Trial and error predicates and the solution to a problem of Mostowski

1965 ◽  
Vol 30 (1) ◽  
pp. 49-57 ◽  
Author(s):  
Hilary Putnam
Keyword(s):  
Finite Number ◽  
Correct Answer ◽  
Decision Procedure ◽  
Finite Sequence ◽  
Decision Procedures ◽  
Turing Machines ◽  
Trial And Error ◽  
Church’S Thesis ◽  
Church's Thesis ◽  
Empirical Means

The purpose of this paper is to present two groups of results which have turned out to have a surprisingly close interconnection. The first two results (Theorems 1 and 2) were inspired by the following question: we know what sets are “decidable” — namely, the recursive sets (according to Church's Thesis). But what happens if we modify the notion of a decision procedure by (1) allowing the procedure to “change its mind” any finite number of times (in terms of Turing Machines: we visualize the machine as being given an integer (or an n-tuple of integers) as input. The machine then “prints out” a finite sequence of “yesses” and “nos”. The last “yes” or “no” is always to be the correct answer.); and (2) we give up the requirement that it be possible to tell (effectively) if the computation has terminated? I.e., if the machine has most recently printed “yes”, then we know that the integer put in as input must be in the set unless the machine is going to change its mind; but we have no procedure for telling whether the machine will change its mind or not.The sets for which there exist decision procedures in this widened sense are decidable by “empirical” means — for, if we always “posit” that the most recently generated answer is correct, we will make a finite number of mistakes, but we will eventually get the correct answer. (Note, however, that even if we have gotten to the correct answer (the end of the finite sequence) we are never sure that we have the correct answer.)

Turing machines

2018 ◽  
Author(s):  
Guglielmo Tamburrini
Keyword(s):  
Finite Number ◽  
Turing Machine ◽  
Finite Alphabet ◽  
Turing Machines ◽  
Church’S Thesis ◽  
Church's Thesis ◽  
Internal Configuration ◽  
Class Of Functions ◽  
Alan Mathison Turing ◽  
Algorithmic Procedure

Turing machines are abstract computing devices, named after Alan Mathison Turing. A Turing machine operates on a potentially infinite tape uniformly divided into squares, and is capable of entering only a finite number of distinct internal configurations. Each square may contain a symbol from a finite alphabet. The machine can scan one square at a time and perform, depending on the content of the scanned square and its own internal configuration, one of the following operations: print or erase a symbol on the scanned square or move on to scan either one of the immediately adjacent squares. These elementary operations are possibly accompanied by a change of internal configuration. Turing argued that the class of functions calculable by means of an algorithmic procedure (a mechanical, stepwise, deterministic procedure) is to be identified with the class of functions computable by Turing machines. The epistemological significance of Turing machines and related mathematical results hinges upon this identification, which later became known as Turing’s thesis; an equivalent claim, Church’s thesis, had been advanced independently by Alonzo Church. Most crucially, mathematical results stating that certain functions cannot be computed by any Turing machine are interpreted, by Turing’s thesis, as establishing absolute limitations of computing agents.

The unsolvability of the uniform halting problem for two state Turing machines

1969 ◽  
Vol 34 (2) ◽  
pp. 161-165 ◽  
Author(s):  
Gabor T. Herman
Keyword(s):  
Finite Number ◽  
Turing Machine ◽  
Decision Procedure ◽  
Turing Machines ◽  
Halting Problem

The uniform halting problem (UH) can be stated as follows:Give a decision procedure which for any given Turing machine (TM) will decide whether or not it has an immortal instantaneous description (ID).An ID is called immortal if it has no terminal successor. As it is generally the case in the literature (see e.g. Minsky [4, p. 118]) we assume that in an ID the tape must be blank except for some finite number of squares. If we remove this restriction the UH becomes the immortality problem (IP).

Randomised allocation of treatments in sequential trials

1980 ◽  
Vol 12 (01) ◽  
pp. 174-182 ◽  
Author(s):  
John Bather
Keyword(s):  
Finite Number ◽  
Finite Sequence ◽  
Decision Procedures ◽  
Bandit Problem ◽  
Expected Number ◽  
Long Run ◽  
Sequential Trials

Given a finite number of different experiments with unknown probabilities p 1, p 2, ···, p k of success, the multi-armed bandit problem is concerned with maximising the expected number of successes in a sequence of trials. There are many policies which ensure that the proportion of successes converges to p = max (p 1, p 2, ···, p k ), in the long run. This property is established for a class of decision procedures which rely on randomisation, at each stage, in selecting the experiment for the next trial. Further, it is suggested that some of these procedures might perform well over any finite sequence of trials.

Randomised allocation of treatments in sequential trials

1980 ◽  
Vol 12 (1) ◽  
pp. 174-182 ◽  
Author(s):  
John Bather
Keyword(s):  
Finite Number ◽  
Finite Sequence ◽  
Decision Procedures ◽  
Bandit Problem ◽  
Expected Number ◽  
Long Run ◽  
Sequential Trials

Given a finite number of different experiments with unknown probabilities p1, p2, ···, pk of success, the multi-armed bandit problem is concerned with maximising the expected number of successes in a sequence of trials. There are many policies which ensure that the proportion of successes converges to p = max (p1, p2, ···, pk), in the long run. This property is established for a class of decision procedures which rely on randomisation, at each stage, in selecting the experiment for the next trial. Further, it is suggested that some of these procedures might perform well over any finite sequence of trials.

scholarly journals Computability and Recursion

1996 ◽  
Vol 2 (3) ◽  
pp. 284-321 ◽  
Author(s):  
Robert I. Soare
Keyword(s):  
Recursive Function ◽  
Conceptual Analysis ◽  
Function Theory ◽  
Computability Theory ◽  
Turing Machines ◽  
Church’S Thesis ◽  
The Future ◽  
Church's Thesis ◽  
The Subject

AbstractWe consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness”. We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas.After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of “computability” and “recursion.” Specifically we recommend that: the term “recursive” should no longer carry the additional meaning of “computable” or “decidable;” functions defined using Turing machines, register machines, or their variants should be called “computable” rather than “recursive;” we should distinguish the intensional difference between Church's Thesis and Turing's Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be “Computability Theory” or simply Computability rather than “Recursive Function Theory.”

Deciding Clause Classes by Semantic Clash Resolution

1993 ◽  
Vol 18 (2-4) ◽  
pp. 163-182
Author(s):  
Alexander Leitsch
Keyword(s):  
Decision Procedure ◽  
Decision Procedures ◽  
General Concept ◽  
Complexity Measure ◽  
Syntactical Structure

It is investigated, how semantic clash resolution can be used to decide some classes of clause sets. Because semantic clash resolution is complete, the termination of the resolution procedure on a class Γ gives a decision procedure for Γ. Besides generalizing earlier results we investigate the relation between termination and clause complexity. For this purpose we define the general concept of atom complexity measure and show some general results about termination in terms of such measures. Moreover, rather than using fixed resolution refinements we define an algorithmic generator for decision procedures, which constructs appropriate semantic refinements out of the syntactical structure of the clause sets. This method is applied to the Bernays – Schönfinkel class, where it gives an efficient (resolution) decision procedure.

Proving Church's Thesis

2000 ◽  
Vol 8 (3) ◽  
pp. 244-258 ◽  
Author(s):  
ROBERT BLACK
Keyword(s):  
Church’S Thesis ◽  
Church's Thesis

scholarly journals Reflections on Church's thesis.

1987 ◽  
Vol 28 (4) ◽  
pp. 490-498 ◽  
Author(s):  
Stephen C. Kleene
Keyword(s):  
Church’S Thesis ◽  
Church's Thesis
Sign in / Sign up

Export Citation Format

Share Document