10,956 research outputs found
Consistency and Completeness of Rewriting in the Calculus of Constructions
Adding rewriting to a proof assistant based on the Curry-Howard isomorphism,
such as Coq, may greatly improve usability of the tool. Unfortunately adding an
arbitrary set of rewrite rules may render the underlying formal system
undecidable and inconsistent. While ways to ensure termination and confluence,
and hence decidability of type-checking, have already been studied to some
extent, logical consistency has got little attention so far. In this paper we
show that consistency is a consequence of canonicity, which in turn follows
from the assumption that all functions defined by rewrite rules are complete.
We provide a sound and terminating, but necessarily incomplete algorithm to
verify this property. The algorithm accepts all definitions that follow
dependent pattern matching schemes presented by Coquand and studied by McBride
in his PhD thesis. It also accepts many definitions by rewriting, containing
rules which depart from standard pattern matching.Comment: 20 page
Specifying Reusable Components
Reusable software components need expressive specifications. This paper
outlines a rigorous foundation to model-based contracts, a method to equip
classes with strong contracts that support accurate design, implementation, and
formal verification of reusable components. Model-based contracts
conservatively extend the classic Design by Contract with a notion of model,
which underpins the precise definitions of such concepts as abstract
equivalence and specification completeness. Experiments applying model-based
contracts to libraries of data structures suggest that the method enables
accurate specification of practical software
Decidability of Univariate Real Algebra with Predicates for Rational and Integer Powers
We prove decidability of univariate real algebra extended with predicates for
rational and integer powers, i.e., and . Our decision procedure combines computation over real algebraic
cells with the rational root theorem and witness construction via algebraic
number density arguments.Comment: To appear in CADE-25: 25th International Conference on Automated
Deduction, 2015. Proceedings to be published by Springer-Verla
Probability Logic for Harsanyi Type Spaces
Probability logic has contributed to significant developments in belief types
for game-theoretical economics. We present a new probability logic for Harsanyi
Type spaces, show its completeness, and prove both a de-nesting property and a
unique extension theorem. We then prove that multi-agent interactive
epistemology has greater complexity than its single-agent counterpart by
showing that if the probability indices of the belief language are restricted
to a finite set of rationals and there are finitely many propositional letters,
then the canonical space for probabilistic beliefs with one agent is finite
while the canonical one with at least two agents has the cardinality of the
continuum. Finally, we generalize the three notions of definability in
multimodal logics to logics of probabilistic belief and knowledge, namely
implicit definability, reducibility, and explicit definability. We find that
S5-knowledge can be implicitly defined by probabilistic belief but not reduced
to it and hence is not explicitly definable by probabilistic belief
Consequences of a Goedel's misjudgment
The fundamental aim of the paper is to correct an harmful way to interpret a
Goedel's erroneous remark at the Congress of Koenigsberg in 1930. Despite the
Goedel's fault is rather venial, its misreading has produced and continues to
produce dangerous fruits, as to apply the incompleteness Theorems to the full
second-order Arithmetic and to deduce the semantic incompleteness of its
language by these same Theorems. The first three paragraphs are introductory
and serve to define the languages inherently semantic and its properties, to
discuss the consequences of the expression order used in a language and some
question about the semantic completeness: in particular is highlighted the fact
that a non-formal theory may be semantically complete despite using a language
semantically incomplete. Finally, an alternative interpretation of the Goedel's
unfortunate comment is proposed. KEYWORDS: semantic completeness, syntactic
incompleteness, categoricity, arithmetic, second-order languages, paradoxesComment: English version, 19 pages. Fixed and improved terminolog
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