Nondeterministic functions and the existence of optimal proof systems

We provide new characterizations of two previously studied questions on nondeterministic function classes: Q1: Do nondeterministic functions admit efficient deterministic refinements? Q2: Do nondeterministic function classes contain complete functions? We show that Q1 for the class is equivalent to the question whether the standard proof system for SAT is p-optimal, and to the assumption that every optimal proof system is p-optimal. Assuming only the existence of a p-optimal proof system for SAT, we show that every set with an optimal proof system has a p-optimal proof system. Under the latter assumption, we also obtain a positive answer to Q2 for the class . An alternative view on nondeterministic functions is provided by disjoint sets and tuples. We pursue this approach for disjoint -pairs and its generalizations to tuples of sets from and with disjointness conditions of varying strength. In this way, we obtain new characterizations of Q2 for the class . Question Q1 for is equivalent to the question of whether every disjoint -pair is easy to separate. In addition, we characterize this problem by the question of whether every propositional proof system has the effective interpolation property. Again, these interpolation properties are intimately connected to disjoint -pairs, and we show how different interpolation properties can be modeled by -pairs associated with the underlying proof system

On symmetric differences of NP-hard sets with weakly P-selective sets

AbstractThe symmetric differences of NP-hard sets with weakly-P-selective sets are investigated. We show that if there exist a weakly-P-selective set A and an NP-⩽Pm-hard set H such that H - AϵPbtt(sparse) and A — HϵPm(sparse) then P = NP. So no NP-⩽Pm-hard set has sparse symmetric difference with any weakly-P-selective set unless P = NP. The proof of our main result is an interesting application of the tree prunning techniques (Fortune 1979; Mahaney 1982). In addition, we show that there exists a P-selective set which has exponentially dense symmetric difference with every set in Pbtt(sparse)

On adaptive versus nonadaptive bounded query machines

AbstractThe polynomial-time adaptive (Turing) and nonadaptive (truth-table) bounded query machines are compared with respect to sparse oracles. A k-query adaptive machine has been found which, relative to a sparse oracle, cannot be simulated by any (2k−2)-query nonadaptive machine, even with a different sparse oracle. Conversely, there is a (3·2k−2)-query nonadaptive machine which, relative to a sparse oracle, cannot be simulated by any k-query adaptive machine, with any sparse oracle

Complexity of equivalence relations and preorders from computability theory

We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations $R, S$, a componentwise reducibility is defined by R\le S \iff \ex f \, \forall x, y \, [xRy \lra f(x) Sf(y)]. Here $f$ is taken from a suitable class of effective functions. For us the relations will be on natural numbers, and $f$ must be computable. We show that there is a $\Pi_1$-complete equivalence relation, but no $\Pi k$-complete for $k \ge 2$. We show that $\Sigma k$ preorders arising naturally in the above-mentioned areas are $\Sigma k$-complete. This includes polynomial time $m$-reducibility on exponential time sets, which is $\Sigma 2$, almost inclusion on r.e.\ sets, which is $\Sigma 3$, and Turing reducibility on r.e.\ sets, which is $\Sigma 4$.Comment: To appear in J. Symb. Logi

Polynomial-time reducibilities and “almost all” oracle sets

AbstractIt is shown for every k>0 and for almost every tally setT, {A|A ⩽Pk−ttT} ≠ {A|A ⩽P(k+1)−ttT}. In contrast, it is shown that for every set A, the following holds: (a) for almost every set B,A ⩽ Pm B if and only if A ⩽ P(logn)−T B; and (b) for almost every set B, A ⩽Ptt B if and only ifA ⩽PTB

Learning Weak Reductions to Sparse Sets

We study the consequences of NP having non-uniform polynomial size circuits of various types. We continue the work of Agrawal and Arvind~\cite{AA:96} who study the consequences of \SAT being many-one reducible to functions computable by non-uniform circuits consisting of a single weighted threshold gate. (\SAT \leq_m^p \LT). They claim that as a consequence \PTIME = \NP follows, but unfortunately their proof was incorrect. We take up this question and use results from computational learning theory to show that if \SAT \leq_m^p \LT then \PH = \PTIME^\NP. We furthermore show that if \SAT disjunctive truth-table (or majority truth-table) reduces to a sparse set then \SAT \leq_m^p \LT and hence a collapse of \PH to \PTIME^\NP also follows. Lastly we show several interesting consequences of \SAT \leq_{dtt}^p \SPARSE