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Saturation-Based Decision Procedures and
Complexity Analysis
Yevgeny Kazakov, MPI für Informatik. Beginning of
studies 01.10.2001. Supervised by Prof. Dr.
Harald Ganzinger and Dr. Hans de Nivelle
More Results
To a problem P one can assign a first-order
formula F such that
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.
Resolution for specification of decision
procedures using clause schemes
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
Combinations of decidable fragments
F is provable ? F implies a falsity ?
To refute a hypothesis F, new conclusions are
derived using inference rules like
The Semantic Web aims is to make a web content
meaningful for computers.
Resolution

• Join recursive definitions

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

• Compute cross-inferences

Add semantic markup to a content ltclause
FOclause gt

• To avoid non-relevant information
• ltSanta Clause 2, The (2002) Movie gt

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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.
An Ontology or terminology TBox defines logical
relationships between concepts using a
Description Logic
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Number restrictions are used to describe
configuration constrains like Client(x)
Computer(x)Æ91y.Connection(x,y) Server(y)
Computer(y)Æ9lt100y.Connection(y,x)
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
It is possible to translate the number
restrictions to a guarded fragment efficiently.

• Reasoning tasks aim for
• Consistency checking
• Are there any hidden contradictions?
• Classification of terminologies
• Does FOclause v Movie ?

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

• Set Constraints can formalize dataflow and type
  inferences in Program Analysis
• entailment of specifications in Model Checking
• and many others . . .

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.
Yevgeny Kazakov
ykazakov_at_mpi-sb.mpg.de R. 615