Alexis Maciel

A new proof of the weak pigeonhole principle

The exact complexity of the weak pigeonhole principle is an old and fundamental problem in proof complexity. Using a diagonalization argument, J. B. Paris et al. (J. Symbolic Logic 53 (1988), 1235-1244) showed how to prove the weak pigeonhole principle with bounded-depth, quasipolynomialsize proofs. Their argument was further refined by J. Krajicek (J. Symbolic Logic 59 (1994), 73-86). In this paper, we present a new proof: we show that the weak pigeonhole principle has quasipolynomial-size LK proofs where every formula consists of a single AND/OR of polylog fan-in. Our proof is conceptually simpler than previous arguments, and is optimal with respect to depth.

Threshold circuits of small majority-depth

We investigate the complexity of computations with constant-depth threshold circuits. Such circuits are composed of gates that determine if the sum of their inputs is greater than a certain threshold. When restricted to polynomial size, these circuits compute exactly the functions in the class TC

Non-automatizability of bounded-depth Frege proofs

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Programs over semigroups of dot-depth one

. These circuits are usually studied by measuring their efficiency in terms of their total depth. Using this point of view, the best division and iterated multiplication circuits have depth three and four, respectively. In this thesis, we propose a different approach. Since threshold gates are much more powerful than AND-OR gates, we allow the explicit use of AND-OR gates and consider the main measure of complexity to be the majority-depth of the circuit, i.e. the maximum number of threshold gates on any path in the circuit. Using this approach, we obtain division and iterated multiplication circuits of total depth four and five, but of majority-depth two and three. The technique used is called Chinese remaindering. We present this technique as a general tool for computing functions with integer values and use it to obtain depth-four threshold circuits of majority-depth two for other arithmetic problems such as the logarithm and power series approximation. We also consider the iterated multiplication problem for integers modulo q and for finite fields. The notion of majority-depth naturally leads to a hierarchy of subclasses of TC

Conditional Lower Bound for a System of Constant-Depth Proofs with Modular Connectives

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Efficient threshold circuits for power series

. We investigate this hierarchy and show that it is closely related to the usual depth hierarchy.

Upper and lower bounds for some depth-3 circuit classes

Abstract.In this paper, we show how to extend the argument due to Bonet, Pitassi and Raz to show that bounded-depth Frege proofs do not have feasible interpolation, assuming that factoring of Blum integers or computing the Diffie–Hellman function is sufficiently hard. It follows as a corollary that bounded-depth Frege is not automatizable; in other words, there is no deterministic polynomial-time algorithm that will output a short proof if one exists. A notable feature of our argument is its simplicity.

On ACC 0 [ p k ] Frege proofs

Abstract The notion of a p-variety arises in the algebraic approach to Boolean circuit complexity. It has great significance, since many known and conjectured lower bounds on circuits are equivalent to the assertion that certain classes of semigroups form p-varieties. In this paper, we prove that semigroups of dot-depth one form a p-variety. This example has the following implication: if a Boolean combination of Σ 1 formulas, using arbitrary numerical predicates, defines a regular language, one can then find an equivalent Σ 1 formula all of whose numerical predicates are regular.

Lifting lower bounds for tree-like proofs

It is known that constant-depth Frege proofs of some tautologies require exponential size. No such lower bound result is known for more general proof systems. We consider sequent calculus proofs in which formulas can contain modular connectives and only the cut formulas are restricted to be of constant depth. Under a plausible hardness assumption concerning small-depth Boolean circuits, we prove an exponential lower bound for such proofs. We prove this lower bound directly from the computational hardness assumption. By using the same approach, we obtain the following additional results. We provide a much simpler proof of a known (unconditional) lower bound in the case where only conjunctions and disjunctions are allowed. We establish a conditional exponential separation between the power of constant-depth proofs that use different modular connectives. Finally, under a plausible hardness assumption concerning the polynomial-time hierarchy, we show that the hierarchy Gi* of quantified propositional proof systems does not collapse

Circuit lower bounds collapse relativized complexity classes

Abstract We show that functions with convergent real power series can be well approximated by two classes of polynomial-size small-weight threshold circuits: depth-three circuits with threshold gates on all levels and depth-four circuits with threshold gates on the first two levels and AND–OR gates on the last two. This is done without restricting the input to a fixed closed subinterval of the interval of convergence of the series. We also point out that rational functions and the logarithm of x in base b can be well approximated by the same classes of circuits when both x and b are given as input.