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Saturation-Based Decision Procedures and Complexity Analysis

Yevgeny Kazakov ykazakov@mpi-sb.mpg.de R. 615

ApplicationsApplications

Add semantic markup to a content:

<clause : FOclause >

-To avoid non-relevant information:

<Santa Clause 2, The (2002) : Movie >

The Semantic Web aims is to make a web content meaningful for computers.

An Ontology or terminology TBox defines logical relationships between concepts using

a Description Logic:

Reasoning tasks aim for:

Consistency checking :

Are there any hidden contradictions?

Classification of terminologies:

Does FOclause v Movie ?

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Set Constraints can formalize dataflow

and type inferences in Program Analysis;

entailment of specifications in Model

Checking and many others . . .

To a problem P one can assign a first-order formula F such that:

P holds F is provable

This allows to use a theorem prover for solving the problems automatically. Saturation-based theorem provers are successfully used for those tasks. A prover is based on a well-known principle of proving by contradiction:

F is provable : F implies a falsity ?

To refute a hypothesis : F, new conclusions are derived using inference rules like:

Resolution

Many refinements have been proposed to reduce the number of inferences needed to refute a formula.

There is no guarantee that an answer can be computed in a finite time, since the first-order logic is undecidable. However the classes of problems often correspond to decidable fragments of first order logic. One of them is the Guarded Fragment:

Resolution for specification of decision procedures using clause schemes:

Example: Guarded formula:

Non-guarded formula:

Transitivity is used to formalize temporal reasoning and ordered structures. How to capture it? – Allow transitive guards!

Theorem [1] There is a resolution-based decision procedure for the guarded with transitive guards that runs in 2EXPTIME.

Combinations of decidable fragments:

Join recursive definitions:

Compute cross-inferences:

Theorem [2] There are resolution-based decision procedures for combinations of two-variable, guarded and monadic fragments without equality having the maximal complexity of the combined fragments.

NoNo

YesYes

Saturated Set

Number restrictions are used to describe configuration constrains like:

Client(x)´ Computer(x)Æ9¸1y.Connection(x,y)

Server(y)´ Computer(y)Æ9<100y.Connection(y,x)

It is possible to translate the number restrictions to a guarded fragment efficiently.

Theorem [3] There is a polynomial translation from the two-variable guarded fragment with number restrictions to the three-variable guarded fragment that produces a formula of linear size even if number restrictions are coded in binary.

[1] Kazakov, Y. & de Nivelle, H. (2004) ‘A resolution decision procedure for the guarded fragment with transitive guards’, IJCAR 2004*

[2] Kazakov, Y. (2004) ‘Combining resolution decision procedures’, unpublished manuscript*

[3] Kazakov, Y. (2004) ‘A polynomial translation of the two-variable guarded fragment with number restrictions to the guarded fragment’, JELIA 2004*

*Available from www.mpi-sb.mpg.de/~ykazakov/

First-order logic is used to specify problems from different areas of mathematics and computer science. To solve these problems automatically, many methods have been developed, the most successful of which are strategies based on saturation.

We demonstrate how saturation-based calculi can be used to specify decision procedures and analyze their complexity.

where G is an atom-guard containing all variables of F

Yevgeny Kazakov, MPI für Informatik. Beginning of studies: 01.10.2001. Supervised by Prof. Dr. Harald Ganzinger† and Dr. Hans de Nivelle