Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
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Transcript of Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 1
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
Using Redundancy and Basicnessfor Obtaining Decision
Proceduresfor Fragments of FO-logic
Using Redundancy and Basicnessfor Obtaining Decision
Proceduresfor Fragments of FO-logic
Yevgeny Kazakov
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 2
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
Plan of the TalkPlan of the Talk
I. Fragments of FO-logic
II. Decision procedures
III. Redundancy and Basicness
I. Fragments of FO-logic
II. Decision procedures
III. Redundancy and Basicness
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 3
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
I. Fragments of FO-logicI. Fragments of FO-logic
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 4
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
Fragments of FO-logicFragments of FO-logicMany problems from different fields can be
naturally represented in FO-logic: Knowledge representation (description logics) Planning Formal linguistics Relational databases …
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 5
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
Fragments of FO-logicFragments of FO-logicExample. The Basic Description Logic:
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 6
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
Fragments of FO-logicFragments of FO-logicDescription logic as FO-fragment:
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 7
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
II. Decision proceduresII. Decision procedures
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 8
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
Decision ProceduresDecision 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.
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 9
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
Decision ProceduresDecision ProceduresExtensions of description-like logics are harder to
handle:
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 10
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
Extensions of description-like logics are harder to handle:
Decision procedures rely on heavy model-theoretic analysis: “Good” model representation property
Decision ProceduresDecision Procedures
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 11
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
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.
Decision ProceduresDecision Procedures
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 12
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
Saturation-based Decision ProceduresSaturation-based Decision Procedures
NoNo
YesYes
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 13
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
Ordered Paramodulation CalculusOrdered Paramodulation Calculus
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 14
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
The Guarded FragmentThe Guarded Fragment
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 15
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
Saturating the Guarded FragmentSaturating the Guarded Fragment
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 16
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
Guarded Fragment With TransitivityGuarded Fragment With TransitivityTransitivity 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.
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 17
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
Guarded Fragment With TransitivityGuarded Fragment With TransitivityIs GF decidable when transitive predicates can
appear in guards only? => [GF+TG]?What is the complexity of monadic-GF[T]?[Szwast,Tendera,2001]: [GF+TG] is in DEXPTIME,
monadic-GF[T] is NEXPTIME-hard.[Kierionski,2002,2003]: [GF+TG!] is EXPSPACE-
hard, [GF+TG] is DEXPTIME-hard.
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Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
III. Redundancy and BasicnessIII. Redundancy and Basicness
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Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
Why Transitivity Is Hard?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!
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 20
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
RedundancyRedundancyAbstract notion of redundancy
[Bachmair,Ganzinger,1990]:
How to show that inference is redundant?
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Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
RedundancyRedundancy
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Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
BasicnessBasicness
This restriction can be strengthen to basicness:.
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 23
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
BasicnessBasicnessOnly paramodulation to the “source” of Skolem
function is needed. Helps to avoid the “dangerous” paramodulation
inferences:
Eligible paramodulation inferences produce redundant clauses only.
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Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic
ConclusionsConclusionsUsing 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.
10-12 December, 2003 AG-2 Logic Seminar, Schloß Ringberg 25
Using Redundancy and Basicness for Obtaining Decision Procedures for Fragments of FO-logic