XII. Elementary Function Yuxi Fu BASICS, Shanghai Jiao Tong University What is the class of arithmetic functions we use in mathematics? Computability Theory, by Y. Fu XII. Elementary Function 1 / 17 Definition The class E of elementary function is constructed from 1. zero, successor, projection, and 2. subtraction x−_ y (for defining conditionals); and is 3. closed under composition and 4. bounded sum/product (bounded recursion). Remark. 1. Addition and multiplication can be defined using bounded sum; (hyper) exponential can be defined using bounded product. 2. Lower elementary functions are constructed by dropping (4). 3. A predicate is elementary if its characterization function is elementary. Computability Theory, by Y. Fu XII. Elementary Function 2 / 17 Bounded Minimalisation is Elementary Fact. E is closed under bounded minimalisation. Proof. Suppose f (x; z) is elementary. Then µz < y:f (x; z) = 0 is X Y sg(f (x; u)): v<y u≤v It is easy to see that sg is elementary. Computability Theory, by Y. Fu XII. Elementary Function 3 / 17 Logical Operation Fact. The set of elementary predicates is closed under negation, conjunction, disjunction and bounded quantifiers. Computability Theory, by Y. Fu XII. Elementary Function 4 / 17 Basic Arithmetic Functions are Elementary y Q 2 1. The exponential x is defined by z<y U1 (x; z). 2. The function px is defined by 2x px = µy < 2 :(x = 0 or y is the xth prime) 0 1 2x X = µy < 2 : @x = Pr(z)A z≤y 0 1 2x X = µy < 2 : @x−_ Pr(z) = 0A : z≤y Computability Theory, by Y. Fu XII. Elementary Function 5 / 17 Bounded Recursion Fact. Let f (x) and g(x; y; z) be elementary and h(x; y) be defined from f ; g via primitive recursion. If h(x; y) ≤ b(x; y) for some elementary function b(x; y), then h(x; y) is elementary. Proof. Observe that h(x;0) h(x;1) h(x;y) Y b(x;z) 2 3 ::: py+1 ≤ pz+1 : z≤y So we can define h(x; y) by Y b(x;z) µe ≤ pz+1 : ((e)1 = f (x) ^ 8z < y:((e)z+2 = g(x; z; (e)z+1))) : z≤y We are done. Computability Theory, by Y. Fu XII. Elementary Function 6 / 17 G¨odelEncoding is Elementary Fact. G¨odelencoding functions are elementary. Computability Theory, by Y. Fu XII. Elementary Function 7 / 17 Kleene's Predicate is Elementary Fact. The Kleene's functions σn, cn and jn are elementary. Computability Theory, by Y. Fu XII. Elementary Function 8 / 17 Elementary Time Functions A computable function φe is in elementary time if te (x) ≤ b(x) almost everywhere for some elementary function b(x). Fact. The elementary time functions are elementary. Proof. φe (x) is almost everywhere computable by the elementary function (cn(e; ex; µt ≤ b(ex):jn(e; ex; t) = 0))1; which implies that φe (x) is elementary. Computability Theory, by Y. Fu XII. Elementary Function 9 / 17 It has been suggested that E contains all practical computable functions. Computability Theory, by Y. Fu XII. Elementary Function 10 / 17 A computable function f (x) is practically computable if it can be computed in 2x 2··· expk (x) = 2 | {z } k exp (x) steps for some k. We let 2 k stand for expk+1(x). Computability Theory, by Y. Fu XII. Elementary Function 11 / 17 Upper Bound of Elementary Functions Theorem. For each elementary function f (ex) there is some k such that f (ex) ≤ expk (maxfexg). Proof. The basic elementary functions satisfy the upper bound. The elementary operations preserves the upper bound. Computability Theory, by Y. Fu XII. Elementary Function 12 / 17 Corollary. expx (x) is primitive recursive but not elementary. Proof. The function expx (x) is defined by g(x; x), where g(x; 0) = x; g(x; y + 1) = 2g(x;y): We are done. Computability Theory, by Y. Fu XII. Elementary Function 13 / 17 Elementary Functions are Elementary Time Lemma. Suppose f (ex) and g(ex; y; z) are in elementary time and h(ex; y) is defined from f ; g via recursion. If h(ex; y) is elementary, then h(ex; y) is in elementary time. Proof. The standard program that calculates h does it in elementary time. Computability Theory, by Y. Fu XII. Elementary Function 14 / 17 Elementary Functions are Elementary Time Theorem. If f (ex) is elementary, then there is a program P for f n such that tP (ex) is elementary. Proof. Use the above lemma. Computability Theory, by Y. Fu XII. Elementary Function 15 / 17 Complexity Theoretical Characterization Theorem. A total function f (ex) is elementary iff it is computable in time expk (maxfexg) for some k. Computability Theory, by Y. Fu XII. Elementary Function 16 / 17 n 2n ELEMENTARY = TIME(2n) [ TIME(22 ) [ TIME(22 ) [ :::: Computability Theory, by Y. Fu XII. Elementary Function 17 / 17.