Logical foundations and complexity of 4QL, a query language with unrestricted negation

The paper discusses properties of 4QL, a DATALOG⌉⌉-like query language, originally outlined by Maluszyński and Szalas (Maluszyński & Szalas, 2011). 4QL allows one to use rules with negation in heads and bodies of rules. It is based on a simple and intuitive semantics and provides uniform tools for “lightweight” versions of known forms of nonmonotonic reasoning. Negated literals in heads of rules may naturally lead to inconsistencies. On the other hand, rules do not have to attach meaning to some literals. Therefore 4QL is founded on a four-valued semantics, employing the logic introduced in (Maluszyński et al., 2008; Vitória et al., 2009) with truth values: ‘true’, ‘false’, ‘inconsistent’ and ‘unknown’. In addition, 4QL is tractable w.r.t. data complexity and captures PTIME queries. Even though DATALOG⌉⌉ is known as a concept for the last 30 years, to our best knowledge no existing approach enjoys these properties. In the current paper we: investigate properties of well-supported models of 4QL prove the correctness of the algorithm for computing well-supported models show that 4QL has PTIME data complexity and captures PTIME.

[1]  Umberto Straccia,et al.  A relevance terminological logic for information retrieval , 1996, SIGIR '96.

[2]  Andrzej Skowron,et al.  Knowledge Representation Techniques - A Rough Set Approach , 2006, Studies in Fuzziness and Soft Computing.

[3]  Letizia Tanca,et al.  What you Always Wanted to Know About Datalog (And Never Dared to Ask) , 1989, IEEE Trans. Knowl. Data Eng..

[4]  Raymond Reiter.  On Closed World Data Bases , 1977, Logic and Data Bases.

[5]  Nuel D. Belnap,et al.  A Useful Four-Valued Logic , 1977 .

[6]  Andrzej Szalas,et al.  Paraconsistent Logic Programs with Four-Valued Rough Sets , 2008, RSCTC.

[7]  Patrick Doherty,et al.  Approximate databases: a support tool for approximate reasoning , 2006, J. Appl. Non Class. Logics.

[8]  Donald Nute,et al.  Defeasible Logic , 1994, INAP.

[9]  Melvin Fitting,et al.  A Kripke-Kleene Semantics for Logic Programs , 1985, J. Log. Program..

[10]  C. Damásio,et al.  A survey of paraconsistent semantics for logic programs , 1998 .

[11]  Frederick Maier,et al.  Extending paraconsistent SROIQ , 2010, RR 2010.

[12]  Sandra de Amo,et al.  A paraconsistent logic programming approach for querying inconsistent databases , 2007, Int. J. Approx. Reason..

[13]  Ofer Arieli,et al.  Paraconsistent Declarative Semantics for Extended Logic Programs , 2010 .

[14]  Itala M. Loffredo D'Ottaviano,et al.  Paraconsistency: The Logical Way to the Inconsistent , 2002 .

[15]  François Fages,et al.  Consistency of Clark's completion and existence of stable models , 1992, Methods Log. Comput. Sci..

[16]  Umberto Straccia,et al.  A Sequent Calculus for Reasoning in Four-Valued Description Logics , 1997, TABLEAUX.

[17]  Andrzej Szalas,et al.  Modeling and Reasoning with Paraconsistent Rough Sets , 2009, Fundam. Informaticae.

[18]  Heinrich Wansing,et al.  Inconsistency-tolerant description logic. Part II: A tableau algorithm for CACLC , 2008, J. Appl. Log..

[19]  Luís Moniz Pereira,et al.  An encompassing framework for Paraconsistent Logic Programs , 2005, J. Appl. Log..

[20]  Andrzej Szalas,et al.  Three-Valued Paraconsistent Reasoning for Semantic Web Agents , 2010, KES-AMSTA.

[21]  Serge Abiteboul,et al.  Foundations of Databases , 1994 .

[22]  Andrzej Szalas,et al.  Living with Inconsistency and Taming Nonmonotonicity , 2010, Datalog.

[23]  Pascal Hitzler,et al.  Paraconsistent Reasoning for OWL 2 , 2009, RR.

[24]  Walter A. Carnielli,et al.  A Taxonomy of C-Systems , 2001 .

[25]  J. Martin Marcos,et al.  Formal inconsistency and evolutionary databases , 2004 .

[26]  Serge Abiteboul,et al.  Complexity of answering queries using materialized views , 1998, PODS.

[27]  Didier Dubois,et al.  On Ignorance and Contradiction Considered as Truth-Values , 2007, Log. J. IGPL.

[28]  W. Carnielli,et al.  Logics of Formal Inconsistency , 2007 .

[29]  Walter Alexandre Carnielli,et al.  A Logical Framework for Integrating Inconsistent Information in Multiple Databases , 2002, FoIKS.

[30]  Michael Gelfond,et al.  Classical negation in logic programs and disjunctive databases , 1991, New Generation Computing.

[31]  as,et al.  Living with inconsistency and taming nonmonotonicity , 2010 .

[32]  Melvin Fitting,et al.  Fixpoint Semantics for Logic Programming a Survey , 2001, Theor. Comput. Sci..

[33]  Chiaki Sakama,et al.  Paraconsistent Stable Semantics for Extended Disjunctive Programs , 1995, J. Log. Comput..