Alexander Knop

On the Limits of Gate Elimination.

Although a simple counting argument shows the existence of Boolean functions of exponential circuit complexity, proving superlinear circuit lower bounds for explicit functions seems to be out of reach of the current techniques. There has been a (very slow) progress in proving linear lower bounds with the latest record of 3 1/86*n-o(n). All known lower bounds are based on the so-called gate elimination technique. A typical gate elimination argument shows that it is possible to eliminate several gates from an optimal circuit by making one or several substitutions to the input variables and repeats this inductively. In this note we prove that this method cannot achieve linear bounds of cn beyond a certain constant c, where c depends only on the number of substitutions made at a single step of the induction.

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.

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 the limits of gate elimination

Abstract Although a simple counting argument shows the existence of Boolean functions of exponential circuit complexity, proving superlinear circuit lower bounds for explicit functions seems to be out of reach of the current techniques. There has been a (very slow) progress in proving linear lower bounds with the latest record of 3 1 86 n − o ( n ) . All known lower bounds are based on the so-called gate elimination technique. A typical gate elimination argument shows that it is possible to eliminate several gates from a circuit by making one or several substitutions to the input variables and repeats this inductively. In this paper we prove that this method cannot achieve linear bounds of cn beyond a certain constant c, where c depends only on the number of substitutions made at a single step of the induction.

Hard Satisfiable Formulas for Splittings by Linear Combinations

Itsykson and Sokolov in 2014 introduced the class of \(\mathrm {DPLL}(\oplus )\) algorithms that solve Boolean satisfiability problem using the splitting by linear combinations of variables modulo 2. This class extends the class of \(\mathrm {DPLL}\) algorithms that split by variables. \(\mathrm {DPLL}(\oplus )\) algorithms solve in polynomial time systems of linear equations modulo 2 that are hard for \(\mathrm {DPLL}\), \(\mathrm {PPSZ}\) and \(\mathrm {CDCL}\) algorithms. Itsykson and Sokolov have proved first exponential lower bounds for \(\mathrm {DPLL}(\oplus )\) algorithms on unsatisfiable formulas.

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.

Circuit Lower Bounds for Average-Case MA

Santhanam (2007) proved that \(\mathbf {MA}/1\) does not have circuits of size \(n^k\). We translate his result to the average-case setting by proving that there is a constant a such that for any k, there is a language in \(\mathrm {Avg}_{ }\mathbf {MA}\) that cannot be solved by circuits of size \(n^k\) on more than the \(1 - \frac{1}{n^a}\) fraction of inputs.