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:.

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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.

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