ON THE POWER OF FORMAL SYSTEMS FOR ANALYZING LINEAR AND POLYNOMIAL TIME PROGRAM BEHAVIOR
Abstract
Recently considerable attention has been given to program behavior in nonstandard models of various arithmetic theories ({P&D-80}, {CSI-80a,b}, {LIP-78}, {LEI 79}, {D&L-79}, {D&L-80}, {LEI-81a,b}) Most of the basic notions of computation and complexity are the same in nonstandard models of full arithmetic as in the standard model; however, if one considers subtheories of arithmetic that are sufficiently weak, then some significant differences arise. Some of the theories that have been investigated recently by computer scientists are in fact weak enough that standard notions of computation and complexity are not preserved. ({LIP-78}, {D&L-79}, {LEI-79}, {D&L-80}, {J&Y-81a,b,c}, {LEI-81a}) This thesis discusses some of these theories and the difficulties that arise when they are used to analyze program complexity and termination. Chapters II and III are devoted to the study of linear and polynomial time program behavior in models of the Theory of Exponential Time, ET, (Chapter III) and Basic Number Theory and T(,(PI)(,2)) (Chapter II). In Chapter II, Basic Number Theory and T(,(PI)(,2)) are investigated as possible formal system for analyzing program behavior. These theories are shown to be inadequate for various reasons: (i) Finite (i.e., bounded) sets are undecidable in some models, (ii) standard r.e. sets that Peano Arithmetic proves undecidable have linear time decision procedures in some models, and (iii) programs which simply loop a bounded number of times may not terminate in some models. The latter demonstrates the inadequacy of Basic Number Theory and T(,(PI)(,2)) for proving program termination. In Chapter III we study two approaches for investigating whether NP-complete sets have fast algorithms. One approach is to ask whether there are arbitrarily long initial segments on which such sets are easily decidable by short programs. The other approach is to ask whether there are weak fragments of arithmetic with which it is consistent to believe P = NP. We show that in a certain sense these questions are related: It is consistent to believe that NP-complete sets and for that matter, all elementary sets, are polynomially decidable in certain models of ET if and only if it is true (in the standard model of computation) that they have very long initial segments that are easily decidable by short programs.
Degree
Ph.D.
Subject Area
Computer science
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