20 research outputs found
ASMs and Operational Algorithmic Completeness of Lambda Calculus
We show that lambda calculus is a computation model which can step by step
simulate any sequential deterministic algorithm for any computable function
over integers or words or any datatype. More formally, given an algorithm above
a family of computable functions (taken as primitive tools, i.e., kind of
oracle functions for the algorithm), for every constant K big enough, each
computation step of the algorithm can be simulated by exactly K successive
reductions in a natural extension of lambda calculus with constants for
functions in the above considered family. The proof is based on a fixed point
technique in lambda calculus and on Gurevich sequential Thesis which allows to
identify sequential deterministic algorithms with Abstract State Machines. This
extends to algorithms for partial computable functions in such a way that
finite computations ending with exceptions are associated to finite reductions
leading to terms with a particular very simple feature.Comment: 37 page
Language, Life, Limits
In the context of second-order polynomial-time computability, we prove that
there is no general function space construction. We proceed to identify
restrictions on the domain or the codomain that do provide a function space
with polynomial-time function evaluation containing all polynomial-time
computable functions of that type.
As side results we show that a polynomial-time counterpart to admissibility
of a representation is not a suitable criterion for natural representations,
and that the Weihrauch degrees embed into the polynomial-time Weihrauch
degrees
Formalizing Turing Machines
Abstract. We discuss the formalization, in the Matita Theorem Prover, of a few, basic results on Turing Machines, up to the existence of a (certified) Universal Machine. The work is meant to be a preliminary step towards the creation of a formal repository in Complexity Theory, and is a small piece in our Reverse Complexity program, aiming to a comfortable, machine independent axiomatization of the field.