Quantifier elimination by dependency sequents

We consider the problem of existential quantifier elimination for Boolean CNF formulas. We present a new method for solving this problem called derivation of dependency-sequents (DDS). A dependency-sequent (D-sequent) is used to record that a set of variables is redundant under a partial assignment. We introduce the join operation that produces new D-sequents from existing ones. We show that DDS is compositional, i.e., if our input formula is a conjunction of independent formulas, DDS automatically recognizes and exploits this information. We introduce an algorithm based on DDS and present experimental results demonstrating its potential.     

A sound and complete R-calculi with respect to contraction and minimal change

AGM postulates are for belief revision (revision by a single belief), and DP postulates are for iterated revision (revision by a finite sequence of beliefs). R-calculus is given for R-configurations Δ|Γ, where Δ is a set of atomic formulas or the negations of atomic formulas, and Γ is a finite set of formulas. We shall give two R-calculi C and M (sets of deduction rules) such that for any finite consistent sets Γ, Δ of formulas in the propositional logic, there is a consistent set Θ ? Γ of formulas such that Δ|Γ ? Δ, Θ is provable and Θ is a contraction of Γ by Δ or a minimal change of Γ by Δ; and prove that C and M are sound and complete with respect to the contraction and the minimal change, respectively.     

Algebraic proofs over noncommutative formulas

We study possible formulations of algebraic propositional proof systems operating with noncommutative formulas. We observe that a simple formulation gives rise to systems at least as strong as Frege, yielding a semantic way to define a Cook-Reckhow (i.e., polynomially verifiable) algebraic analog of Frege proofs, different from that given in Buss et al. (1997) and Grigoriev and Hirsch (2003). We then turn to an apparently weaker system, namely, polynomial calculus (PC) where polynomials are written as ordered formulas (PC over ordered formulas, for short). Given some fixed linear order on variables, an arithmetic formula is ordered if for each of its product gates the left subformula contains only variables that are less-than or equal, according to the linear order, than the variables in the right subformula of the gate. We show that PC over ordered formulas (when the base field is of zero characteristic) is strictly stronger than resolution, polynomial calculus and polynomial calculus with resolution (PCR), and admits polynomial-size refutations for the pigeonhole principle and Tseitin?s formulas. We conclude by proposing an approach for establishing lower bounds on PC over ordered formulas proofs, and related systems, based on properties of lower bounds on noncommutative formulas (Nisan, 1991).The motivation behind this work is developing techniques incorporating rank arguments (similar to those used in arithmetic circuit complexity) for establishing lower bounds on propositional proofs.     

Post-quantum signature algorithms on noncommutative algebras,using difficulty of solving systems of quadratic equations

A recently proposed new concept for constructing algebraic signature schemes with a hidden group is used to develop two new post-quantum signature algorithms on four-dimensional and six-dimensional finite noncommutative associative algebras. As in the case of the post-quantum signature schemes of multivariate cryptography, the security of the introduced algorithms is based on the computational difficulty of solving systems of many quadratic equations (44 and 42) with many unknowns (40 and 36). The signature represents a pair of a natural number e and a vector S. The latter enters three times in the verification equation used, providing resistance to the forging signature attacks by using the value S as a fitting parameter. A public key is generated in the form of a set of vectors, each of which is calculated as the product of triples of secret vectors. With a special choice of these triples, a signer has the possibility of calculating a signature that satisfies the verification equation. The developed post-quantum signature algorithms are practical, having sufficiently small signatures (97 and 109 bytes), public keys (387 and 291 bytes), and secret keys (315 and 451 bytes). A significant difference from the public-key algorithms of multivariate cryptography is that in the developed signature schemes, the system of quadratic equations is derived from the formulas for generating the public-key elements in the form of a set of vectors of m-dimensional finite noncommutative algebra with an associative vector multiplication operation. The formulas define the system of n quadratic vector equations, which reduces to the system of m quadratic equations over a finite field.     

Learning a subclass of k-quasi-Horn formulas with membership queries

Boolean formulas have been widely studied in the field of learning theory. We focus on the model of learning with queries, and study a restriction of the class of k-quasi-Horn formulas, that is, conjunctive normal form formulas where the number of unnegated literals per clause is at most k. This class is known to be as hard to learn as the general class CNF of conjunctive normal form formulas. By imposing some constraints, we define a fragment of this logic that can be learned using only membership queries. Also we prove that none of these constraints makes by itself the class learnable.     

A practical approach to model checking Duration Calculus using Presburger Arithmetic

This paper investigates the feasibility of reducing a model-checking problem K???? for discrete time Duration Calculus to the decision problem for Presburger Arithmetic. Theoretical results point at severe limitations of this approach: (1) the reduction in Fränzle and Hansen (Int J Softw Inform 3(2–3):171–196, 2009) produces Presburger formulas whose sizes grow exponentially in the chop-depth of ?, where chop is an interval modality originating from Moszkowski (IEEE Comput 18(2):10–19, 1985), and (2) the decision problem for Presburger Arithmetic has a double exponential lower bound and a triple exponential upper bound. The generated Presburger formulas have a rich Boolean structure, many quantifiers and quantifier alternations. Such formulas are simplified using so-called guarded formulas, where a guard provides a context used to simplify the rest of the formula. A normal form for guarded formulas supports global effects of local simplifications. Combined with quantifier-elimination techniques, this normalization gives significant reductions in formula sizes and in the number of quantifiers. As an example, we solve a configuration problem using the SMT-solver Z3 as backend. Benefits and the current limits of the approach are illustrated by a family of examples.     

Logic of knowledge and discovery via interacting agents - Decision algorithm for true and satisfiable statements

In this paper we study logical properties of the operation chance discovery (CD) via structures based on special Kripke/Hintikka models. These models use as bases partially ordered sets of indexes (indexes of steps in a computation, or ones indicating time points in a time flow), and clusters of states associated to each index. The language chosen to build the logical formulas includes modal/temporal operations, operations for the agents’ knowledge, local and global operations for CD, operation of local common knowledge, and an operation for chance of discovery via agents’ interactions. We introduce and study a logic (of knowledge and discovery via interaction of agents), LDK_a, which is defined by semantics, as the set of all formulas that are valid in all suggested models. The paper provides an algorithm to recognize logical laws (and satisfiable formulas) of LDK_a. The algorithm replaces a formula with a rule in a special, so-called reduced normal form, and, then it verifies the validity of this rule in specific models of exponential size in the size of the rule. We show that the problem of computing the true logical laws of LDK_a is decidable.     

Theory of truth degrees of propositions in the logic system L n *

By means of infinite product of uniformly distributed probability spaces of cardinal n the concept of truth degrees of propositions in the n-valued generalized Lukasiewicz propositional logic system L _n ^* is introduced in the present paper. It is proved that the set consisting of truth degrees of all formulas is dense in [0, 1], and a general expression of truth degrees of formulas as well as a deduction rule of truth degrees is then obtained. Moreover, similarity degrees among formulas are proposed and a pseudo-metric is defined therefrom on the set of formulas, and hence a possible framework suitable for developing approximate reasoning theory in n-valued generalized Lukasiewicz propositional logic is established.     

Exact learning of DNF formulas using DNF hypotheses

We show the following: (a) For any ε>0, log^(3+ε)n-term DNF cannot be polynomial-query learned with membership and strongly proper equivalence queries. (b) For sufficiently large t, t-term DNF formulas cannot be polynomial-query learned with membership and equivalence queries that use t^1+ε-term DNF formulas as hypotheses, for some ε<1 (c) Read-thrice DNF formulas are not polynomial-query learnable with membership and proper equivalence queries. (d) logn-term DNF formulas can be polynomial-query learned with membership and proper equivalence queries. (This complements a result of Bshouty, Goldman, Hancock, and Matar that -term DNF can be so learned in polynomial time.)Versions of (a)-(c) were known previously, but the previous versions applied to polynomial-time learning and used complexity theoretic assumptions. In contrast, (a)-(c) apply to polynomial-query learning, imply the results for polynomial-time learning, and do not use any complexity-theoretic assumptions.     

Convergence of quadratures for Cauchy principal value integrals

Quadrature formulas based on the “practical” abscissasx _k=cos(k π/n),k=0(1)n, are obtained for the numerical evaluation of the weighted Cauchy principal value integrals $$\mathop {\rlap{--} \smallint }\limits_{ - 1}^1 (1 - x)^\alpha (1 + x)^\beta (f(x))/(x - a)){\rm E}dx,$$ where α,β>?1 andaε(?1, 1). An interesting problem concerning these quadrature formulas is their convergence for a suitable class of functions. We establish convergence of these quadrature formulas for the class of functions which are Hölder-continuous on [?1, 1].