1,499 research outputs found
A Bi-Directional Refinement Algorithm for the Calculus of (Co)Inductive Constructions
The paper describes the refinement algorithm for the Calculus of
(Co)Inductive Constructions (CIC) implemented in the interactive theorem prover
Matita. The refinement algorithm is in charge of giving a meaning to the terms,
types and proof terms directly written by the user or generated by using
tactics, decision procedures or general automation. The terms are written in an
"external syntax" meant to be user friendly that allows omission of
information, untyped binders and a certain liberal use of user defined
sub-typing. The refiner modifies the terms to obtain related well typed terms
in the internal syntax understood by the kernel of the ITP. In particular, it
acts as a type inference algorithm when all the binders are untyped. The
proposed algorithm is bi-directional: given a term in external syntax and a
type expected for the term, it propagates as much typing information as
possible towards the leaves of the term. Traditional mono-directional
algorithms, instead, proceed in a bottom-up way by inferring the type of a
sub-term and comparing (unifying) it with the type expected by its context only
at the end. We propose some novel bi-directional rules for CIC that are
particularly effective. Among the benefits of bi-directionality we have better
error message reporting and better inference of dependent types. Moreover,
thanks to bi-directionality, the coercion system for sub-typing is more
effective and type inference generates simpler unification problems that are
more likely to be solved by the inherently incomplete higher order unification
algorithms implemented. Finally we introduce in the external syntax the notion
of vector of placeholders that enables to omit at once an arbitrary number of
arguments. Vectors of placeholders allow a trivial implementation of implicit
arguments and greatly simplify the implementation of primitive and simple
tactics
Light Logics and the Call-by-Value Lambda Calculus
The so-called light logics have been introduced as logical systems enjoying
quite remarkable normalization properties. Designing a type assignment system
for pure lambda calculus from these logics, however, is problematic. In this
paper we show that shifting from usual call-by-name to call-by-value lambda
calculus allows regaining strong connections with the underlying logic. This
will be done in the context of Elementary Affine Logic (EAL), designing a type
system in natural deduction style assigning EAL formulae to lambda terms.Comment: 28 page
Explicit Substitutions for Contextual Type Theory
In this paper, we present an explicit substitution calculus which
distinguishes between ordinary bound variables and meta-variables. Its typing
discipline is derived from contextual modal type theory. We first present a
dependently typed lambda calculus with explicit substitutions for ordinary
variables and explicit meta-substitutions for meta-variables. We then present a
weak head normalization procedure which performs both substitutions lazily and
in a single pass thereby combining substitution walks for the two different
classes of variables. Finally, we describe a bidirectional type checking
algorithm which uses weak head normalization and prove soundness.Comment: In Proceedings LFMTP 2010, arXiv:1009.218
A Lambda Term Representation Inspired by Linear Ordered Logic
We introduce a new nameless representation of lambda terms inspired by
ordered logic. At a lambda abstraction, number and relative position of all
occurrences of the bound variable are stored, and application carries the
additional information where to cut the variable context into function and
argument part. This way, complete information about free variable occurrence is
available at each subterm without requiring a traversal, and environments can
be kept exact such that they only assign values to variables that actually
occur in the associated term. Our approach avoids space leaks in interpreters
that build function closures.
In this article, we prove correctness of the new representation and present
an experimental evaluation of its performance in a proof checker for the
Edinburgh Logical Framework.
Keywords: representation of binders, explicit substitutions, ordered
contexts, space leaks, Logical Framework.Comment: In Proceedings LFMTP 2011, arXiv:1110.668
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
Nominal Unification from a Higher-Order Perspective
Nominal Logic is a version of first-order logic with equality, name-binding,
renaming via name-swapping and freshness of names. Contrarily to higher-order
logic, bindable names, called atoms, and instantiable variables are considered
as distinct entities. Moreover, atoms are capturable by instantiations,
breaking a fundamental principle of lambda-calculus. Despite these differences,
nominal unification can be seen from a higher-order perspective. From this
view, we show that nominal unification can be reduced to a particular fragment
of higher-order unification problems: Higher-Order Pattern Unification. This
reduction proves that nominal unification can be decided in quadratic
deterministic time, using the linear algorithm for Higher-Order Pattern
Unification. We also prove that the translation preserves most generality of
unifiers
Deduction modulo theory
This paper is a survey on Deduction modulo theor
Analytical learning and term-rewriting systems
Analytical learning is a set of machine learning techniques for revising the representation of a theory based on a small set of examples of that theory. When the representation of the theory is correct and complete but perhaps inefficient, an important objective of such analysis is to improve the computational efficiency of the representation. Several algorithms with this purpose have been suggested, most of which are closely tied to a first order logical language and are variants of goal regression, such as the familiar explanation based generalization (EBG) procedure. But because predicate calculus is a poor representation for some domains, these learning algorithms are extended to apply to other computational models. It is shown that the goal regression technique applies to a large family of programming languages, all based on a kind of term rewriting system. Included in this family are three language families of importance to artificial intelligence: logic programming, such as Prolog; lambda calculus, such as LISP; and combinatorial based languages, such as FP. A new analytical learning algorithm, AL-2, is exhibited that learns from success but is otherwise quite different from EBG. These results suggest that term rewriting systems are a good framework for analytical learning research in general, and that further research should be directed toward developing new techniques
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