Abstract
A homomorphism φ of logic programs from P to P′ is a function mapping Atoms(P) to Atoms(P′) and it preserves complements and program clause. For each definite program clause a ← a1, ..., a n ∈ P it implies that φ(a) ← φ(a1), ..., φ(a n ) is a program clauses of P′. A homomorphism φ is an isomorphism if φ is a bijection. In this paper, the complexity of the decision problems on homomorphism and isomorphism for definite logic programs is studied. It is shown that the homomorphism problem (HOM-LP) for definite logic programs is NP–complete, and the isomorphism problem (ISO-LP) is equivalent to the graph isomorphism problem (GI).
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References
Cook S A, Reckhow R A. The relative efficiency of propositional proof system. Journal of Symbolic Logic, 1979, 44(1): 36–50, 2001.
Krishnamurthy B. Short proofs for tricky formulas. Acta Informatica, 1985, 22: 253–275.
Szeider S. How to Prove Unsatisfiability by Homomorphisms. Elsevier Preprint.
Szeider S. NP-completeness of refutability by literal-once resolution. Lecture Notes in Artificial Intelligence 2083, Springer Verlag, Draft version, 2001.
Urquhart A. The complexity of propositional proofs. The Bulletin of Symbolic Logic, 1995, 1(4): 425–467.
Urquhart A. The symmetry rule in propositional logic. Discrete Applied Mathematics, 1999, 96–97: 177–193.
Davis M, Putnam H. A computing procedure for quantification theory. Journal of the ACM, 1960, 7: 201–215.
Daoyun Xu. On the complexity of renamings and homomorphisms for minimal unsatisfiable formulas [Dissertation]. Nanjing University, 2002.
Szeider S. Homomorphisms of conjunctive normal forms. Discrete Applied Mathematics, 2003, 130(2): 351–356.
Papadimitriou C H, Wolfe D. The complexity of facets resolved. Journal of Computer and System Sciences, 1988, 37(1): 2–13.
Aharoni R. Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas. Journal of Combinatorial Theory, Series A, 1996, 43(A): 196–204.
G Davydov, I Davydova, H Kleine Büning. An efficient algorithm for the minimal unsatisfiability problem for a subclass of CNF. Annals of Mathematics and Artificial Intelligence, 1998, 23(3-4): 229–245.
Fleischner H, Kullmann O, Szeider S. Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference. Theoretical Computer Science, 2002, 289(1): 503–516.
H Kleine Büning, Xishun Zhao. Polynomial time algorithms for computing a representation for minimal unsatisfiable formulas with fixed deficiency. Information Processing Letters, 2002, 84(3): 147–151.
H Kleine Büning, Daoyun Xu. The complexity of homomorphisms and renamings for minimal unsatisfiable formulas. Annals of Mathematics and Artificial Intelligence, 2005, 43(1–4): 113–127.
Köbler J, Schöning J, Toran J. The Graph Isomorphism Problem: Its Structural Complexity. Birkhäuser Verlag, 1993.
Hell P, Nešetřil J. On the complexity of H-coloring. Journal of Combinatorial Theory, Series B, 1990, 48: 92–110.
Bondy J A, Murty U S R. Graph Theory with Applications. London: Macmillan, 1976.
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The work is supported by the National Natural Science Foundation of China (Grant Nos. 60310213 and 60463001), the Special Foundation for Improving Scientific Research Condition of Guizhou, and the Government Foundation of Guizhou Province.
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Xu, DY., Tao, ZH. Complexities of Homomorphism and Isomorphism for Definite Logic Programs. J Comput Sci Technol 20, 758–762 (2005). https://doi.org/10.1007/s11390-005-0758-x
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DOI: https://doi.org/10.1007/s11390-005-0758-x