Polynomial interpretations and their generalizations like quasi-interpretations have been used in the setting of first-order functional languages to design criteria ensuring statically some complexity bounds on programs [1]. This fits in the area of implicit computational complexity, which aims at giving machine-free characterizations of complexity classes. Here we extend this approach to the h...

An attractive and challenging problem in computational number theory is to count in an e*cient manner the number of solutions to a multivariate polynomial equation over a -nite -eld. One desires an algorithm whose time complexity is a small polynomial function of some appropriate measure of the size of the polynomial. A natural measure of size is d logðqÞ for a polynomial of total degree d in n...

We investigate the computational complexity of finding the minimum-area circumscribed rectangle of a given two-dimensional domain. We study this problem in the polynomial-time complexity theory of real functions based on the oracle Turing machine model. We show that a bounded domain S with a polynomial-time computable Jordan curve Γ as the boundary may not have a computable minimum-area circums...

Tavenas has recently proved that any nO(1)-variate and degree n polynomial in VP can be computed by a depth-4 ΣΠ[O( p n)]ΣΠ[ p n] circuit of size 2O( p n log n) [Tav13]. So to prove VP 6= VNP, it is sufficient to show that an explicit polynomial ∈ VNP of degree n requires 2ω( p n log n) size depth-4 circuits. Soon after Tavenas’s result, for two different explicit polynomials, depth-4 circuit s...

We present iSAPP (Imperative Static Analyser for Probabilistic Polynomial Time), a complexity verifier tool that is sound and extensionally complete for the Probabilistic Polynomial Time (PP) complexity class. iSAPP works on an imperative programming language for stack machines. The certificate of polynomiality can be built in polytime, with respect to the number of stacks used. 1998 ACM Subjec...