Revisiting Synthesis for One-Counter Automata

We study the (parameter) synthesis problem for one-counter automata with parameters. One-counter automata are obtained by extending classical finite-state automata with a counter whose value can range over non-negative integers and be tested for zero. The updates and tests applicable to the counter can further be made parametric by introducing a set of integer-valued variables called parameters. The synthesis problem for such automata asks whether there exists a valuation of the parameters such that all infinite runs of the automaton satisfy some omega-regular property. Lechner showed that (the complement of) the problem can be encoded in a restricted one-alternation fragment of Presburger arithmetic with divisibility. In this work (i) we argue that said fragment, called AERPADPLUS, is unfortunately undecidable. Nevertheless, by a careful re-encoding of the problem into a decidable restriction of AERPADPLUS, (ii) we prove that the synthesis problem is decidable in general and in N2EXP for several fixed omega-regular properties. Finally, (iii) we give a polynomial-space algorithm for the special case of the problem where parameters can only be used in tests, and not updates, of the counter

On the computation of rational points of a hypersurface over a finite field

We design and analyze an algorithm for computing rational points of hypersurfaces defined over a finite field based on searches on "vertical strips", namely searches on parallel lines in a given direction. Our results show that, on average, less than two searches suffice to obtain a rational point. We also analyze the probability distribution of outputs, using the notion of Shannon entropy, and prove that the algorithm is somewhat close to any "ideal" equidistributed algorithm.Comment: 31 pages, 5 table

On the value set of small families of polynomials over a finite field, II

We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of Fq[T] of degree d for which s consecutive coefficients a_{d-1},...,a_{d-s} are fixed. Our estimate asserts that \mathcal{V}(d,s,\bfs{a})=\mu_d\,q+\mathcal{O}(q^{1/2}), where \mathcal{V}(d,s,\bfs{a}) is such an average cardinality, \mu_d:=\sum_{r=1}^d{(-1)^{r-1}}/{r!} and \bfs{a}:=(a_{d-1},...,a_{d-s}). We also prove that \mathcal{V}_2(d,s,\bfs{a})=\mu_d^2\,q^2+\mathcal{O}(q^{3/2}), where that \mathcal{V}_2(d,s,\bfs{a}) is the average second moment on any family of monic polynomials of Fq[T] of degree d with s consecutive coefficients fixed as above. Finally, we show that \mathcal{V}_2(d,0)=\mu_d^2\,q^2+\mathcal{O}(q), where \mathcal{V}_2(d,0) denotes the average second moment of all monic polynomials in Fq[T] of degree d with f(0)=0. All our estimates hold for fields of characteristic p>2 and provide explicit upper bounds for the constants underlying the \mathcal{O}--notation in terms of d and s with "good" behavior. Our approach reduces the questions to estimate the number of Fq--rational points with pairwise--distinct coordinates of a certain family of complete intersections defined over Fq. A critical point for our results is an analysis of the singular locus of the varieties under consideration, which allows to obtain rather precise estimates on the corresponding number of Fq--rational points.Comment: 36 page

Efficient implementation of a van der Waals density functional: Application to double-wall carbon nanotubes

We present an efficient implementation of the van der Waals density functional of Dion et al [Phys. Rev. Lett. 92, 246401 (2004)], which expresses the nonlocal correlation energy as a double spacial integral. We factorize the integration kernel and use fast Fourier transforms to evaluate the selfconsistent potential, total energy, and atomic forces, in N log(N) operations. The resulting overhead in total computational cost, over semilocal functionals, is very moderate for medium and large systems. We apply the method to calculate the binding energies and the barriers for relative translation and rotation in double-wall carbon nanotubes.Comment: 4 pages, 1 figure, 1 tabl