The complexity of promise problems with applications to public-key cryptography
Information and Control
Semirings, automata, languages
Semirings, automata, languages
Some observations on the connection between counting and recursion
Theoretical Computer Science
Structural complexity 1
Promise problems complete for complexity classes
Information and Computation
Finite monoids and the fine structure of NC1
Journal of the ACM (JACM)
Time/space trade-offs for reversible computation
SIAM Journal on Computing
The theory of semirings with applications in mathematics and theoretical computer science
The theory of semirings with applications in mathematics and theoretical computer science
Gap-definable counting classes
Journal of Computer and System Sciences
SIAM Journal on Computing
Two remarks on the power of counting
Proceedings of the 6th GI-Conference on Theoretical Computer Science
One complexity theorist's view of quantum computing
Theoretical Computer Science - Algorithms,automata, complexity and games
The complexity of tensor calculus
Computational Complexity
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Through the study of gate arrays we develop a unified framework to deal with probabilistic and quantum computations, where the former is shown to be a natural special case of the latter. On this basis we show how to encode a probabilistic or quantum gate array into a sum-free tensor formula which satisfies the conditions of the partial trace problem, and vice-versa; that is, given a tensor formula F of order n × 1 over a semiring I plus a positive integer k, deciding whether the kth partial trace of the matrix valIn,n (F ċ FT) fulfills a certain property. We use this to show that a certain promise version of the sum-free partial trace problem is complete for the class pr- BPP (promise BPP) for formulas over the semiring (Q+, +, ċ) of the positive rational numbers, for pr-BQP (promise BQP) in the case of formulas defined over the field (Q+, +, ċ), and if the promise is given up, then completeness for PP is shown, regardless whether tensor formulas over positive rationals or rationals in general are used. This suggests that the difference between probabilistic and quantum polytime computers may ultimately lie in the possibility, in the latter case, of having destructive interference between computations occurring in parallel. Moreover, by considering variants of this problem, classes like OP, NP, C=P, its complement co-C=P, the promise version of Valiant's class UP, its generalization promise SPP, and unique polytime US can be characterized by carrying the problem properties and the underlying semiring.