Parameterized bounded-depth Frege is not optimal

A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [9]. There the authors concentrate on tree-like Parameterized Resolution-a parameterized version of classical Resolution-and their gap complexity theorem implies lower bounds for that system. The main result of the present paper significantly improves upon this by showing optimal lower bounds for a parameterized version of bounded-depth Frege. More precisely, we prove that the pigeonhole principle requires proofs of size n in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in [9]. In the opposite direction, we interpret a well-known technique for FPT algorithms as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNF's

Reducing the complexity of reductions

We build on the recent progress regarding isomorphisms of complete sets that was reported in Agrawal et al. (1998). In that paper, it was shown that all sets that are complete under (non-uniform) AC0 reductions are isomorphic under isomorphisms computable and invertible via (non-uniform) depth-three AC0 circuits. One of the main tools in proving the isomorphism theorem in Agrawal et al. (1998) is a "Gap Theorem", showing that all sets complete under AC0 reductions are in fact already complete under NC0 reductions. The following questions were left open in that paper: ¶1. Does the "gap" between NC0 and AC0 extend further? In particular, is every set complete under polynomial-time reducibility already complete under NC0 reductions? ¶2. Does a uniform version of the isomorphism theorem hold? ¶3. Is depth-three optimal, or are the complete sets isomorphic under isomorphisms computable by depth-two circuits? ¶ We answer all of these questions. In particular, we prove that the Berman-Hartmanis isomorphism conjecture is true for P-uniform AC0 reductions. More precisely, we show that for any class closed under uniform TC0-computable many-one reductions, the following three theorems hold: ¶1. If contains sets that are complete under a notion of reduction at least as strong as Dlogtime-uniform AC0[mod 2] reductions, then there are such sets that are not complete under (even non-uniform) AC0 reductions. ¶2. The sets complete for under P-uniform AC0 reductions are all isomorphic under isomorphisms computable and invertible by P-uniform AC0 circuits of depth-three. ¶3. There are sets complete for under Dlogtime-uniform AC0 reductions that are not isomorphic under any isomorphism computed by (even non-uniform) AC0 circuits of depth two. ¶To prove the second theorem, we show how to derandomize a version of the switching lemma, which may be of independent interest. (We have recently learned that this result is originally due to Ajtai and Wigderson, but it has not been published.

Stochastic Boolean Satisfiability

. Satisfiability problems and probabilistic models are core topics of artificial intelligence and computer science. This paper looks at the rich intersection between these two areas, opening the door for the use of satisfiability approaches in probabilistic domains. The paper examines a generic stochastic satisfiability problem, SSat, which can function for probabilistic domains as Sat does for deterministic domains. It shows the connection between SSat and well studied problems in belief network inference and planning under uncertainty, and defines algorithms, both systematic and stochastic, for solving SSat instances. These algorithms are validated on random SSat formulae generated under the fixed-clause model. In spite of the large complexity gap between SSat (PSPACE) and Sat (NP), the paper suggests that much of what we've learned about Sat transfers to the probabilistic domain. 1. Introduction There has been a recent focus in artificial intelligence (AI) on solving problems exh..

Diagnostic factors in pediatric primary headache

Primary headaches are frequent in children. They are difficult to diagnose because there is much disagrrement about the interpretation of the historical data and the use of signs and/or symptoms in diagnosis. It would be useful, therefore, to standardize this procedure. We used linear discriminant analysis to determine a classification rule capable of diagnosing new cases of chronic and recurrent primary headache in children. We considered 23 symptoms in 118 patients. Through discriminant anlysis we chose five variables: frequency of the attacks, type of pain, neurologic deficits, nausea, and vomiting. With this classification rule, we obtained a total correct classification of 84.7% for migraine, psychogenic headache, and non-defined headache in respect to the diagnoses formulated by a pediatrician and a child neuropsychiatrist after 3 months of follow-up. Our method for diagnosing migraine has a sensitivity of 95% and a specificity of 100%. The algorithm, validated on another 105 pediatric patients, produced a total correct result of 82.9%