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Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic

This presentation discusses the use of redundancy and basicness in obtaining decision procedures for fragments of FO-logic. Various fragments of FO-logic are explored, and their applications in areas such as knowledge representation, planning, formal linguistics, and relational databases are discussed. The talk also covers decision procedures and their computational properties, as well as the challenges faced when handling extensions of description-like logics. The presentation concludes with the importance of basicness and the potential for designing effective decision procedures using redundancy and basicness.

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Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic

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  1. Using Redundancy and Basicnessfor Obtaining Decision Proceduresfor Fragments of FO-logic Yevgeny Kazakov AG-2 Logic Seminar, Schloß Ringberg

  2. Plan of the Talk • Fragments of FO-logic • Decision procedures • Redundancy and Basicness AG-2 Logic Seminar, Schloß Ringberg

  3. I. Fragments of FO-logic AG-2 Logic Seminar, Schloß Ringberg

  4. Fragments of FO-logic • Many problems from different fields can be naturally represented in FO-logic: • Knowledge representation (description logics) • Planning • Formal linguistics • Relational databases • … AG-2 Logic Seminar, Schloß Ringberg

  5. Fragments of FO-logic • Example. The Basic Description Logic: AG-2 Logic Seminar, Schloß Ringberg

  6. Fragments of FO-logic • Description logic as FO-fragment: AG-2 Logic Seminar, Schloß Ringberg

  7. II. Decision procedures AG-2 Logic Seminar, Schloß Ringberg

  8. Decision Procedures • How to explain good computational properties of description logics? • “Good” model properties: • Finite model property • Tree model property • Basis for Tableau-based decision procedures. AG-2 Logic Seminar, Schloß Ringberg

  9. Decision Procedures • Extensions of description-like logics are harder to handle: AG-2 Logic Seminar, Schloß Ringberg

  10. Decision Procedures • Extensions of description-like logics are harder to handle: • Decision procedures rely on heavy model-theoretic analysis: • “Good” model representation property AG-2 Logic Seminar, Schloß Ringberg

  11. Decision Procedures • Alternative approach: use general theorem provers for FO-logic. • Advantage: • No need to invent anything; • Soundness and completeness are guarantied; • Easy to implement: just write a translator to FO-logic and use existing theorem provers. • However: • Still need to prove termination. • Relatively slow in comparison to specialized decision procedures. AG-2 Logic Seminar, Schloß Ringberg

  12. Yes No Saturation-based Decision Procedures AG-2 Logic Seminar, Schloß Ringberg

  13. Ordered Paramodulation Calculus AG-2 Logic Seminar, Schloß Ringberg

  14. The Guarded Fragment AG-2 Logic Seminar, Schloß Ringberg

  15. Saturating the Guarded Fragment AG-2 Logic Seminar, Schloß Ringberg

  16. Guarded Fragment With Transitivity • Transitivity and functionality axioms are outside the Guarded Fragment. • Does GF loose decidability when some predicates are allowed to be transitive, or functional? • YES[Grädel,1999]: GF3 with one functional or transitive predicate is undecidable. • How to explain decidability of modal and description logics with transitivity? • [Ganzinger et al.,1999]: GF2[T] is undecidable, but monadic-GF2[T] is decidable. AG-2 Logic Seminar, Schloß Ringberg

  17. Guarded Fragment With Transitivity • Is GFdecidable when transitive predicates can appear in guards only? =>[GF+TG]? • What is the complexity of monadic-GF[T]? • [Szwast,Tendera,2001]: [GF+TG] is inDEXPTIME, monadic-GF[T] is NEXPTIME-hard. • [Kierionski,2002,2003]: [GF+TG!] is EXPSPACE-hard, [GF+TG] is DEXPTIME-hard. AG-2 Logic Seminar, Schloß Ringberg

  18. III. Redundancy and Basicness AG-2 Logic Seminar, Schloß Ringberg

  19. Why Transitivity Is Hard? • Consider the resolution inferences with transitivity: • The clause 4 can be obtained another way: • With the smaller instance of transitivity clause! AG-2 Logic Seminar, Schloß Ringberg

  20. Redundancy • Abstract notion of redundancy [Bachmair,Ganzinger,1990]: • How to show that inference is redundant? AG-2 Logic Seminar, Schloß Ringberg

  21. Redundancy AG-2 Logic Seminar, Schloß Ringberg

  22. This restriction can be strengthen to basicness:. Basicness AG-2 Logic Seminar, Schloß Ringberg

  23. Basicness • Only paramodulation to the “source” of Skolem function is needed. • Helps to avoid the “dangerous” paramodulation inferences: • Eligible paramodulation inferences produce redundant clauses only. AG-2 Logic Seminar, Schloß Ringberg

  24. Conclusions • Using advanced refinements of saturation-based procedures it is possible establish decidability and complexity results for very expressive fragments of FO-logic. • In particular, decidability of [GF+TG] can be established using redundancy and basicness. • Basicness is important: allowing conjunctions of transitive relations in guards leads to undecidability. • New perspectives for designing saturation-based decision procedures. AG-2 Logic Seminar, Schloß Ringberg

  25. Thank you ! AG-2 Logic Seminar, Schloß Ringberg

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