Complexity Theory and the Operational Structure of Algebraic Programming Systems

An algebraic programming system is a language built from a fixed algebraic data abstraction and a selection of deterministic, and non-deterministic, assignment and control constructs. First, we give a detailed analysis of the operational structure of an algebraic data type, one which is designed to classify programming systems in terms of the complexity of their implementations. Secondly, we test our operational description by comparing the computations in deterministic and non-deterministic programming systems under certain space and time restrictions

Metric interpretations of infinite trees and semantics of non deterministic recursive programs

AbstractIn order to define semantics of non deterministic recursive programs we are led to consider infinite computations and to replace the structure of cpo on computation domain by the structure of complete metric space. In this setting we prove the two main theorems of semantics: 1.(i) equivalence between operational and denotational semantics, where this last one is defined2.as a greatest fixed point for inclusion,3.(ii) the one-many function computed by a program is the image of the set of trees computed by4.the scheme associated with it

Renormalization and Computation II: Time Cut-off and the Halting Problem

This is the second installment to the project initiated in [Ma3]. In the first Part, I argued that both philosophy and technique of the perturbative renormalization in quantum field theory could be meaningfully transplanted to the theory of computation, and sketched several contexts supporting this view. In this second part, I address some of the issues raised in [Ma3] and provide their development in three contexts: a categorification of the algorithmic computations; time cut--off and Anytime Algorithms; and finally, a Hopf algebra renormalization of the Halting Problem.Comment: 28 page

Matrix Code

Matrix Code gives imperative programming a mathematical semantics and heuristic power comparable in quality to functional and logic programming. A program in Matrix Code is developed incrementally from a specification in pre/post-condition form. The computations of a code matrix are characterized by powers of the matrix when it is interpreted as a transformation in a space of vectors of logical conditions. Correctness of a code matrix is expressed in terms of a fixpoint of the transformation. The abstract machine for Matrix Code is the dual-state machine, which we present as a variant of the classical finite-state machine.Comment: 39 pages, 19 figures; extensions and minor correction