Related papers: On variables with few occurrences in conjunctive n…

We investigate connections between SAT (the propositional satisfiability problem) and combinatorics, around the minimum degree (number of occurrences) of variables in various forms of redundancy-free boolean conjunctive normal forms…

This is the report-version of a mini-series of two articles on the foundations of satisfiability of conjunctive normal forms with non-boolean variables, to appear in Fundamenta Informaticae, 2011. These two parts are here bundled in one…

Starting with Aharoni and Linial in 1986, the deficiency delta(F) = c(F) - n(F) >= 1 for minimally unsatisfiable clause-sets F, the difference of the number of clauses and the number of variables, is playing an important role in…

The topic of this paper is the Finiteness Conjecture for minimally unsatisfiable clause-sets (MUs), stating that for each fixed deficiency (number of clauses minus number of variables) there are only finitely many patterns, given a certain…

We establish a new bridge between propositional logic and elementary number theory. The main objects are "minimally unsatisfiable clause-sets", short "MUs", unsatisfiable conjunctive normal forms rendered satisfiable by elimination of any…

A pair of unit clauses is called conflicting if it is of the form $(x)$, $(\bar{x})$. A CNF formula is unit-conflict free (UCF) if it contains no pair of conflicting unit clauses. Lieberherr and Specker (J. ACM 28, 1981) showed that for…

We determine the thresholds for the number of variables, number of clauses, number of clause intersection pairs and the maximum clause degree of a k-CNF formula that guarantees satisfiability under the assumption that every two clauses…

We aim at providing a foundation of a theory of "good" SAT representations F of boolean functions f. We argue that the hierarchy UC_k of unit-refutation complete clause-sets of level k, introduced by the authors, provides the most basic…

We study the non-canonical method for solving the Satisfiability problem which given by a formula in the form of the conjunctive normal form. The essence of this method consists in counting the number of tuples of Boolean variables, on…

We call a CNF formula linear if any two clauses have at most one variable in common. Let m(k) be the largest integer m such that any linear k-CNF formula with <= m clauses is satisfiable. We show that 4^k / (4e^2k^3) <= m(k) < ln(2) k^4…

For investigations into the structure of MU, i.e., minimally unsatisfiable clause-sets or conjunctive normal forms, singular DP-reduction is a fundamental tool, applying DP-reduction F -> DP_v(F) in case variable v occurs in one polarity…

The Frankl conjecture, also known as the union-closed sets conjecture, states that in any finite non-empty union-closed family, there exists an element in at least half of the sets. From an optimization point of view, one could instead…

We call a CNF formula linear if any two clauses have at most one variable in common. Let Linear k-SAT be the problem of deciding whether a given linear k-CNF formula is satisfiable. Here, a k-CNF formula is a CNF formula in which every…

For random CNF formulae with m clauses, n variables and an unrestricted number of literals per clause the transition from high to low satisfiability can be determined exactly for large n. The critical density m/n turns out to be strongly…

A boolean formula in a conjuctive normal form is called a (k,s)-formula if every clause contains exactly k variables and every variable occurs in at most s clauses. We prove the existence of a (k, 4 * (2^k/k))-CNF formula which is…

Maximum satisfiability is a canonical NP-hard optimization problem that appears empirically hard for random instances. Let us say that a Conjunctive normal form (CNF) formula consisting of $k$-clauses is $p$-satisfiable if there exists a…

Sparse representation over redundant dictionaries constitutes a good model for many classes of signals (e.g., patches of natural images, segments of speech signals, etc.). However, despite its popularity, very little is known about the…

The minimum number of clauses in a CNF representation of the parity function $x_1 \oplus x_2 \oplus \dotsb \oplus x_n$ is $2^{n-1}$. One can obtain a more compact CNF encoding by using non-deterministic variables (also known as guess or…

A superredundant clause is a clause that is redundant in the resolution closure of a formula. The converse concept of superirredundancy ensures membership of the clause in all minimal CNF formulae that are equivalent to the given one. This…

We consider several novel aspects of unique factorization in formal languages. We reprove the familiar fact that the set uf(L) of words having unique factorization into elements of L is regular if L is regular, and from this deduce an…