MA5219 LOGIC AND FOUNDATION OF MATHEMATICS I. Lecture 1: Programs. Tin Lok Wong 13 August, 2018

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1 MA5219 LOGIC AND FOUNDATION OF MATHEMATICS I Lecture 1: Programs Tin Lok Wong 13 August, 2018 The aim of this lecture is to characterize S N : some algorithm can tell whether n N belongs to S}. By an algorithm we mean here an idealized algorithm with no memory or time restriction. Convention 1.1. The set of all natural numbers is N = 0, 1, 2,... }. Arithmetic means arithmetic on N. Unless otherwise stated, the domain of quantification is N. If the length of a tuple x 1, x 2,..., x n is understood or unimportant, then we often abbreviate the tuple as x. Let us start by considering some simple decision algorithms on N. These algorithms are specified in a programming language that we call L A (exp). Definition. The notion of an L A (exp) term is defined by recursion as follows. Every variable is an L A (exp) term. The symbols 0 and 1 are L A (exp) terms. If t, s are L A (exp) terms, then (t + s), (t s) and (2 t ) are L A (exp) terms. Every L A (exp) term is obtained by applying the rules above finitely many times. To make the notation lighter, we may omit some brackets by representing by juxtaposition, e.g., by writing x y as xy; and adopting the usual precedence rules for arithmetic operations, e.g., by writing (x+(y z))+x as x + yz + x. If x 1, x 2,..., x k is a list of variables with no repetition, and all the variables in an L A (exp) term t appear in this list, then we may write t as t(x 1, x 2,..., x k ). Example 1.2. If w, x, z are variables, then 2 x+1 z + 2 x + w is an L A (exp) term. Definition. An L A (exp) term t(x 1, x 2,..., x k ) can be evaluated on inputs a 1, a 2,..., a k N to give a value t(a 1, a 2,..., a k ) N. This evaluation is defined by recursion on (the number of steps in the construction of) the term t as follows: whenever t(x 1, x 2,..., x k ), s(x 1, x 2,..., x k ) are L A (exp) terms and a 1, a 2,..., a k N, if t(x 1, x 2,..., x k ) = x i, then t(a 1, a 2,..., a k ) = a i ; if t(x 1, x 2,..., x k ) = 0, then t(a 1, a 2,..., a k ) = 0; if t(x 1, x 2,..., x k ) = 1, then t(a 1, a 2,..., a k ) = 1; (t + s)(a 1, a 2,..., a k ) = t(a 1, a 2,..., a k ) + s(a 1, a 2,..., a k ); (t s)(a 1, a 2,..., a k ) = t(a 1, a 2,..., a k ) s(a 1, a 2,..., a k ); (2 t )(a 1, a 2,..., a k ) = 2 t(a1,a2,...,a k). 1

2 Example 1.3. If t(w, x, z) = 2 x+1 z + 2 x + w, then t(0, 1, 2) = = 10. Definition. Each (decision) program has finitely many input variables. We sometimes indicate the input variables x 1, x 2,..., x k of a program θ in the notation by writing θ(x 1, x 2,..., x k ). When using this notation, we implicitly assume that the variables x 1, x 2,..., x k are mutually distinct. Given inputs a 1, a 2,..., a k N, an L A (exp) program θ(x 1, x 2,..., x k ) may or may not return an answer, but if it does, then the answer must be either true or false. The output of an L A (exp) program θ(x 1, x 2,..., x k ) on inputs a 1, a 2,..., a k N is defined to be true, if θ returns true on inputs a 1, a 2,..., a k ; [θ(a 1, a 2,..., a k )] = false, if θ returns false on inputs a 1, a 2,..., a k ; The notion of an L A (exp) program is defined by recursion as follows. Let x be variables. The L A (exp) program ( x) is return true For all inputs ā N, we have [ (ā)] = true. Let t( x), s( x) be L A (exp) terms. The L A (exp) program (t = s)( x) is if t( x) = s( x) [(t = s)(ā)] = true, false, if t(ā) = s(ā); Let t( x), s( x) be L A (exp) terms. The L A (exp) program (t < s)( x) is if t( x) < s( x) [(t < s)(ā)] = true, false, if t(ā) < s(ā); If θ( x) is an L A (exp) program, then so is θ( x), which is defined to be if [θ( x)] = true then return false else return true true, [ θ(ā)] = false, UNDEF, if [[θ(ā)] = false; if [[θ(ā)] = true; If θ( x, z), η(ȳ, z) are L A (exp) programs, where x} ȳ} =. then so is (θ η)( x, ȳ, z), which is defined to be 2

3 if [θ( x, z)] = true par-or [η(ȳ, z)] = true For inputs ā, b, c N of appropriate lengths, true, if [[θ(ā, c)] = true or [[η( b, c)] = true; [(θ η)(ā, b, c)] = false, if [[θ(ā, c)] = [η( b, c)] = false; Let t( x) be an L A (exp) term. If θ( x, y) is an L A (exp) program, then so is ( y<t θ)( x), which is defined to be par-for y 0, 1,..., t( x) 1 do if [θ( x, y)] = true return false true, [( y<t θ)(ā)] = false, UNDEF, if [[θ(ā, b)] = true for some b < t(ā); if [[θ(ā, b)] = false for all b < t(ā); If θ( x, y) be an L A (exp) program, then so is ( y θ)( x), which is defined to be initialize y 0 par-while [θ( x, y)] = false do y y + 1 end par-while return true [( y θ)(ā)] = true, if [[θ(ā, b)] = true for some b N; Every L A (exp) program is obtained by applying the construction rules above finitely many times. We also introduce the following shorthand. Let t, s be L A (exp) terms. t s stands for t < s + 1. t s stands for (t = s). Let θ, η be L A (exp) programs. =. θ η = ( θ η). θ η = θ η. θ η = (θ η) (η θ). Let θ( x, y) be an L A (exp) program and t( x) be an L A (exp) term. 3

4 y<t θ = y<t θ. y θ = y θ. Here,, are more binding than,, which are in turn more binding than,. In this module, since we will not study the run-times of programs, we could specify a program entirely using its input output pairs. Nevertheless, a program written in terms of if, then, parfor, par-while,... is usually easier to understand, especially when unravelled informally. Example 1.4. Let θ(x, y) be the L A (exp) program z<y w<2 x (y = 2 x+1 z + 2 x + w). Unravelling the definitions, one sees that θ(x, y) roughly corresponds to par-for z 0, 1,..., y 1 par-for w 0, 1,..., 2 x 1 if y = 2 x+1 z + 2 x + w return false For instance, notice 10 = On the one hand, by considering (z, w) = (2, 0), we see that [θ(1, 10)] = true. On the other hand, as one can directly verify, we have [θ(2, 10)] = false. More generally, for all inputs a, b N, true, if the ath digit in the binary representation of b is 1; [θ(a, b)] = false, if the ath digit in the binary representation of b is 0. As in the example above, many programs do not need par-while. These programs are particularly well-behaved for instance, they always return an answer. The availability of par-while makes the language strictly stronger, but it seems difficult to find a simple example demonstrating this. Definition. A 0 (exp) program is an L A (exp) program with no par-while. A Σ 1 program is an L A (exp) program of the form ȳ θ( x, ȳ), where θ( x, ȳ) is a 0 (exp) program and ȳ is a possibly empty tuple of variables. Although Σ 1 programs are only a very special kind of computer algorithms, they are extremely powerful. In fact, it is postulated that, as far as subsets of N are concerned, the intuitive notion of algorithms can be captured by Σ 1 programs. This postulate is commonly accepted, but it cannot be proved or refuted mathematically because it involves the intuitive notion of algorithms which by nature cannot have a rigorous definition. Church Turing Thesis. For any k N and S N k, the following are equivalent. There is an algorithm A such that S = (a 1, a 2,..., a k ) N k : A on input ā returns true}. There is a Σ 1 program θ(x 1, x 2,..., x k ) such that S = (a 1, a 2,..., a k ) N k : [θ(ā)] = true}. At various points, we will appeal to the Church Turing Thesis to obtain Σ 1 programs that replace the algorithms we informally describe. Challenge. Find a counterexample to the Church Turing Thesis with reasonable justification. 4

5 Definition. Let k N. A set S N k is recursively enumerable, or r.e. for short, if there is a Σ 1 program θ(x 1, x 2,..., x k ) such that S = (a 1, a 2,..., a k ) N k : [θ(ā)] = true}. The set S N k is recursive if both S and N k \ S are r.e. Remark 1.5. Some authors write computably enumerable, c.e., and computable for recursively enumerable, r.e., and recursive respectively. Strictly speaking, it would be more appropriate to call our Σ 1 programs Σ 1 (exp), but the two notions actually coincide in the cases we will consider. Assignment 1.6. A 0 program is a 0 (exp) program in which no L A (exp) term contains a subterm of the form 2 t, where t is a term. For example, the following are 0 programs: divides(x, y) = z y (y = xz); prime(y) = y 2 x y ( divides(x, y) x = 1 x = y ). Using these examples or otherwise, find a 0 program θ(w) such that a N : [θ(a)] = true} = 2 b N : b N}. [5 points] It is evident that the output of a program θ is closely related to the truth of θ as a formula. In the next lecture, we will investigate this relationship. 5