Showing papers on "Quantum complexity theory published in 1980"

An information-theoretic approach to time bounds for on-line computation (preliminary version)

Abstract: Static, descriptional complexity (program size) [16, 9] can be used to obtain lower bounds on dynamic, computational complexity (such as running time). We describe and discuss this “information-theoretic approach” in the following section. Paul introduced it in [13], to obtain restricted lower bounds on the time complexity of sorting. We use the approach here to obtain lower time bounds for on-line simulation of one abstract storage unit by another. A major goal of our work is to promote the approach.

Succinct representation random strings, and complexity classes

Abstract: A general paradigm for relating measures of succinctness of representation and complexity theory is presented. The measures are based on the new Private and Blindfold Alternation machines. These measures are used to indicate the inherent information (or "randomness") of a string, but with respect to time and space complexity classes. These measures are then used to show that the existence of strings which are random with respect to one measure but not to another can show the relationship between the corresponding complexity classes. The basic hierarchy theorems given allow different and possibly more powerful approaches to these problems.

Complexity of implementations on the level of algebraic specifications

Abstract: The aim of this paper is to study implementations of abstract data types and their complexity within the framework of algebraic specifications. An implementation of an abstract data type ADTO by an abstract data type ADT1 is defined on a syntactical and on a semantical level where data and operations of ADTO are simulated by those of ADT1. In order to investigate complexity of implemented operations and to compair different implementations we axiomatically introduce complexity measures for the operations in ADTO with respect to a given implementation of ADTO by ADT1. A most natural interesting class of complexity measures which satisfy our axioms is compatible with time complexity of Turing Machines. This is shown by specification and implementation of timebounded Turing Machines within our algebraic framework and the simulation of a general nondeterministic interpreter for algebraic implementations by a nondeterministic Turing Machine. This relationship allows to show the existence of solutions and of upper and lower bounds for the complexity of a broad class of implementation problems in our sense. A corollary shows that there are algebraic specifications for all those recursive functions which are bounded in time by algebraically specifyable functions. Since the complexity measure for ADTO-operations in a given implementation IMPL1 of ADTO by ADT1 in general depends on the complexity of the ADT1-operations it is possible to substitute the complexity of the ADT1-operations which may be obtained from an implementation IMPL2 of ADT1 by ADT2. On the other hand the composite implementation IMPL3 of ADTO by ADT2 can be constructed and it is shown that the composite complexity of IMPL3 is less or equal to the substituted complexity considered above provided that some natural consistency conditions are satisfied.