CHURCH

WITHOUT

DOGMA

Wilfried Sieg

Carnegie Mellon

Effective Calculability

Gödel on Turing (1963)

In consequence of later advances, in particular of the fact that due to A.M. Turing’s work a precise and unquestionably adequate definition of the general notion of formal system can now be given, a completely general version of Theorems VI and XI is now possible. (Collected Works I, p. 1955)

Overview

Part 1. Church Canons: Gödel’s perspectives

Part 2. Turing Analysis: Boundedness and locality

Part 3. Axiomatics: Representation theorems

PART 1

Church Canons: Gödel’s perspectives

Doubts (Gödel in 1934)

… at the time of these lectures I was not at

all convinced that my concept of recursion

comprised all possible recursions …

Herbrand’s definition

These axioms will satisfy the following conditions:(i) The defining axioms for fn contain, besides fn, only functions of lesser index.(ii) These axioms contain only constants and free variables.(iii) We must be able to show, by means of intuitionistic proofs, that with these axioms it is possible to compute the value of the functions univocally for each specified system of values of their arguments.

Gödel’s equational calculus

Church’s Identification

Church 1935:

In this paper a definition of recursive function of positive integers which is essentially Gödel's is adopted. And it is maintained that the notion of an effectively calculable function of positive integers should be identified with that of a recursive function, since other plausible definitions of effective calculability turn out to yield notions that are either equivalent to or weaker than recursiveness.

Absoluteness (Gödel 1936)

It can, moreover, be shown that a function computable in one of the systems Si, or even in a system of transfinite order, is computable already in S1. Thus the notion ‘computable’ is in a certain sense ‘absolute’, while almost all metamathematical notions otherwise known (for example, provable, definable, and so on) quite essentially depend upon the system adopted. (Collected Works I, p. 399)

Absoluteness (Gödel 1946)

Tarski has stressed … the great importance of the concept of general recursiveness (or Turing computability). It seems to me that this importance is largely due to the fact that with this concept one has for the first time succeeded in giving an absolute definition of an interesting epistemological notion, i.e., one not depending on the formalism chosen.

(Gödel, Princeton Bicentennial, 1946)

Gödel 193?

That this is really the correct definition of mechanical computability was established beyond any doubt by Turing.

Gödel 193? Ctd.

He [Turing] has shown that the computable functions defined in this way [via the equational calculus] are exactly those for which you can construct a machine with a finite number of parts which will do the following thing. If you write down any number n1, … nr on a slip of paper and put the slip of paper into the machine and turn the crank, then after a finite number of turns the machine will stop and the value of the function for the argument n1, … nr will be printed on the paper. (Collected Works

III, p. 168)

Gödel on Turing (1951)

The most satisfactory way … [of arriving at such a definition] is that of reducing the concept of finite procedure to that of a machine with a finite number of parts, as has been done by the British mathematician Turing. (Collected Works III, pp. 304-5)

Gödel on Turing (1964)

Turing’s work gives an analysis of the concept of ”mechanical procedure” (alias “algorithm” or “computation procedure” or “finite combinatorial procedure”). This concept is shown to be equivalent with that of a “Turing machine”.

PART 2

Turing Analysis: Boundedness and locality

Behmann 1921For the nature of the problem it is of fundamental significance that as auxiliary means … only the completely mechanical reckoning according to a given prescription [Vorschrift] is admitted, i.e., without any thinking in the proper sense of the word. If one wanted to, one could speak of mechanical or machine-like thinking. (Perhaps it can later even be carried out by a machine.)

Turing analysis (1936)

We may now construct a machine to do the work of the computer … The machines just described [string machines] do not differ very essentially from computing machines as defined in section 2 [letter machines] and corresponding to any machine of this type a computing machine can be constructed to compute the same sequence, that is to say the sequence computed by the computer.

Boundedness & Locality

(B) There is a finite bound on the number of configurations a computor can immediately recognize.

(L) A computor can change only immediately recognizable (sub-) configurations.

Gödel on the psychology …

Gödel asserts in the 1946 Princeton lecture that certain aspects of the concept of definability

“would involve some extramathe-matical element concerning the psychology of the being who deals with mathematics”.

Limited result

If computors satisfy (B) and (L), but also operate on strings, then string machines codify their computational behavior and letter machines can provably carry out their calculations.

Methodological dilemma ?

Calculability of number-theoretic functions

Calculability by computor satisfy-ing boundedness and locality conditions

Computability by string machine

Computability by letter machine

Turing’s Thesis Equivalence proof

1 2

Turing 1954

This statement is still somewhat lacking in definiteness, and will remain so. … The statement is moreover one which one does not attempt to prove. Propaganda is more appropriate to it than proof, for its status is something between a theorem and a definition. In so far as we know a priori what is a puzzle and what is not, the statement is a theorem. In so far as we do not know what puzzles are, the statement is a definition which tells us something about what they are.

PART 3

Axiomatics: representation theorems

Gödel 1934

Gödel viewed Church’s proposal as “thoroughly unsatisfactory”and made a counterproposal, namely,

“to state a set of axioms which would embody the generally accepted properties of this notion [i.e., effective calculability], and to do something on that basis”.

General features• Computors operate on finite configurations

• They recognize immediately only a bounded number of different patterns in these configurations

• They operate locally on one such pattern at a time

• They assemble from the original configuration and the result of the local operation the next configuration

Discrete dynamical systems

Hereditarily finite sets

Structurality

Picture

Turing computor

Definition: M = <S; T, G> is a Turing Computor on S, where S is a structural class, T a finite set of patterns, and G a structural operation on T, if and only if, for every xS there is a zS, such that(L.0) ( ! y) yCn(x)(L.1) ( ! v Dr(z,x)) vxG(cn(x));(A.1) z = (x\Cn(x)) Dr(z,x).

Facts I

• Any Turing machine is a Turing computor, i.e. satisfies the axioms.

• Any Turing computor is reducible to a Turing machine.

Addition to picture

Assembling

Facts II

• Any Turing machine is a Gandy machine, i.e. satisfies the axioms.

• Any Gandy machine is reducible to a Turing machine.

CONCLUDING REMARKS

So what?

Church on Turing (1937)

… As a matter of convenience, certain further restrictions are imposed on the character of the machine, but these are of such a nature as obviously to cause no loss of generality - in particular, a human calculator … can be regarded as a kind of Turing machine.

Gödel on Turing (1964)

Turing’s work gives an analysis of the concept of ”mechanical procedure” (alias “algorithm” or “computation procedure” or “finite combinatorial procedure”). This concept is shown to be equivalent with that of a “Turing machine”.

References

All the classical papers mentioned in the talk.Sieg (2006): Gödel on computability; Philosophia Mathematica.

Sieg (to appear): Church without Dogma - axioms for computability.

Gandy machineDefinition: M = <S; T1, G1, T2, G2> is a Gandy Machine on S, where S is a structural class, Ti a finite set of stereotypes, Gi a structural operation on Ti, if and only if, for every xS there is a zS, such thatL.1: (yCn1(x))(!vDr1(z,x))vxG1(y)

L.2: (yCn2(x))(vDr2(z,x)) vxG2(y)

A.1: ( C) [C Dr1(z,x)) & {Sup(v)A(z,x)| vC}

( wDr2(z,x)) ( vC) v<*w ];A.2: z = Dr1(z,x).