99,890 research outputs found

    On the Complexity of Solving Zero-Dimensional Polynomial Systems via Projection

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    Given a zero-dimensional polynomial system consisting of n integer polynomials in n variables, we propose a certified and complete method to compute all complex solutions of the system as well as a corresponding separating linear form l with coefficients of small bit size. For computing l, we need to project the solutions into one dimension along O(n) distinct directions but no further algebraic manipulations. The solutions are then directly reconstructed from the considered projections. The first step is deterministic, whereas the second step uses randomization, thus being Las-Vegas. The theoretical analysis of our approach shows that the overall cost for the two problems considered above is dominated by the cost of carrying out the projections. We also give bounds on the bit complexity of our algorithms that are exclusively stated in terms of the number of variables, the total degree and the bitsize of the input polynomials

    On Quasi-Interpretations, Blind Abstractions and Implicit Complexity

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    Quasi-interpretations are a technique to guarantee complexity bounds on first-order functional programs: with termination orderings they give in particular a sufficient condition for a program to be executable in polynomial time, called here the P-criterion. We study properties of the programs satisfying the P-criterion, in order to better understand its intensional expressive power. Given a program on binary lists, its blind abstraction is the nondeterministic program obtained by replacing lists by their lengths (natural numbers). A program is blindly polynomial if its blind abstraction terminates in polynomial time. We show that all programs satisfying a variant of the P-criterion are in fact blindly polynomial. Then we give two extensions of the P-criterion: one by relaxing the termination ordering condition, and the other one (the bounded value property) giving a necessary and sufficient condition for a program to be polynomial time executable, with memoisation.Comment: 18 page

    On monotone circuits with local oracles and clique lower bounds

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    We investigate monotone circuits with local oracles [K., 2016], i.e., circuits containing additional inputs yi=yi(x⃗)y_i = y_i(\vec{x}) that can perform unstructured computations on the input string x⃗\vec{x}. Let μ∈[0,1]\mu \in [0,1] be the locality of the circuit, a parameter that bounds the combined strength of the oracle functions yi(x⃗)y_i(\vec{x}), and Un,k,Vn,k⊆{0,1}mU_{n,k}, V_{n,k} \subseteq \{0,1\}^m be the set of kk-cliques and the set of complete (k−1)(k-1)-partite graphs, respectively (similarly to [Razborov, 1985]). Our results can be informally stated as follows. 1. For an appropriate extension of depth-22 monotone circuits with local oracles, we show that the size of the smallest circuits separating Un,3U_{n,3} (triangles) and Vn,3V_{n,3} (complete bipartite graphs) undergoes two phase transitions according to μ\mu. 2. For 5≤k(n)≤n1/45 \leq k(n) \leq n^{1/4}, arbitrary depth, and μ≤1/50\mu \leq 1/50, we prove that the monotone circuit size complexity of separating the sets Un,kU_{n,k} and Vn,kV_{n,k} is nΘ(k)n^{\Theta(\sqrt{k})}, under a certain restrictive assumption on the local oracle gates. The second result, which concerns monotone circuits with restricted oracles, extends and provides a matching upper bound for the exponential lower bounds on the monotone circuit size complexity of kk-clique obtained by Alon and Boppana (1987).Comment: Updated acknowledgements and funding informatio
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