Comparing (8) and (9), the

recursive function can be obtained as

Remark: This result can also be obtained as a consequence of a theorem of Ehrenfeucht and Mycielski which states that, if T + [logical not][alpha] is undecidable, then there is no

recursive function f such that [W.sub.T]([phi]) [less than or equal to] f([W.sub.t+[alpha]]([phi])) holds for all theorems [phi] of T; here, [W.sub.S] is a measure for the complexity of the shortest proof of formula in the theory S.

The

recursive function r used in R finds a path where it is possible to move to up, down, left and right from the current position (x,y), moves to that position, and finds the next allowable path.

First, we need to update the probability distribution of the knee point as well as the residual unsorted list as two parameters for the

recursive function. According to Lemma 3, the update of the probability distribution of the knee point yields

And Kurt GE[micro]del worked on the incompleteness theory and

recursive function theory.

With the EP data model that is semantically equivalent to a class of total

recursive function, a monolithic architecture becomes available again for database applications.

In Section V of his 1981 paper, Tait argues that every primitive

recursive function f : A [right arrow] B is finitist in the sense that the finitist can accept it as a construction of a B from an arbitrary A.

Notice that the

recursive function m is called only at two places: one by function map with C as the return continuation, one inside K with Q as the return continuation.

The hash function will be decomposed into a truly

recursive function followed by a nearly trivial "address" function that maps the result of the recursive part into the appropriate address format.

FIGURE 7 The Fibonacci numbers A 1 1 2 1 3 2 4 3 5 5 6 8 7 13 8 21 9 34 10 55 11 89 12 144 13 233 14 377 15 610 16 987 17 1597 18 2584 19 4181 20 6765 21 10946 22 17711 23 28657 24 46368 25 75025 26 121393 27 196418 28 317811 29 514229 30 832040 31 1646269 FIGURE 8 Patterns with the Fibonacci numbers A B C 1 1 2 10 2 1 2 10 3 2 4 20 4 3 6 30 5 5 10 50 6 8 16 80 7 13 26 130 8 21 42 210 9 34 68 340 10 55 110 550 (Note that the students were developing an informal understanding of

recursive functions. A formal definition of a

recursive function is beyond this level.

Furthermore, if we conceive of Church's Thesis as asserting that a function is 'intuitively' computable if and only if it is a partial

recursive function (and this is surely a common conception of Church's Thesis), then the presupposition in Young [1977] amounts to no more than the application of the if direction of Church's Thesis to the resource bounded computations of complexity theory.