148 research outputs found
The complexity of completions in partial combinatory algebra
We discuss the complexity of completions of partial combinatory algebras, in
particular of Kleene's first model. Various completions of this model exist in
the literature, but all of them have high complexity. We show that although
there do not exist computable completions, there exists completions of low
Turing degree. We use this construction to relate completions of Kleene's first
model to complete extensions of PA. We also discuss the complexity of pcas
defined from nonstandard models of PA
A Structural Approach to Reversible Computation
Reversibility is a key issue in the interface between computation and
physics, and of growing importance as miniaturization progresses towards its
physical limits. Most foundational work on reversible computing to date has
focussed on simulations of low-level machine models. By contrast, we develop a
more structural approach. We show how high-level functional programs can be
mapped compositionally (i.e. in a syntax-directed fashion) into a simple kind
of automata which are immediately seen to be reversible. The size of the
automaton is linear in the size of the functional term. In mathematical terms,
we are building a concrete model of functional computation. This construction
stems directly from ideas arising in Geometry of Interaction and Linear
Logic---but can be understood without any knowledge of these topics. In fact,
it serves as an excellent introduction to them. At the same time, an
interesting logical delineation between reversible and irreversible forms of
computation emerges from our analysis.Comment: 30 pages, appeared in Theoretical Computer Scienc
Categorical combinators
Our main aim is to present the connection between Ī»-calculus and Cartesian closed categories both in an untyped and purely syntactic setting. More specifically we establish a syntactic equivalence theorem between what we call categorical combinatory logic and Ī»-calculus with explicit products and projections, with Ī² and Ī·-rules as well as with surjective pairing. āCombinatory logicā is of course inspired by Curry's combinatory logic, based on the well-known S, K, I. Our combinatory logic is ācategoricalā because its combinators and rules are obtained by extracting untyped information from Cartesian closed categories (looking at arrows only, thus forgetting about objects). Compiling Ī»-calculus into these combinators happens to be natural and provokes only n log n code expansion. Moreover categorical combinatory logic is entirely faithful to Ī²-reduction where combinatory logic needs additional rather complex and unnatural axioms to be. The connection easily extends to the corresponding typed calculi, where typed categorical combinatory logic is a free Cartesian closed category where the notion of terminal object is replaced by the explicit manipulation of applying (a function to its argument) and coupling (arguments to build datas in products). Our syntactic equivalences induce equivalences at the model level. The paper is intended as a mathematical foundation for developing implementations of functional programming languages based on a ācategorical abstract machine,ā as developed in a companion paper (Cousineau, Curien, and Mauny, in āProceedings, ACM Conf. on Functional Programming Languages and Computer Architecture,ā Nancy, 1985)
A Semantic Approach to Illative Combinatory Logic
This work introduces the theory of illative combinatory algebras,
which is closely related to systems of illative combinatory logic. We
thus provide a semantic interpretation for a formal framework in which
both logic and computation may be expressed in a unified
manner. Systems of illative combinatory logic consist of combinatory
logic extended with constants and rules of inference intended to
capture logical notions. Our theory does not correspond strictly to
any traditional system, but draws inspiration from many. It differs
from them in that it couples the notion of truth with the notion of
equality between terms, which enables the use of logical formulas in
conditional expressions. We give a consistency proof for first-order
illative combinatory algebras. A complete embedding of classical
predicate logic into our theory is also provided. The translation is
very direct and natural
The Internal Operads of Combinatory Algebras
We argue that operads provide a general framework for dealing with
polynomials and combinatory completeness of combinatory algebras, including the
classical -algebras, linear -algebras, planar
-algebras as well as the braided -algebras. We show that every extensional combinatory algebra gives rise to
a canonical closed operad, which we shall call the internal operad of the
combinatory algebra. The internal operad construction gives a left adjoint to
the forgetful functor from closed operads to extensional combinatory algebras.
As a by-product, we derive extensionality axioms for the classes of combinatory
algebras mentioned above
On a conjecture of Bergstra and Tucker
AbstractBergstra and Tucker (1983) conjectured that a semicomputable (abstract) data type has a finite hidden enrichment specification under its initial algebra semantics. In a previous paper (1987) we tried to solve the entire conjecture and we found a weak solution. Here, following the line and the proof techniques of the previous paper, we examine a nontrivial case in which the conjecture has a positive answer
Term rewriting systems from Church-Rosser to Knuth-Bendix and beyond
Term rewriting systems are important for computability theory of abstract data types, for automatic theorem proving, and for the foundations of functional programming. In this short survey we present, starting from first principles, several of the basic notions and facts in the area of term rewriting. Our treatment, which often will be informal, covers abstract rewriting, Combinatory Logic, orthogonal systems, strategies, critical pair completion, and some extended rewriting formats
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