A Formalization of the Church-Turing Thesis

Church's thesis, "consistency", "formalization", "proof theory" : dictionary entries

Ray tracing - computing the incomputable?

We recall from previous work a model-independent framework of computational complexity theory. Notably for the present paper, the framework allows formalization of the issues of precision that present themselves when one considers physical, error-prone (especially analogue rather than digital) computational systems. We take as a case study the ray-tracing problem, a Turing-machineincomputable p...

A Formalization of the Church-Turing Thesis for State-Transition Models

Our goal is to formalize the Church-Turing Thesis for a very large class of computational models. Specifically, the notion of an “effective model of computation” over an arbitrary countable domain is axiomatized. This is accomplished by modifying Gurevich’s “Abstract State Machine” postulates for statetransition systems. A proof is provided that all models satisfying our axioms, regardless of u...

Reduction Relations

The goal of the article is to start the formalization of Knuth-Bendix completion method (see [2], [10] or [1]; see also [11],[9]), i.e. to formalize the concept of the completion of a reduction relation. The completion of a reduction relation R is a complete reduction relation equivalent to R such that convertible elements have the same normal forms. The theory formalized in the article include...

Hypercomputation and the Physical Church-Turing Thesis

A version of the Church-Turing Thesis states that every effectively realizable physical system can be defined by Turing Machines (‘Thesis P’); in this formulation the Thesis appears an empirical, more than a logico-mathematical, proposition. We review the main approaches to computation beyond Turing definability (‘hypercomputation’): supertask, non-well-founded, analog, quantum, and retrocausal...