Dmitry Sokolov

Dag-Like Communication and Its Applications

In 1990 Karchmer and Widgerson considered the following communication problem \(\mathtt {Bit}\): Alice and Bob know a function \(f: \{0, 1\}^n \rightarrow \{0, 1\}\), Alice receives a point \(x \in f^{-1}(1)\), Bob receives \(y \in f^{-1}(0)\), and their goal is to find a position i such that \(x_i \ne y_i\). Karchmer and Wigderson proved that the minimal size of a boolean formula for the function f equals the size of the smallest communication protocol for the \(\mathtt {Bit}\) relation. In this paper we consider a model of dag-like communication complexity (instead of classical one where protocols correspond to trees). We prove an analogue of Karchmer-Wigderson Theorem for this model and boolean circuits. We also consider a relation between this model and communication PLS games proposed by Razborov in 1995 and simplify the proof of Razborov’s analogue of Karchmer-Wigderson Theorem for PLS games.

Influence of Usnic Acid and its Derivatives on the Activity of Mammalian Poly(ADP-ribose)polymerase 1 and DNA Polymerase β

The influence of a number of usnic acid derivatives on auto(polyADP-ribosyl)ation catalyzed by PARP1 and DNA synthesis catalyzed by DNA polymerase β was studied. The derivatives of usnic acid containing aromatic substituents were shown to be moderate inhibitors of PARP1. The presence of both usnic acid tricyclic structure and aromatic substituent at any position of the molecule is a key factor for the inhibitory action. In the case of DNA polymerase β, no relationship between the structure and inhibitory properties has been found with the only exception. Derivatives with modified ring A showed mild activation of DNA synthesis catalyzed by DNA polymerase β.

Lower bounds for myopic DPLL algorithms with a cut heuristic

The paper is devoted to lower bounds on the time complexity of DPLL algorithms that solve the satisfiability problem using a splitting strategy. Exponential lower bounds on the running time of DPLL algorithms on unsatisfiable formulas follow from the lower bounds for resolution proofs. Lower bounds on satisfiable instances are also known for some classes of DPLL algorithms; this lower bounds are usually based on reductions to unsatisfiable instances. In this paper we consider DPLL algorithms with a cut heuristic that may decide that some branch of the splitting tree will not be investigated. DPLL algorithms with a cut heuristic always return correct answer on unsatisfiable formulas while they may err on satisfiable instances. We prove the theorem about effectiveness vs. correctness trade-off for deterministic myopic DPLL algorithms with cut heuristic. Myopic algorithms can see formulas with erased signs of negations; they may also request a small number of clauses to read them precisely. We construct a family of unsatisfiable formulas Φ(n) and a polynomial time samplable ensemble of distributions Qn concentrated on satisfiable formulas such that every deterministic myopic algorithm that gives a correct answer with probability 1−o(1) on a random formula from the ensemble Qn runs exponential time on the formulas Φ(n).

How overproduction of foreign proteins affects physiology of the recombinant strains ofHansenula polymorpha

Changes in the activity of key enzymes of the methanol utilization pathway of the recombinant strains of methylotrophic yeastHansenula polymorpha R22-2B and LAC-56 were studied at different rates of chemostat growth on methanol containing mineral media. It was shown that the strain R22-2B, initially having a 10-fold increased activity of dihydroxyacetone kinase (DHAK, a key enzyme of formaldehyde assimilation) acquired increased activity of formaldehyde dehydrogenase (FADH, a key enzyme of formaldehyde dissimilation) which resulted in the enhanced oxidation of formaldehyde to CO2. Strain LAC-56, overproducingEscherichia coli β-galactosidase, acquired the decreased intracellular concentration of ATP which resulted in the decrease of the efficiency of formaldehyde assimilation catalyzed by DHAK and resulted in accumulation of toxic formaldehyde. As a consequence some biochemical responses occurred in cells that were directed to a diminishing of the toxic effect of accumulated formaldehyde, namely, the decreasing of methanol oxidase activity (to reduce the rate of formaldehyde synthesis), and the increasing of FADH activity (to increase the rate of formaldehyde oxidation).

Lower Bounds for Splittings by Linear Combinations

A typical DPLL algorithm for the Boolean satisfiability problem splits the input problem into two by assigning the two possible values to a variable; then it simplifies the two resulting formulas. In this paper we consider an extension of the DPLL paradigm. Our algorithms can split by an arbitrary linear combination of variables modulo two. These algorithms quickly solve formulas that explicitly encode linear systems modulo two, which were used for proving exponential lower bounds for conventional DPLL algorithms.

Heuristic Time Hierarchies via Hierarchies for Sampling Distributions

We introduce a new framework for proving the time hierarchy theorems for heuristic classes. The main ingredient of our proof is a hierarchy theorem for sampling distributions recently proved by Watson [11]. Class \(\mathrm {Heur}_{\epsilon }{\mathbf {FBPP}}\) consists of functions with distributions on their inputs that can be computed in randomized polynomial time with bounded error on all except \(\epsilon \) fraction of inputs. We prove that for every a, \(\delta \) and integer k there exists a function \({F: \{0, 1\}^* \rightarrow \{0, 1, \dots , k - 1\}}\) such that \((F, U) \in \mathrm {Heur}_{\epsilon }{\mathbf {FBPP}}\) for all \(\epsilon > 0\) and for every ensemble of distributions \(D_n\) samplable in \(n^a\) steps, \((F, D) \notin \mathrm {Heur}_{1 - \frac{1}{k} - \delta }{\mathbf {FBPTime}}[n^a]\). This extends a previously known result for languages with uniform distributions proved by Pervyshev [9] by handling the case \(k > 2\). We also prove that \({\mathbf {P}}\not \subseteq \mathrm {Heur}_{\frac{1}{2} - \epsilon }{\mathbf {BPTime}}[n^k]\) if one-way functions exist.

Resolution complexity of perfect matching principles for sparse graphs

The resolution complexity of the perfect matching principle was studied by Razborov [Raz04], who developed a technique for proving its lower bounds for dense graphs. We construct a constant degree bipartite graph \(G_n\) such that the resolution complexity of the perfect matching principle for \(G_n\) is \(2^{\varOmega (n)}\), where n is the number of vertices in \(G_n\). This lower bound is tight up to some polynomial. Our result implies the \(2^{\varOmega (n)}\) lower bounds for the complete graph \(K_{2n+1}\) and the complete bipartite graph \(K_{n, O(n)}\) that improves the lower bounds following from [Raz04]. Our results also imply the well-known exponential lower bounds on the resolution complexity of the pigeonhole principle, the functional pigeonhole principle and the pigeonhole principle over a graph.

Monotone circuit lower bounds from resolution

For any unsatisfiable CNF formula F that is hard to refute in the Resolution proof system, we show that a gadget-composed version of F is hard to refute in any proof system whose lines are computed by efficient communication protocols—or, equivalently, that a monotone function associated with F has large monotone circuit complexity. Our result extends to monotone real circuits, which yields new lower bounds for the Cutting Planes proof system.

Reordering rule makes OBDD proof systems stronger

Atserias, Kolaitis, and Vardi showed that the proof system of Ordered Binary Decision Diagrams with conjunction and weakening, OBDD(∧, weakening), simulates CP* (Cutting Planes with unary coefficients). We show that OBDD(∧, weakening) can give exponentially shorter proofs than dag-like cutting planes. This is proved by showing that the Clique-Coloring tautologies have polynomial size proofs in the OBDD(∧, weakening) system. The reordering rule allows changing the variable order for OBDDs. We show that OBDD(∧, weakening, reordering) is strictly stronger than OBDD(∧, weakening). This is proved using the Clique-Coloring tautologies, and by transforming tautologies using coded permutations and orification. We also give CNF formulas which have polynomial size OBDD(∧) proofs but require superpolynomial (actually, quasipolynomial size) resolution proofs, and thus we partially resolve an open question proposed by Groote and Zantema. Applying dag-like and tree-like lifting techniques to the mentioned results, we completely analyze which of the systems among CP*, OBDD(∧), OBDD(∧, reordering), OBDD(∧, weakening) and OBDD(∧, weakening, reordering) polynomially simulate each other. For dag-like proof systems, some of our separations are quasipolynomial and some are exponential; for tree-like systems, all of our separations are exponential.

On OBDD-Based Algorithms and Proof Systems That Dynamically Change Order of Variables

In 2004 Atserias, Kolaitis and Vardi proposed OBDD-based propositional proof systems that prove unsatisfiability of a CNF formula by deduction of identically false OBDD from OBDDs representing clauses of the initial formula. All OBDDs in such proofs have the same order of variables. We initiate the study of OBDD based proof systems that additionally contain a rule that allows to change the order in OBDDs. At first we consider a proof system OBDD(and, reordering) that uses the conjunction (join) rule and the rule that allows to change the order. We exponentially separate this proof system from OBDD(and)-proof system that uses only the conjunction rule. We prove two exponential lower bounds on the size of OBDD(and, reordering)-refutations of Tseitin formulas and the pigeonhole principle. The first lower bound was previously unknown even for OBDD(and)-proofs and the second one extends the result of Tveretina et al. from OBDD(and) to OBDD(and, reordering). In 2004 Pan and Vardi proposed an approach to the propositional satisfiability problem based on OBDDs and symbolic quantifier elimination (we denote algorithms based on this approach as OBDD(and, exists)-algorithms. We notice that there exists an OBDD(and, exists)-algorithm that solves satisfiable and unsatisfiable Tseitin formulas in polynomial time. In contrast, we show that there exist formulas representing systems of linear equations over F_2 that are hard for OBDD(and, exists, reordering)-algorithms. Our hard instances are satisfiable formulas representing systems of linear equations over F_2 that correspond to some checksum matrices of error correcting codes.