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Register a new userResolution with Symmetry Rule applied to Linear Equations
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Added by
Constantin Seebach
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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This paper considers the length of resolution proofs when using Krishnamurthys classic symmetry rules. We show that inconsistent linear equation systems of bounded width over a fixed finite field $mathbb{F}_p$ with $p$ a prime have, in their standard encoding as CNFs, polynomial length resolutions when using the local symmetry rule (SRC-II). As a consequence it follows that the multipede instances for the graph isomorphism problem encoded as CNF formula have polynomial length resolution proofs. This contrasts exponential lower bounds for individualization-refinement algorithms on these graphs. For the Cai-Furer-Immerman graphs, for which Toran showed exponential lower bounds for resolution proofs (SAT 2013), we also show that already the global symmetry rule (SRC-I) suffices to allow for polynomial length proofs.
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We show that there are CNF formulas which can be refuted in resolution in both small space and small width, but for which any small-width proof must have space exceeding by far the linear worst-case upper bound. This significantly strengthens the space-width trade-offs in [Ben-Sasson 09]}, and provides one more example of trade-offs in the supercritical regime above worst case recently identified by [Razborov 16]. We obtain our results by using Razborovs new hardness condensation technique and combining it with the space lower bounds in [Ben-Sasson and Nordstrom 08].
The study of regular linear conjunctive normal form (LCNF) formulas is of interest because exact satisfiability (XSAT) is known to be NP-complete for this class of formulas. In a recent paper it was shown that the subclass of regular exact LCNF formulas (XLCNF) is of sub-exponential complexity, i.e. XSAT can be determined in sub-exponential time. Here I show that this class is just a subset of a larger class of LCNF formulas which display this very kind of complexity. To this end I introduce the property of disjointedness of LCNF formulas, measured, for a single clause C, by the number of clauses which have no variable in common with C. If for a given LCNF formula F all clauses have the same disjointedness d we call F d-disjointed and denote the class of such formulas by dLCNF. XLCNF formulas correspond to the special cased=0. One main result of the paper is that the class of all monotone l-regular LCNF formulas which are d-disjointed, with d smaller than some upper bound D, is of sub-exponential complexity. This result can be generalized to show that all monotone, l-regular LCNF formulas F which have a bounded mean disjointedness, are of sub-exponential XSAT-complexity, as well.
We consider non-linear stochastic field equations such as the KPZ equation for deposition and the noise driven Navier-Stokes equation for hydrodynamics. We focus on the Fourier transform of the time dependent two point field correlation, $Phi_{bf{k}}(t)$. We employ a Lagrangian method aimed at obtaining the distribution function of the possible histories of the system in a way that fits naturally with our previous work on the static distribution. Our main result is a non-linear integro-differential equation for $Phi_{bf{k}}(t)$, which is derived from a Peierls-Boltzmann type transport equation for its Fourier transform in time $Phi_{bf{k}, omega}$. That transport equation is a natural extension of the steady state transport equation, we previously derived for $Phi_{bf{k}}(0)$. We find a new and remarkable result which applies to all the non-linear systems studied here. The long time decay of $Phi_{bf{k}}(t)$ is described by $Phi_{bf{k}}(t) sim exp(-a|{bf k}|t^{gamma})$, where $a$ is a constant and $gamma$ is system dependent.
An automata network (AN) is a finite graph where each node holds a state from a finite alphabet and is equipped with a local map defining the evolution of the state of the node depending on its neighbors. The global dynamics of the network is then induced by an update scheme describing which nodes are updated at each time step. We study how update schemes can compensate the limitations coming from symmetric local interactions. Our approach is based on intrinsic simulations and universality and we study both dynamical and computational complexity. By considering several families of concrete symmetric AN under several different update schemes, we explore the edge of universality in this two-dimensional landscape. On the way, we develop a proof technique based on an operation of glueing of networks, which allows to produce complex orbits in large networks from compatible pseudo-orbits in small networks.
Many papers in the field of integer linear programming (ILP, for short) are devoted to problems of the type $max{c^top x colon A x = b,, x in mathbb{Z}^n_{geq 0}}$, where all the entries of $A,b,c$ are integer, parameterized by the number of rows of $A$ and $|A|_{max}$. This class of problems is known under the name of ILP problems in the standard form, adding the word bounded if $x leq u$, for some integer vector $u$. Recently, many new sparsity, proximity, and complexity results were obtained for bounded and unbounded ILP problems in the standard form. In this paper, we consider ILP problems in the canonical form $$max{c^top x colon b_l leq A x leq b_r,, x in mathbb{Z}^n},$$ where $b_l$ and $b_r$ are integer vectors. We assume that the integer matrix $A$ has the rank $n$, $(n + m)$ rows, $n$ columns, and parameterize the problem by $m$ and $Delta(A)$, where $Delta(A)$ is the maximum of $n times n$ sub-determinants of $A$, taken in the absolute value. We show that any ILP problem in the standard form can be polynomially reduced to some ILP problem in the canonical form, preserving $m$ and $Delta(A)$, but the reverse reduction is not always possible. More precisely, we define the class of generalized ILP problems in the standard form, which includes an additional group constraint, and prove the equivalence to ILP problems in the canonical form. We generalize known sparsity, proximity, and complexity bounds for ILP problems in the canonical form. Additionally, sometimes, we strengthen previously known results for ILP problems in the canonical form, and, sometimes, we give shorter proofs. Finally, we consider the special cases of $m in {0,1}$. By this way, we give specialised sparsity, proximity, and complexity bounds for the problems on simplices, Knapsack problems and Subset-Sum problems.
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