17,075 research outputs found
Formalization of Universal Algebra in Agda
In this work we present a novel formalization of universal algebra in Agda. We show that heterogeneous signatures can be elegantly modelled in type-theory using sets indexed by arities to represent operations. We prove elementary results of heterogeneous algebras, including the proof that the term algebra is initial and the proofs of the three isomorphism theorems. We further formalize equational theory and prove soundness and completeness. At the end, we define (derived) signature morphisms, from which we get the contravariant functor between algebras; moreover, we also proved that, under some restrictions, the translation of a theory induces a contra-variant functor between models.Fil: Gunther, Emmanuel. Universidad Nacional de Córdoba. Facultad de Matemática, AstronomÃa y FÃsica; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas; ArgentinaFil: Gadea, Alejandro Emilio. Universidad Nacional de Córdoba. Facultad de Matemática, AstronomÃa y FÃsica; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas; ArgentinaFil: Pagano, Miguel Maria. Consejo Nacional de Investigaciones CientÃficas y Técnicas; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, AstronomÃa y FÃsica; Argentin
A Divergence Critic for Inductive Proof
Inductive theorem provers often diverge. This paper describes a simple
critic, a computer program which monitors the construction of inductive proofs
attempting to identify diverging proof attempts. Divergence is recognized by
means of a ``difference matching'' procedure. The critic then proposes lemmas
and generalizations which ``ripple'' these differences away so that the proof
can go through without divergence. The critic enables the theorem prover Spike
to prove many theorems completely automatically from the definitions alone.Comment: See http://www.jair.org/ for any accompanying file
A Logic for Natural Language
This paper describes a language called £N whose structure mirrors that of natural language. £N is characterized by absence of variables and individual constants. Singular predicates assume the role of both individual constants and free variables. The role of bound variables is played by predicate functors called selection operators. Like natural languages, £N is implicitly many-sorted. £N does not have an identity relation. Its expressive power lies between the predicate calculus without identity and the predicate calculus with identity. The loss in expressiveness relative to the predicate calculus with identity however is not significant. Deduction in £N is intended to parallel reasoning in natural language, and therefore is termed surface reasoning. In contrast to deduction in a disparate underlying logic such as clausal form, each step of a proof in £N has a direct counterpart in the surface language. A sound and complete axiomatization is given. Derived rules, corresponding to monotonicity and conservativity of quantifiers and to unification and resolution in conventional logic, are presented. Several problems are worked to illustrate reasoning in £N
Canonical Abstract Syntax Trees
This paper presents Gom, a language for describing abstract syntax trees and
generating a Java implementation for those trees. Gom includes features
allowing the user to specify and modify the interface of the data structure.
These features provide in particular the capability to maintain the internal
representation of data in canonical form with respect to a rewrite system. This
explicitly guarantees that the client program only manipulates normal forms for
this rewrite system, a feature which is only implicitly used in many
implementations
A Focused Sequent Calculus Framework for Proof Search in Pure Type Systems
Basic proof-search tactics in logic and type theory can be seen as the
root-first applications of rules in an appropriate sequent calculus, preferably
without the redundancies generated by permutation of rules. This paper
addresses the issues of defining such sequent calculi for Pure Type Systems
(PTS, which were originally presented in natural deduction style) and then
organizing their rules for effective proof-search. We introduce the idea of
Pure Type Sequent Calculus with meta-variables (PTSCalpha), by enriching the
syntax of a permutation-free sequent calculus for propositional logic due to
Herbelin, which is strongly related to natural deduction and already well
adapted to proof-search. The operational semantics is adapted from Herbelin's
and is defined by a system of local rewrite rules as in cut-elimination, using
explicit substitutions. We prove confluence for this system. Restricting our
attention to PTSC, a type system for the ground terms of this system, we obtain
the Subject Reduction property and show that each PTSC is logically equivalent
to its corresponding PTS, and the former is strongly normalising iff the latter
is. We show how to make the logical rules of PTSC into a syntax-directed system
PS for proof-search, by incorporating the conversion rules as in
syntax-directed presentations of the PTS rules for type-checking. Finally, we
consider how to use the explicitly scoped meta-variables of PTSCalpha to
represent partial proof-terms, and use them to analyse interactive proof
construction. This sets up a framework PE in which we are able to study
proof-search strategies, type inhabitant enumeration and (higher-order)
unification
An Integrated First-Order Theory of Points and Intervals over Linear Orders (Part I)
There are two natural and well-studied approaches to temporal ontology and
reasoning: point-based and interval-based. Usually, interval-based temporal
reasoning deals with points as a particular case of duration-less intervals. A
recent result by Balbiani, Goranko, and Sciavicco presented an explicit
two-sorted point-interval temporal framework in which time instants (points)
and time periods (intervals) are considered on a par, allowing the perspective
to shift between these within the formal discourse. We consider here two-sorted
first-order languages based on the same principle, and therefore including
relations, as first studied by Reich, among others, between points, between
intervals, and inter-sort. We give complete classifications of its
sub-languages in terms of relative expressive power, thus determining how many,
and which, are the intrinsically different extensions of two-sorted first-order
logic with one or more such relations. This approach roots out the classical
problem of whether or not points should be included in a interval-based
semantics
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