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**1 - 4**of**4**### LOWER BOUNDS FOR COPRIMENESS AND OTHER DECISION PROBLEMS IN ARITHMETIC

"... This talk was about some joint work with Lou van den Dries, in which we try to derive lower bounds for the worst-case, time complexity of functions and decision problems in arithmetic which apply to all—or, in any case, to as many as possible— algorithms. The relevant papers are listed in the biblio ..."

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This talk was about some joint work with Lou van den Dries, in which we try to derive lower bounds for the worst-case, time complexity of functions and decision problems in arithmetic which apply to all—or, in any case, to as many as possible— algorithms. The relevant papers are listed in the bibliography and [3] gives a brief account of how we came to these questions, as well as a fairly complete exposition of what we have proved. Here I will confine myself to very few precise statements of specific results, since my main aim is to describe the methods that we use. 1 1. One conjecture and two results The ancient Euclidean algorithm computes the greatest common divisor of two natural numbers using iterated division. It can be expressed succinctly by the recursive equation (1) gcd(a, b) = if (rem(a, b) = 0) then b else gcd(b, rem(a, b)) (a ≥ b ≥ 1), and its natural complexity measure is cε(a, b) = the number of divisions required to compute gcd(m, n) using (1). It is easy to check that

### On the Polynomial Transparency of Resolution 4

"... In this paper a framework is developed for measuring and computationally perspicuous manner. As a notion of central importance appears the so-called polynomial transparency of a calculus. If a logic calculus possesses this property, then the complexity of any deduction can be correctly measured in t ..."

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In this paper a framework is developed for measuring and computationally perspicuous manner. As a notion of central importance appears the so-called polynomial transparency of a calculus. If a logic calculus possesses this property, then the complexity of any deduction can be correctly measured in terms of its inference steps. The resolution calculus lacks this property. It is proven that the number of inference steps of a resolution proof does not give a representative measure of the actual complexity of the proof, even if only shortest proofs are considered. We use a class of formulae which have proofs with a polynomial number of inference steps, but for which the size of any proof is exponential. The polynomial intransparency of resolution is due to the renaming of derived clauses, which is a fundamental deduction mechanism. This result motivates the development of new data structures for the representation of logical formulae. 1

### Quantum Computation: A Computer Science Perspective

, 2005

"... The theory of quantum computation is presented in a self contained way from a computer science perspective. The basics of classical computation and quantum mechanics is reviewed. The circuit model of quantum computation is presented aspects of computation and the interplay between them. This report ..."

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The theory of quantum computation is presented in a self contained way from a computer science perspective. The basics of classical computation and quantum mechanics is reviewed. The circuit model of quantum computation is presented aspects of computation and the interplay between them. This report is presented as a Master’s thesis at the department of Computer Science and Engineering at Göteborg University, Göteborg, Sweden. The text is part of a larger work that is planned to include chapters on quantum algorithms, the quantum Turing machine model and abstract approaches to quantum computation.

### Memoization for Unary Logic Programming: Characterizing PTIME

"... Abstract—We give a characterization of deterministic polyno-mial time computation based on an algebraic structure called the resolution semiring, whose elements can be understood as logic programs or sets of rewriting rules over first-order terms. More precisely, we study the restriction of this fra ..."

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Abstract—We give a characterization of deterministic polyno-mial time computation based on an algebraic structure called the resolution semiring, whose elements can be understood as logic programs or sets of rewriting rules over first-order terms. More precisely, we study the restriction of this framework to terms (and logic programs, rewriting rules) using only unary sym-bols. We prove it is complete for polynomial time computation, using an encoding of pushdown automata. We then introduce an algebraic counterpart of the memoization technique in order to show its PTIME soundness. We finally relate our approach and complexity results to complexity of logic programming. As an application of our techniques, we show a PTIME-completeness result for a class of logic programming queries which use only unary function symbols.