Unsatisfiable Formulas Research Articles

Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable

Recognition of minimal unsatisfiable CNF formulas (unsatisfiable CNF formulas which become satisfiable if any clause is removed) is a classical D P -complete problem. It was shown recently that minimal unsatisfiable formulas with n variables and n+ k clauses can be recognized in time n O(k) . We improve this result and present an algorithm with time complexity O(2 kn 4) ; hence the problem turns out to be fixed-parameter tractable (FTP) in the sense of Downey and Fellows (Parameterized Complexity, 1999). Our algorithm gives rise to a fixed-parameter tractable parameterization of the satisfiability problem: If for a given set of clauses F, the number of clauses in each of its subsets exceeds the number of variables occurring in the subset at most by k, then we can decide in time O(2 kn 3) whether F is satisfiable; k is called the maximum deficiency of F and can be efficiently computed by means of graph matching algorithms. Known parameters for fixed-parameter tractable satisfiability decision are tree-width or related to tree-width. Tree-width and maximum deficiency are incomparable in the sense that we can find formulas with constant maximum deficiency and arbitrarily high tree-width, and formulas where the converse prevails.

On the structure of some classes of minimal unsatisfiable formulas

We investigate classes of minimal unsatisfiable formulas which are closed under splitting. For marginal formulas the equivalence to some natural classes of formulas is proved. Further, we show that maximal formulas are closely related to the so-called hitting formulas. That are formulas for which any two clauses contain a pair of complementary literals.

Homomorphisms of conjunctive normal forms

We study homomorphisms of propositional formulas in CNF generalizing symmetries considered by Krishnamurthy. If ϕ : H→F is a homomorphism, then unsatisfiability of H implies unsatisfiability of F. Homomorphisms from F to a subset F′ of F (endomorphisms) are of special interest, since in such cases F and F′ are satisfiability-equivalent. We show that the smallest subsets F′ of a formula F for which an endomorphism F→ F′ exists are mutually isomorphic. Furthermore, we study connections between homomorphisms and autark assignments. We introduce the concept of “proof by homomorphism” which is based on the observation that there exist sets Γ of unsatisfiable formulas such that (i) formulas in Γ can be recognized in polynomial time, and (ii) for every unsatisfiable formula F there exist some H∈ Γ and a homomorphism ϕ : H→F . We identify several sets Γ of unsatisfiable formulas satisfying (i) and (ii) for which proofs by homomorphism w.r.t. Γ and tree resolution proofs can be simulated by each other in polynomial time.

Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference

A formula (in conjunctive normal form) is said to be minimal unsatisfiable if it is unsatisfiable and deleting any clause makes it satisfiable. The deficiency of a formula is the difference of the number of clauses and the number of variables. It is known that every minimal unsatisfiable formula has positive deficiency. Until recently, polynomial-time algorithms were known to recognize minimal unsatisfiable formulas with deficiency 1 and 2. We state an algorithm which recognizes minimal unsatisfiable formulas with any fixed deficiency in polynomial time.

Polynomial time algorithms for computing a representation for minimal unsatisfiable formulas with fixed deficiency

We say a propositional formula F in conjunctive normal form is represented by a formula H and a homomorphism φ, if φ( H)= F. A homomorphism is a mapping consisting of a renaming and an identification of literals. The deficiency of a formula is the difference between the number of clauses and the number of variables. We show that for fixed k⩾1 and t⩾1 each minimal unsatisfiable formula with deficiency k can be represented by a formula H with deficiency t and a homomorphism and such a representation can be computed in polynomial time.

Partial Instantiation Methods for Inference in First-Order Logic

Satisfiability algorithms for propositional logic have improved enormously in recently years. This improvement increases the attractiveness of satisfiability methods for first-order logic that reduce the problem to a series of ground-level satisfiability problems. R. Jeroslow introduced a partial instantiation method of this kind that differs radically from the standard resolution-based methods. This paper lays the theoretical groundwork for an extension of his method that is general enough and efficient enough for general logic programming with indefinite clauses. In particular we improve Jeroslow's approach by (1) extending it to logic with functions, (2) accelerating it through the use of satisfiers, as introduced by Gallo and Rago, and (3) simplifying it to obtain further speedup. We provide a similar development for a “dual” partial instantiation approach defined by Hooker and suggest a primal–dual strategy. We prove correctness of the primal and dual algorithms for full first-order logic with functions, as well as termination on unsatisfiable formulas. We also report some preliminary computational results.

The Complexity of Read-Once Resolution

We investigate the complexity of deciding whether a propositional formula has a read-once resolution proof. We give a new and general proof of Iwama–Miynano's theorem which states that the problem whether a formula has a read-once resolution proof is i>NP-complete. Moreover, we show for fixed i>kg2 that the additional restriction that in each resolution step one of the parent clauses is a i>k-clause preserves the i>NP-completeness. If we demand that the formulas are minimal unsatisfiable and read-once refutable then the problem remains i>NP-complete. For the subclasses i>MU(i>k) of minimal unsatisfiable formulas we present a pol-time algorithm deciding whether a i>MU(i>k)-formula has a read-once resolution proof. Furthermore, we show that the problems whether a formula contains a i>MU(i>k)-subformula or a read-once refutable i>MU(i>k)-subformula are i>NP-complete.

The Efficiency of Resolution and Davis--Putnam Procedures

We consider several problems related to the use of resolution-based methods for determining whether a given boolean formula in conjunctive normal form is satisfiable. First, building on the work of Clegg, Edmonds, and Impagliazzo in [Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, Philadelphia, PA, 1996, ACM, New York, 1996, pp. 174--183], we give an algorithm for unsatisfiability that when given an unsatisfiable formula of F finds a resolution proof of F. The runtime of our algorithm is subexponential in the size of the shortest resolution proof of F. Next, we investigate a class of backtrack search algorithms for producing resolution refutations of unsatisfiability, commonly known as Davis--Putnam procedures, and provide the first asymptotically tight average-case complexity analysis for their behavior on random formulas. In particular, for a simple algorithm in this class, called ordered DLL, we prove that the running time of the algorithm on a randomly generated k-CNF formula with n variables and m clauses is $2^{\Theta(n(n/m)^{1/(k-2)})}$ with probability $1-o(1)$. Finally, we give new lower bounds on $\mbox{res}(F)$, the size of the smallest resolution refutation of F, for a class of formulas representing the pigeonhole principle and for randomly generated formulas. For random formulas, Chvatal and Szemeredi [J. ACM, 35 (1988), pp. 759--768] had shown that random 3-CNF formulas with a linear number of clauses require exponential size resolution proofs, and Fu [On the Complexity of Proof Systems, Ph. D. thesis, University of Toronto, Toronto, ON, Canada, 1995] extended their results to k-CNF formulas. These proofs apply only when the number of clauses is $\Omega(n \log n)$. We show that a lower bound of the form $2^{n^{\gamma}}$ holds with high probability even when the number of clauses is $n^{(k+2)/4-\epsilon}$.

Space Bounds for Resolution

We introduce a new way to measure the space needed in resolution refutations of CNF formulas in propositional logic. With the former definition (1994, B. H. Kleine and T. Lettman, “Aussangenlogik: Deduktion und Algorithmen, Teubner, Stuttgart) the space required for the resolution of any unsatisfiable formula in CNF is linear in the number of clauses. The new definition allows a much finer analysis of the space in the refutation, ranging from constant to linear space. Moreover, the new definition allows us to relate the space needed in a resolution proof of a formula to other well-studied complexity measures. It coincides with the complexity of a pebble game in the resolution graphs of a formula and, as we show, has relationships to the size of the refutation. We also give upper and lower bounds on the space needed for the resolution of unsatisfiable formulas. We show that Tseitin formulas associated to a certain kind of expander graphs of n nodes need resolution space n−c for some constant c. Measured on the number of clauses, this result is the best possible. We also show that the formulas expressing the general pigeonhole principle with n holes and more than n pigeons need space n+1 independent of the number of pigeons. Since a matching space upper bound of n+1 for these formulas exists, the obtained bound is exact. We also point to a possible connection between resolution space and resolution width, another measure for the complexity of resolution refutations.

Restoring Satisfiability or Maintaining Unsatisfiability by finding small Unsatisfiable Subformulae

In several applicative fields, the generic system or structure to be designed can be encoded as a CNF formula, which should have a well-defined satisfiability property (either to be satisfiable or to be unsatisfiable). Within a complete solution framework, we develop an heuristic procedure which is able, for unsatisfiable instances, to locate a set of clauses causing unsatisfiability. That corresponds to the part of the system that we respectively need to re-design or to keep when we respectively want a satisfiable or unsatisfiable formula. Such procedure can guarantee to find an unsatisfiable subformula, and is aimed to find an approximation of a minimum unsatisfiable subformula. Successful results on both real life data collecting problems and Dimacs problems are presented.

On subclasses of minimal unsatisfiable formulas

We consider the minimal unsatisfiablity problem MU( k) for propositional formulas in conjunctive normal form (CNF) over n variables and n+ k clauses, where k is fixed. k is called the difference. Any formula in MU( k) can be split into two minimal unsatisfiable formula. For such splittings we investigate the size of the differences of the resulting formulas in comparison to the difference of the initial formula. Based on these results we prove that MU( k) for fixed k is in NP, and for MU(2) we present a simple and unique characterization.

The proof complexity of analytic and clausal tableaux

It is widely believed that a family Σ n of unsatisfiable formulae proposed by Cook and Reckhow in their landmark paper (Proc. ACM Symp. on Theory of Computing, 1974) can be used to give a lower bound of 2 Ω(2 n) on the proof size with analytic tableaux. This claim plays a key role in the proof that tableaux cannot polynomially simulate tree resolution. We exhibit an analytic tableau proof for Σ n for whose size we prove an upper bound of O(2 n 2 ) , which, although not polynomial in the size O(2 n) of the input formula, is exponentially shorter than the claimed lower bound. An analysis of the proofs published in the literature reveals that the pitfall is the blurring of n-ary (clausal) and binary versions of tableaux. A consequence of this analysis is that a second widely held belief falls too: clausal tableaux are not just a more efficient notational variant of analytic tableaux for formulae in clausal normal form. Indeed clausal tableaux (and thus model elimination without lemmaizing) cannot polynomially simulate analytic tableaux.

An efficient algorithm for the minimal unsatisfiability problem for a subclass of CNF

We consider the minimal unsatisfiability problem for propositional formulas over n variables with n+k clauses for fixed k. We will show that in case of at most n clauses no formula is minimal unsatisfiable. For n+1 clauses the minimal unsatisfiability problem is solvable in quadratic time. Further, we present a characterization of minimal unsatisfiable formulas with n+1 clauses in terms of a certain form of matrices.

The maximum length of prime implicates for instances of 3-SAT

Schrag and Crawford (1996) present strong experimental evidence that the occurrence of prime implicates of varying lengths in random instances of 3-SAT exhibits behaviour similar to the well-known phase transition phenomenon associated with satisfiability. Thus, as the ratio of number of clauses ( m) to number of prepositional variables ( n) increases, random instances of 3-SAT progress from formulae which are generally satisfiable through to formulae which are generally not satisfiable, with an apparent sharp threshold being crossed when m/n ∼ 4.2. For instances of 3-SAT, Schrag and Crawford (1996) examine with what probability the longest prime implicate has length k (for k ⩾ 0)—unsatisfiable formulae correspond to those having only a prime implicate of length 0—demonstrating that similar behaviour arises. It is observed by Schrag and Crawford (1996) that experiments failed to identify any instance of 3-SAT over nine propositional variables having a prime implicate of length 7 or greater, and it is conjectured that no such instances are possible. In this note we present a combinatorial argument establishing that no 3-SAT instance on n variables can have a prime implicate whose length exceeds max {[n/2] + 1, [2n/3]}, validating this conjecture for the case n = 9. We further show that these bounds are the best possible. An easy corollary of the latter constructions is that for all k > 3, instances of k-SAT on n variables can be formed, that have prime implicates of length n − o(n).

Decomposition of tautologies into regular formulas and strong completeness of connection-graph resolution

This paper addresses and answers a fundamental question about resolution. Informally, what is gained with respect to the search for a proof by performing a single resolution step? It is first shown that any unsatisfiable formula may be decomposed into regular formulas provable in linear time (by resolution). A relevant resolution step strictly reduces at least one of the formulas in the decomposition while an irrelevant one does not contribute to the proof in any way. the relevance of this insight into the nature of resolution and of the unsatisfiability problem for the development of proof strategies and for complexity considerations are briefly discussed. The decomposition also provides a technique for establishing completeness proofs for refinements of resolution. As a first application, connection-graph resolution is shown to be strongly complete. This settles a problem that remained open for two decades despite many proff attempts. The result is relevant for theorem proving because without strong completeness a connection graph resolution prover might run into an infinite loop even on the ground level.

A Threshold for Unsatisfiability

A propositional formula is in 2-CNF (2-conjunctive normalform) iff it is the conjunction of clauses each of which has exactly two literals. We show: IfC=1+ε, whereε>0 is fixed andq(n)⩾C·n, then almost all formulas in 2-CNF withq(n) different clauses, wherenis the number of variables, are unsatisfiable. IfC=1−εandq(n)⩽C·n, then almost all formulas withq(n) clauses are satisfiable. By “almost all” we mean that the probability of the set of unsatisfiable or satisfiable formulas among all formulas withq(n) clauses approaches 1 asn→∞. SoC=1 gives us a threshold separating satisfiability and unsatisfiability of formulas in 2-CNF in a probabilistic, asymptotic sense. To prove our result we translate the satisfiability problem for formulas in 2-CNF into a graph theoretical question. Then we apply techniques from the theory of random graphs.

The global properties of valid formulas in modal logic K

Global property is the necessary condition which must be satisfied by the provable formulas. It can help to find out some unprovable formula that does not satisfy some global property before proving it using formal automated reasoning systems, thus the efficiency of the whole system is improved. This paper presents some global properties of valid formulas in modal logic K. Such properties are structure characters of formulas, so they are simple and easy to check. At the same time, some global properties of K unsatisfiable formula set are also given.

Regular Resolution Versus Unrestricted Resolution

A resolution proof of an unsatisfiable propositional formula is called regular if and only if no variable is eliminated (with the resolution rule) twice on any branch of the proof tree representing the resolution proof. An infinite family of unsatisfiable propositional formulas is constructed and the following shown: These formulas have polynomial size unrestricted resolution proofs, whereas all regular resolution proofs of these formulas are of superpolynomial length.

On generalized Horn formulas and k-resolution

A hierarchy of classes H k of propositional formulas has been introduced by Yamasaki and Doshita (1983) and Gallo and Scutella (1988). The classes H k have polynomially solvable satisfiability problem, and the basic class is the set of Horn formulas. We show that the k-resolution is complete and sound for H k , where k-resolution is a restriction of ordinary resolution, in that at least one of the parent clauses has at most k literals. Furthermore, we discuss to what extent the class of unsatisfiable formulas in H k and the class of unsatisfiable formulas for which a k-resolution refutation exists coincide.

Counting classes: thresholds, parity, mods, and fewness

Counting classes consist of languages defined in terms of the number of accepting computations of nondeterministic polynomial-time Turing machines. Well-known examples of counting classes are NP, co-NP, ⊕ P, and PP. Every counting class is a subset of P #P[1], the class of languages computable in polynomial time using a single call to an oracle capable of determining the number of accepting paths of an NP machine. Using closure properties of #P, we systematically develop a complexity theory for counting classes defined in terms of thresholds and moduli. An unexpected result is that MOD k i P = MOD k P for prime k. Finally, we improve a result of Cai and Hemachandra (1990) by showing that recognizing languages in the class Few is as easy as distinguishing uniquely satisfiable formulas from unsatisfiable formulas (or detecting unique solutions, as in Valiant and Vazirani (1986)).