Some observations on the connection between counting and recursion

Klaus W. Wagner
1986 Theoretical Computer Science  
Based on Valiant's class ~'P of all functions counting the number of accepting computations of nondeterministic polynomial-time Turing machines, the polynomial-time hierarchy of counting functions is introduced. The class PHCF of all functions of this hierarchy and some of its subclasses are characterized by recursion-theoretic means. It turns out that, from the recursiontheoretic point of view, PHCF is an analogue to Kalmfir's class E of elementary functions, to the class PSPACE of
more » ... pace computable functions as well as to the class P of polynomialtime computable functions. 0304-3975/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland) 132 IC W. Wagner In order to valuate the main result of this paper (Fact B), the comparison with the other facts stated below might be interesting: Fact A. The class P can be generated from the basic functions +, -,., • by substitution, weak sum, weak bounded primitive recursion and weak product. Fact B. The class PHCF can be generated fiom the basic functions +, -,., " by substitution, sum, weak bounded primitive recursion and weak product. Fact C. The class PSPACE can be generated from the basic functions +, -,., • by substitution, sum, bounded primitive recursion and weak product. Fact D. The class E can be generated from the basic functions +, -, .," by substitution, sum, bounded primitive recursion and product.
doi:10.1016/0304-3975(86)90141-6 fatcat:zcgojve77zeojmt3pzbqij5imm