Literal Projection for First-Order Logic

The computation of literal projection generalizes predicate quantifier elimination by permitting, so to speak, quantifying upon an arbitrary sets of ground literals, instead of just (all ground literals with) a given predicate symbol. Literal projection allows, for example, to express predicate quantification upon a predicate just in positive or negative polarity. Occurrences of the predicate in literals with the complementary polarity are then considered as unquantified predicate symbols. We present a formalization of literal projection and related concepts, such as literal forgetting, for first-order logic with a Herbrand semantics, which makes these notions easy to access, since they are expressed there by means of straightforward relationships between sets of literals. With this formalization, we show properties of literal projection which hold for formulas that are free of certain links, pairs of literals with complementary instances, each in a different conjunct of a conjunction, both in the scope of a universal first-order quantifier, or one in a subformula and the other in its context formula. These properties can justify the application of methods that construct formulas without such links to the computation of literal projection. Some tableau methods and direct methods for second-order quantifier elimination can be understood in this way.

[1]  Neil V. Murray,et al.  Tableaux, Path Dissolution, and Decomposable Negation Normal Form for Knowledge Compilation , 2003, TABLEAUX.

[2]  Valentin Goranko,et al.  Algorithmic correspondence and completeness in modal logic. I. The core algorithm SQEMA , 2006, Log. Methods Comput. Sci..

[3]  Roy Dyckhoff.  Automated Reasoning with Analytic Tableaux and Related Methods , 2000, Lecture Notes in Computer Science.

[4]  W. Ackermann.  Untersuchungen über das Eliminationsproblem der mathematischen Logik , 1935 .

[5]  Valentin Goranko,et al.  Algorithmic correspondence and completeness in modal logic. I. The core algorithm SQEMA , 2006, Log. Methods Comput. Sci..

[6]  Kenneth L. McMillan,et al.  Applying SAT Methods in Unbounded Symbolic Model Checking , 2002, CAV.

[7]  Dov M. Gabbay,et al.  Second-Order Quantifier Elimination - Foundations, Computational Aspects and Applications , 2008, Studies in logic : Mathematical logic and foundations.

[8]  Frank Wolter,et al.  Semi-qualitative Reasoning about Distances: A Preliminary Report , 2000, JELIA.

[9]  Christoph Wernhard.  Automated deduction for projection elimination , 2009 .

[10]  Christoph Wernhard,et al.  Semantic Knowledge Partitioning , 2004, JELIA.

[11]  Dov M. Gabbay,et al.  Quantifier Elimination in Second-Order Predicate Logic , 1992, KR.

[12]  Andrzej Szalas.  On the Correspondence between Modal and Classical Logic: An Automated Approach , 1993, J. Log. Comput..

[13]  Alan Robinson,et al.  The Inverse Method , 2001, Handbook of Automated Reasoning.

[14]  Adnan Darwiche,et al.  DPLL with a Trace: From SAT to Knowledge Compilation , 2005, IJCAI.

[15]  Josef Urban,et al.  MPTP 0.2: Design, Implementation, and Initial Experiments , 2006, Journal of Automated Reasoning.

[16]  Carsten Lutz,et al.  Did I Damage My Ontology? A Case for Conservative Extensions in Description Logics , 2006, KR.

[17]  Patrick Doherty,et al.  Computing Circumscription Revisited: A Reduction Algorithm , 1997, Journal of Automated Reasoning.

[18]  Hans Hahn.  Einführung in die Mathematische Logik , 1932 .

[19]  Adnan Darwiche,et al.  Decomposable negation normal form , 2001, JACM.