CHURCH WITHOUT DOGMA Wilfried Sieg Carnegie Mellon

Effective Calculability

G ö del 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 (G ö del 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 f n contain, besides f n, 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.

G ö del ’ 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 (G ö del 1936) It can, moreover, be shown that a function computable in one of the systems S i, or even in a system of transfinite order, is computable already in S 1. 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 (G ö del 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)

G ö del 193? That this is really the correct definition of mechanical computability was established beyond any doubt by Turing.

G ö del 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 n 1, … n r 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 n 1, … n r will be printed on the paper. ( Collected Works III, p. 168)

G ö del 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 )

G ö del 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 1921 For 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.

G ö del 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 ?

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

G ö del 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 = 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 x  S there is a z  S, such that (L.0) (  ! y) y  Cn(x) (L.1) (  ! v  Dr(z,x)) v  x G(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.

G ö del 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 machine Definition: M = is a Gandy Machine on S, where S is a structural class, T i a finite set of stereotypes, G i a structural operation on T i, if and only if, for every x  S there is a z  S, such that L.1: (  y  Cn 1 (x))(  !v  Dr 1 (z,x))v  x G 1 (y) L.2: (  y  Cn 2 (x))(  v  Dr 2 (z,x)) v  x G 2 (y) A.1: (  C) [C  Dr 1 (z,x)) &  {Sup(v)  A(z,x)| v  C}    (  w  Dr 2 (z,x)) (  v  C) v<*w ]; A.2: z =  Dr 1 (z,x).