Characterizing complexity classes by higher type
Higher type primitive recursive definitions (also known as Gödel's system T) defining first-order functions (i.e. functions of type ind→...→ind→ind, ind for individuals, higher types occur in between) can be classified into an infinite syntactic hierarchy: A definition is in the n'th stage of this hierarchy, a so called rank-n-definition, iff n is an upper bound on the levels of the types occurring in it.
We interpret these definitions over finite structures and show for n≥1: Rank-(2n+2)-definitions characterize (in the sense of [Gu83], say) the complexity class DTIME(expn(poly)) whereas rank-(2n+3)-definitions characterize DSPACE(expn(poly)) (here exp0(x) = x, expn+1(x)=2expnx). This extends the results that rank-1-definitions characterize LOGSPACE [Gu83], rank-2-definitions characterize PTIME, rank-3-definitions characterize PSPACE, rank-4-definitions characterize EXPTIME [Go89a].
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- [Ba84]Barendregt, H., The lambda calculus. North Holland 1984.Google Scholar
- [Go89a]Goerdt, A., Characterizing complexity classes by higher type primitive recursive definitions, LICS (1989), 364–374.Google Scholar
- [Go89b]Goerdt, A., Characterizing complexity classes by finitely typed λ-terms, Workshop Computer Science Logic 1988, LNCS, to appear.Google Scholar
- [Gu83]Gurevich, Y., Algebras of feasible functions, 24th FOCS (1983), 210–214.Google Scholar
- [Gu84]Gurevich, Y., Toward logic taylored for computational complexity, computation and proof theory, LNM 1104 (1984), 175–216.Google Scholar
- [Im87]Immerman, N., Expressibility as a complexity measure: results and directions, 2nd Conf. Structure in Complexity Theory (1987), 194–202.Google Scholar
- [KfTiUr87]Kfoury, A., Tiuryn, J.,Urzyczyn, P., The hierarchy of finitely typed functional programs, LICS (1987), 225–235.Google Scholar
- [Lei87]Leivant, D., Characterization of complexity classes in higher order logic, 2nd conf. Structure in Complexity Theory (1987), 203–218.Google Scholar
- [Schw75]Schwichtenberg, H., Elimination of higher type levels in definitions of primitive recursive functions by means of transfinite recursion, Logic Colloquium 1973, North Holland.Google Scholar
- [Sa80]Sazonov, V., Polynomial computability and recursivity in finite domains, EIK 16 (1980), 319–323.Google Scholar
- [Te82]Terlouw, J., On the definition trees of ordinal recursive functionals: reduction of recursion orders by means of type level raising, J. Symbolic Logic 47 (1982), 395–403.Google Scholar
- [Ti86]Tiuryn, J., Higher order arrays and stacks in programming: An application of complexity theory to logics of programs, 12th MFCS (1986), LNCS 233, 177–198.Google Scholar
- [TiUr83]Tiuryn, J.,Urzyczyn, P., Some connections between logic of programs and complexity theory. 24th FOCS (1983), 180–184.Google Scholar