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XII. Elementary Function 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.
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