Related papers: The Recursion Theorem from a Different Angle
Since many real-world problems arising in the fields of compiler optimisation, automated software engineering, formal proof systems, and so forth are equivalent to the Halting Problem--the most notorious undecidable problem--there is a…
We position Turing's result regarding the undecidability of the halting problem as a result about programs rather than machines. The mere requirement that a program of a certain kind must solve the halting problem for all programs of that…
We introduce a set of eight universal Rules of Inference by which computer programs with known properties (axioms) are transformed into new programs with known properties (theorems). Axioms are presented to formalize a segment of Number…
The halting problem is considered to be an essential part of the theoretical background to computing. That halting is not in general computable has supposedly been proved in many text books and taught on many computer science courses, in…
There are numbers k and s and a URM program A(n,m) satisfying the following conditions. 1. If A(n,m) halts, then Cn(m) diverges. 2. For all n, C_k(n) = A(n,n) and C_s(n) = C_k(s). 3. A(k,s) halts and for all n, A(s,n) diverges. Here C_n(_)…
Typical arguments for results like Kleene's Second Recursion Theorem and the existence of self-writing computer programs bear the fingerprints of equational reasoning and combinatory logic. In fact, the connection of combinatory logic and…
A devil's advocate is one who argues against a claim, not as a committed opponent but in order to determine the validity of the claim. We are interested in a devil's advocate that argues against termination of a program. He does so by…
This paper discusses limitations of reflexive and diagonal arguments as methods of proof of limitative theorems (e.g. G\"odel's theorem on Entscheidungsproblem, Turing's halting problem or Chaitin-G\"odel's theorem). The fact, that a formal…
The Halting problem of a quantum computer is considered. It is shown that if halting of a quantum computer takes place the associated dynamics is described by an irreversible operator.
Although the halting problem is undecidable, imperfect testers that fail on some instances are possible. Such instances are called hard for the tester. One variant of imperfect testers replies "I don't know" on hard instances, another…
Disjunctive finitary programs are a class of logic programs admitting function symbols and hence infinite domains. They have very good computational properties, for example ground queries are decidable while in the general case the stable…
We show how to reverse a while language extended with blocks, local variables, procedures and the interleaving parallel composition. Annotation is defined along with a set of operational semantics capable of storing necessary reversal…
Many programs allow the user to input data several times during its execution. If the program runs forever the user may input data infinitely often. A program terminates if it terminates no matter what the user does. We discuss various ways…
Most modern (classical) programming languages support recursion. Recursion has also been successfully applied to the design of several quantum algorithms and introduced in a couple of quantum programming languages. So, it can be expected…
Every function over the natural numbers has an infinite subdomain on which the function is non-decreasing. Motivated by a question of Dzhafarov and Schweber, we study the reverse mathematics of variants of this statement. It turns out that…
We introduce a method of reversing the execution of imperative concurrent programs. Given an irreversible program, we describe the process of producing two versions. The first performs forward execution and saves information necessary for…
We present a new approach to proving non-termination of non-deterministic integer programs. Our technique is rather simple but efficient. It relies on a purely syntactic reversal of the program's transition system followed by a…
By the sometimes so-called 'Main Theorem' of Recursive Analysis, every computable real function is necessarily continuous. We wonder whether and which kinds of HYPERcomputation allow for the effective evaluation of also discontinuous…
In this paper we use the Recursion Theorem to show the existence of various infinite sequences and sets. Our main result is that there is an increasing sequence e_0, e_1, e_2 .. such that W_{e_n}={e_{n+1}} for every n. Similarly, we prove…
This paper proposes a modal typing system that enables us to handle self-referential formulae, including ones with negative self-references, which on one hand, would introduce a logical contradiction, namely Russell's paradox, in the…