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We consider the problem of recognizing important properties of logical calculi and find complexity bounds for some decidable properties. For a given logical system L, a property P of logical calculi is called decidable over L if there is an algorithm which for any finite set Ax of new axiom schemes decides whether the calculus L + Ax has the property P or not. In [11] the complexity of tabularity, pre-tabularity. and interpolation problems over the intuitionistic logic Int and over modal logic S4 was studied, also we found the complexity of amalgamation problems in varieties of Heyting algebras and closure algebras.In the present paper we deal with positive calculi. We prove NP-completeness of tabularity, DP-hardness of pretabularity and PSPACE-completeness of interpolation problem over Int+. In addition to above-mentioned properties, we consider Beth's definability properties. Also we improve some complexity bounds for properties of superintuitionistic calculi.

We give the first super-polynomial separation in the power of bounded-depth boolean formulas vs. circuits. Specifically, we consider the problem Distance $k(n)$ Connectivity, which asks whether two specified nodes in a graph of size $n$ are connected by a path of length at most $k(n)$. This problem is solvable (by the recursive doubling technique) on {\bf circuits} of depth $O(\log k)$ and size $O(kn^3)$. In contrast, we show that solving this problem on {\bf formulas} of depth $\log n/(\log\log n)^{O(1)}$ requires size $n^{\Omega(\log k)}$ for all $k(n) \leq \log\log n$. As corollaries: (i) It follows that polynomial-size circuits for Distance $k(n)$ Connectivity require depth $\Omega(\log k)$ for all $k(n) \leq \log\log n$. This matches the upper bound from recursive doubling and improves a previous $\Omega(\log\log k)$ lower bound of Beame, Pitassi and Impagliazzo [BIP98]. (ii) We get a tight lower bound of $s^{\Omega(d)}$ on the size required to simulate size-$s$ depth-$d$ circuits by depth-$d$ formulas for all $s(n) = n^{O(1)}$ and $d(n) \leq \log\log\log n$. No lower bound better than $s^{\Omega(1)}$ was previously known for any $d(n) \nleq O(1)$. Our proof technique is centered on a new notion of pathset complexity, which roughly speaking measures the minimum cost of constructing a set of (partial) paths in a universe of size $n$ via the operations of union and relational join, subject to certain density constraints. Half of our proof shows that bounded-depth formulas solving Distance $k(n)$ Connectivity imply upper bounds on pathset complexity. The other half is a combinatorial lower bound on pathset complexity.

The polynomial hierarchy is a grading of problems by difficulty, including P, NP and coNP as the best known classes. The promise polynomial hierarchy is similar, but extended to include promise problems. It turns out that the promise polynomial hierarchy is considerably simpler to work with, and many open questions about the polynomial hierarchy can be resolved in the promise polynomial hierarchy. Our main theorem is that, in the world of promise problems, if phi has a weak (Turing, Cook) reduction to SAT then phi has a strong (Karp, many-one) reduction to UVAL2, where UVAL2(f) is the promise problem of finding the unique x such that f(x,y)=1 for all y. We also give a complete promise problem for the promise problem equivalent of UP intersect coUP., 12 pages

The Inverse 3-SAT problem is known to be coNP Complete. This article shows a new interesting way to solve directly the problem by using closure under resolution and partial assignment properties. An algorithm is proposed which lets solve the (co)Inverse 3-SAT problem., Comment: 8 pages

The long code is a central tool in hardness of approximation, especially in questions related to the unique games conjecture. We construct a new code that is exponentially more efficient, but can still be used in many of these applications. Using the new code we obtain exponential improvements over several known results, including the following: 1. For any eps > 0, we show the existence of an n vertex graph G where every set of o(n) vertices has expansion 1 - eps, but G's adjacency matrix has more than exp(log^delta n) eigenvalues larger than 1 - eps, where delta depends only on eps. This answers an open question of Arora, Barak and Steurer (FOCS 2010) who asked whether one can improve over the noise graph on the Boolean hypercube that has poly(log n) such eigenvalues. 2. A gadget that reduces unique games instances with linear constraints modulo K into instances with alphabet k with a blowup of K^polylog(K), improving over the previously known gadget with blowup of 2^K. 3. An n variable integrality gap for Unique Games that that survives exp(poly(log log n)) rounds of the SDP + Sherali Adams hierarchy, improving on the previously known bound of poly(log log n). We show a connection between the local testability of linear codes and small set expansion in certain related Cayley graphs, and use this connection to derandomize the noise graph on the Boolean hypercube., 45 pages

Building upon works of Hironaka, Bierstone-Milman, Villamayor and Wlodarczyk. We give an a priori estimate for the complexity of the simplified Hironaka algorithm.

Arc-annotated sequences are useful in representing the structural information of RNA and protein sequences. The longest arc-preserving common subsequence problem has been introduced as a framework for studying the similarity of arc-annotated sequences. In this paper, we consider arc-annotated sequences with various arc structures. We consider the longest arc preserving common subsequence problem. In particular, we show that the decision version of the 1-{\sc fragment LAPCS(crossing,chain)} and the decision version of the 0-{\sc diagonal LAPCS(crossing,chain)} are {\bf NP}-complete for some fixed alphabet $\Sigma$ such that $|\Sigma| = 2$. Also we show that if $|\Sigma| = 1$, then the decision version of the 1-{\sc fragment LAPCS(unlimited, plain)} and the decision version of the 0-{\sc diagonal LAPCS(unlimited, plain)} are {\bf NP}-complete.