbols and the set of natural numbers \({\mathbb {N}}\) into the alphabet of propositional logic and adjusting the underlying language accordingly. In the extended logic counting constraints are naturally formulated, while many important aspects, such as Boolean nature and the soundness and completeness theorems, are kept preserved.

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Notes

  1. 1.

    Counting constraints are Pseudo-Boolean constraints with positive coefficients.

  2. 2.

    This addition is easily distinguished from the addition of formulas that appears in definition 2.1.

  3. 3.

    Zorns’s lemma If every chain in a nonempty partially ordered set S is bounded then S has a maximal element.

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Acknowledgements

I would like to thank Prof. Steffen Hölldobler, Director of the International Center for Computational Logic, Dresden, Germany. I have learned from him how to rationalize logically rather than just thinking mathematically. Without his influence this work would not have been established.

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Correspondence to Ahmad-Saher Azizi-Sultan.

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Azizi-Sultan, AS. Constrained Pseudo-Propositional Logic. Log. Univers. 14, 523–535 (2020). https://doi.org/10.1007/s11787-020-00266-x

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Keywords

  • Propositional logic
  • Proof system
  • Pseudo-Boolean constraints
  • SAT

Mathematics Subject Classification

  • Primary 03B05
  • Secondary 03B22
  • 03F03
  • 05A05