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Approximate formulae for a logic that capture classes of computational

Summary: Approximate formulae for a logic
that capture classes of computational
ARGIMIRO ARRATIA, Dpto. de Matem´atica Aplicada Facultad de
Ciencias Universidad de Valladolid Valladolid 47005, Spain
E-mail: arratia@mac.uva.es
CARLOS E. ORTIZ, Department of Mathematics and Computer Science
Arcadia University 450 S. Easton Road, Glenside, PA 19038-3295, U.S.A.
E-mail: ortiz@arcadia.edu
This paper presents a syntax of approximate formulae suited for the logic with counting quantifiers SOLP. This
logic was formalised by us in [1] where, among other properties, we showed the following facts: (i) In the presence
of a built­in (linear) order, SOLP can describe NP­complete problems and some of its fragments capture the
classes P and NL; (ii) weakening the ordering relation to an almost order we can separate meaningful fragments,
using a combinatorial tool adapted to these languages.
The purpose of our approximate formulae is to provide a syntactic approximation to the logic SOLP, enhanced
with a built-in order, that should be complementary of the semantic approximation based on almost orders, by
means of producing logics where problems are syntactically described within a small counting error. We introduce a
concept of strong expressibility based on approximate formulae, and show that for many fragments of SOLP with
built-in order, including ones that capture P and NL, expressibility and strong expressibility are equivalent. We


Source: Arratia, Argimiro A. - Departamento de Matemática Aplicada, Universidad de Valladolid


Collections: Mathematics