Decidability of admissibility:On a problem by friedman and its solution by rybakov

Rybakov (1984) proved that the admissible rules of IPC are decidable. We give a proof of the same theorem, using the same core idea, but couched in the many notions that have been developed in the mean time. In particular, we illustrate how the argument can be interpreted as using refinements of the notions of exactness and extendibility

Almost structural completeness; an algebraic approach

A deductive system is structurally complete if its admissible inference rules are derivable. For several important systems, like modal logic S5, failure of structural completeness is caused only by the underivability of passive rules, i.e. rules that can not be applied to theorems of the system. Neglecting passive rules leads to the notion of almost structural completeness, that means, derivablity of admissible non-passive rules. Almost structural completeness for quasivarieties and varieties of general algebras is investigated here by purely algebraic means. The results apply to all algebraizable deductive systems. Firstly, various characterizations of almost structurally complete quasivarieties are presented. Two of them are general: expressed with finitely presented algebras, and with subdirectly irreducible algebras. One is restricted to quasivarieties with finite model property and equationally definable principal relative congruences, where the condition is verifiable on finite subdirectly irreducible algebras. Secondly, examples of almost structurally complete varieties are provided Particular emphasis is put on varieties of closure algebras, that are known to constitute adequate semantics for normal extensions of S4 modal logic. A certain infinite family of such almost structurally complete, but not structurally complete, varieties is constructed. Every variety from this family has a finitely presented unifiable algebra which does not embed into any free algebra for this variety. Hence unification in it is not unitary. This shows that almost structural completeness is strictly weaker than projective unification for varieties of closure algebras

Spacetime deployments parametrized by gravitational and electromagnetic fields

On the basis of a "Punctual" Equivalence Principle of the general relativity context, we consider spacetimes with measurements of conformally invariant physical properties. Then, applying the Pfaff theory for PDE to a particular conformally equivariant system of differential equations, we make explicit the dependence of any kind of function describing a "spacetime deployment", on n(n+1) parametrizing functions, denoting by n the spacetime dimension. These functions, appearing in a linear differential Spencer sequence and determining gauge fields of spacetime deformations relatively to a "substrat spacetime", can be consistently ascribed to unified electromagnetic and gravitational fields, at any spacetime dimensions n greater or equal to 4.Comment: 26 pages, LaTeX2e, file macro "suppl.sty", correction in the definition of germs and local ring

Matrix Model and Ginsparg-Wilson Relation

We discuss that the Ginsparg-Wilson relation, which has the key role in the recent development of constructing lattice chiral gauge theory, can play an important role to define chiral structures in finite matrix models and noncommutative geometries.Comment: Latex 3 pages, To appear in Nucl.Phys.Proc.Suppl. of Lattice2003(chiral), Tsukuba, Japan, Jul.15-19, 200

Relating Nominal and Higher-order Abstract Syntax Specifications

Nominal abstract syntax and higher-order abstract syntax provide a means for describing binding structure which is higher-level than traditional techniques. These approaches have spawned two different communities which have developed along similar lines but with subtle differences that make them difficult to relate. The nominal abstract syntax community has devices like names, freshness, name-abstractions with variable capture, and the new-quantifier, whereas the higher-order abstract syntax community has devices like lambda-binders, lambda-conversion, raising, and the nabla-quantifier. This paper aims to unify these communities and provide a concrete correspondence between their different devices. In particular, we develop a semantics-preserving translation from alpha-Prolog, a nominal abstract syntax based logic programming language, to G-, a higher-order abstract syntax based logic programming language. We also discuss higher-order judgments, a common and powerful tool for specifications with higher-order abstract syntax, and we show how these can be incorporated into G-. This establishes G- as a language with the power of higher-order abstract syntax, the fine-grained variable control of nominal specifications, and the desirable properties of higher-order judgments.Comment: To appear in PPDP 201