The universe of propositional approximations

The idea of approximate entailment has been proposed by Schaerf and Cadoli [Tractable reasoning via approximation, Artif. Intell. 74(2) (1995) 249-310] as a way of modelling the reasoning of an agent with limited resources. In that framework, a family of logics, parameterised by a set of propositional letters, approximates classical logic as the size of the set increases.The original proposal dealt only with formulas in clausal form, but in Finger and Wassermann [Approximate and limited reasoning: semantics, proof theory, expressivity and control, J. Logic Comput. 14(2) (2004) 179-204], one of the approximate systems was extended to deal with full propositional logic, giving the new system semantics, an axiomatisation, and a sound and complete proof method based on tableaux. In this paper, we extend another approximate system by Schaerf and Cadoli, presented in a subsequent work [M. Cadoli, M. Schaerf, The complexity of entailment in propositional multivalued logics, Ann. Math. Artif. Intell. 18(1) (1996) 29-50] and then take the idea further, presenting a more general approximation framework of which the previous ones are particular cases, and show how it can be used to formalise heuristics used in theorem proving.

[1]  David A. McAllester.  Truth Maintenance , 1990, AAAI.

[2]  Marcelo Finger.  Polynomial Approximations of Full Propositional Logic via Limited Bivalence , 2004, JELIA.

[3]  J. van Leeuwen,et al.  Theoretical Computer Science , 2003, Lecture Notes in Computer Science.

[4]  Frank van Harmelen,et al.  Computing Approximate Diagnoses By Using Approximate Entailment , 1996, KR.

[5]  Marcelo Finger,et al.  Approximate and Limited Reasoning: Semantics, Proof Theory, Expressivity and Control , 2004, J. Log. Comput..

[6]  Renata Wassermann,et al.  Approximate Belief Revision , 2001, Log. J. IGPL.

[7]  Marcello D'Agostino,et al.  Are tableaux an improvement on truth-tables? , 1992, J. Log. Lang. Inf..

[8]  Mukesh Dalal,et al.  Semantics of an Anytime Family of Reasoners , 1996, ECAI.

[9]  Francesco Scarcello,et al.  Semantical and Computational Aspects of Horn Approximations , 1993, IJCAI.

[10]  Marcelo Finger,et al.  Logics for Approximate Reasoning: Approximating Classical Logic "From Above" , 2002, SBIA.

[11]  Marcelo Finger,et al.  Expressivity and Control in Limited Reasoning , 2002, ECAI.

[12]  Bart Selman,et al.  Knowledge compilation and theory approximation , 1996, JACM.

[13]  Bart Selman,et al.  Knowledge Compilation using Horn Approximations , 1991, AAAI.

[14]  Dov M. Gabbay,et al.  A generalization of analytic deduction via labelled deductive systems. Part I: Basic substructural logics , 1994, Journal of Automated Reasoning.

[15]  Richard C. T. Lee,et al.  Symbolic logic and mechanical theorem proving , 1973, Computer science classics.

[16]  Nuel D. Belnap,et al.  Entailment : the logic of relevance and necessity , 1975 .

[17]  Mukesh Dalal.  Anytime Families of Tractable Propositional Reasoners Introduction , .

[18]  Marcelo Finger.  Towards Polynomial Approximations of Full Propositional Logic , 2004, SBIA.

[19]  Krysia Broda,et al.  Labelled Natural Deduction for Substructural Logics , 1999, Log. J. IGPL.

[20]  Marco Schaerf,et al.  Approximation in Concept Description Languages , 1992, KR.

[21]  Marco Schaerf,et al.  Tractable Reasoning via Approximation , 1995, Artif. Intell..

[22]  Hector J. Levesque,et al.  A Logic of Implicit and Explicit Belief , 1984, AAAI.

[23]  Marco Schaerf,et al.  On the complexity of entailment in propositional multivalued logics , 1996, Annals of Mathematics and Artificial Intelligence.