Abstract not found

: The aspect of purity versus impurity that we address involves the absence versus presence of mutation: the use of primitives (RPLACA and RPLACD in Lisp, set-car! and set-cdr! in Scheme) that change the state of pairs without creating new pairs. It is well known that cyclic list structures can be created by impure programs, but not by pure ones. In this sense, impure Lisp is "more powerful" than pure Lisp. If the inputs and outputs of programs are restricted to be sequences of atomic symbols, however, this difference in computability disappears. We shall show that if the temporal sequence of input and output operations must be maintained (that is, if computations must be "online "), then a difference in complexity remains: for a pure program to do what an impure program does in n steps, O(n log n) steps are sufficient, and in some cases\Omega\Gamma n log n) steps are necessary. * This research was partially supported by an NSERC Operating Grant. 1. Introduction The programming la...

. We show that a Turing machine with two single-head one-dimensional tapes cannot recognize the set f x2x 0 j x 2 f0; 1g and x 0 is a prefix of x g in real time, although it can do so with three tapes, two two-dimensional tapes, or one two-head tape, or in linear time with just one tape. In particular, this settles the longstanding conjecture that a two-head Turing machine can recognize more languages in real time if its heads are on the same one-dimensional tape than if they are on separate one-dimensional tapes. 1. Introduction The Turing machines commonly used and studied in computer science have separate tapes for input/output and for storage, so that we can conveniently study both storage as a dynamic resource and the more complex storage structures required for efficient implementation of practical algorithms [HS65]. Early researchers [MRF67] asked specifically whether two-head storage is more powerful if both heads are on the same one-dimensional storage tape than if t...

Connections are shown between two properties of a machine model: linear speed-up and polynomial vicinity . In the context of the author's Block Move (BM) model, these relate to: "How long does it take to simulate a finite transducer S on a given input z?" This question is related to the century-old problem of finding economical representations for finite groups. Under some cost measures for computing S(z), the BM enjoys the linear speed-up property, but under more-realistic measures, and subject to a reasonable but unproved hypothesis, it has the antithetical property of a constant-factor time hierarchy . 1 Speed-Up and Vicinity Hartmanis and Stearns [HS65] proved that the standard multitape Turing machine (TM) model enjoys the following property, for which we give a general statement: Definition 1.1. A machine model M has the linear speed-up property if there exists k 0 ? 0 such that for every ffl ? 0 and t(n) time-bounded M-machine M , there exists a M-machine M 0 that computes...

This paper first surveys the near-total lack of superlinear lower bounds in complexity theory, for “natural” computational problems with respect to many models of computation. We note that the dividing line between models where such bounds are known and those where none are known comes when the model allows non-local communication with memory at unit cost. We study a model that imposes a “fair cost ” for non-local communication, and obtain modest superlinear lower bounds for some problems via a Kolmogorov-complexity argument. Then we look to the larger picture of what it will take to prove really striking lower bounds, and pull from ours and others’ work a concept of information vicinity that may offer new tools and modes of analysis to a young field that rather lacks them.

A “big-ideas ” approach to an undergraduate Computation Theory course is described. The aim of this approach to the Theory is to focus the student on those of the Theory’s concepts and tools that are more likely to be relevant to a student’s non-theoretical endeavors. By explaining why these are the “big ” concepts, the course also prepares the student to assimilate these concepts into his/her conceptual toolkit.