Satisfiability Of Formulas Research Articles

Decidability of ∀*∀‐Sentences in Membership Theories

AbstractThe problem is addressed of establishing the satisfiability of prenex formulas involving a single universal quantifier, in diversified axiomatic set theories. A rather general decision method for solving this problem is illustrated through the treatment of membership theories of increasing strength, ending with a subtheory of Zermelo‐Fraenkel which is already complete with respect to the ∀*∀ class of sentences. NP‐hardness and NP‐completeness results concerning the problems under study are achieved and a technique for restricting the universal quantifier is presented.Mathematics Subject Classification: 03B25, 03E30.

Convergence properties of optimization algorithms for the SAT problem

The satisfiability (SAT) problem is a basic problem in computing theory. Presently, an active area of research on SAT problem is to design efficient algorithms to find a solution for a satisfiable conjunctive normal form (CNF) formula. A new formulation, the universal SAT problem model, which transforms the SAT problem on Boolean space into an optimization problem on real space has been developed (J. Gu, 1988; 1992; 1994). Many optimization techniques, such as the steepest descent method, Newton's method, and the coordinate descent method, can be used to solve the universal SAT problem. We prove that when the initial solution is sufficiently close to the optimal solution, the steepest descent method has a linear convergence ratio /spl beta/<1, Newton's method has a convergence ratio of order two, and the convergence ratio of the steepest descent method is approximately (1-/spl beta//m) for the universal SAT problem with m variables. An algorithm based on the coordinate descent method for the universal SAT problem is also presented. Experimental results show that this algorithm is more efficient than some previous ones in finding a solution for certain classes of the satisfiable CNF formulas.

Randomness-Optimal Unique Element Isolation with Applications to Perfect Matching and Related Problems

In this paper, we precisely characterize the randomness complexity of the unique element isolation problem, a crucial step in the $RNC$ algorithm for perfect matching due to Mulmuley, Vazirani, and Vazirani [Combinatorica, 7 (1987), pp. 105--113] and in several other applications. Given a set $S$ and an unknown family ${\cal F} \subseteq 2^S$ with $|{\cal F}| \leq Z$, we present a scheme to assign polynomially bounded weights to the elements of $S$, using only $O(\log Z + \log |S|)$ random bits, such that the minimum weight set in $\cal F$ is unique with high probability. This generalizes the solution of Mulmuley, Vazirani, and Vazirani who use $O(S\log S)$ bits, independent of $Z$. We also provide a matching lower bound for the randomness complexity of this problem. The new weight assignment scheme yields a randomness-efficient $RNC^2$ algorithm for perfect matching which uses $O(\log Z + \log n)$ random bits where $Z$ is any given upper bound on the number of perfect matchings in the input graph. This generalizes the result of Grigoriev and Karpinski [Proc. IEEE Symposium on Foundations of Computer Science, 1987, pp. 166--172], who present an $NC^3$ algorithm when $Z$ is polynomial and also improves the running time in this case. The worst-case randomness complexity of our algorithm is $O(n \log(m/n))$ random bits improving on the previous bound of $O(m \log n)$. Our scheme also gives randomness-efficient solutions for several problems where unique element isolation is used, such as $RNC$ algorithms for variants of matching and basic problems on linear matroids. We obtain a randomness-efficient random reduction from $SAT$ to $USAT$, the language of uniquely satisfiable formulas which can be derandomized in the case of languages in $FewP$ to yield new proofs of the results $FewP\subseteq \oplus P$ and $FewP\subseteq C_{\!\!=}P$.

The computational complexity of the satisfiability of modal Horn clauses for modal propositional logics

This paper presents complexity results about the satisfiability of modal Horn clauses for several modal propositional logics. Almost all these results are negative in the sense that restricting the input formula to modal Horn clauses does not decrease the inherent complexity of the satisfiability problem. We first show that, when restricted to modal Horn clauses, the satisfiability problem for any modal logic clauses as well as the satisfiability of unrestricted formulas for any of K, T, B, ans S4 is PSPACE-complete. This result refutes the expectation (Fariñas del Cerro and Penttonen 1987) of getting a polynomial-time algorithm for the satisfiability of modal Horn clauses for these logics as long as P ≠ PSPACE. Next, we consider S4.3 and extensions of K5 including K5, KD5, K45, KD45 and S5, the satisfiability problem for each of which in general is known to be NP-complete, and show that for each extension of K5, a polynomial-time algorithm for the satisfiability of modal Horn clauses can be obtained; but for S4.3, together with some linear tense logics closely related to S4.3 like Cl, SL and PL, the satisfiability of modal Horn clauses still remains NP-complete.

Model checking for action-based logics

A model checker is described that supports proving logical properties of concurrent systems. The logical properties can be described in different action-based logics (variants of Hennessy-Milner logic). The tools is based on the EMC model checker for the logic CTL. It therefore employs a set of translation functions from the considered logics to CTL, as well as a model translation function from labeled transition systems (models of the action-based logics) to Kripke structures (models for CTL). The obtained tool performs model checking in linear time complexity, and its correctness is guaranteed by the proof that the set of translation functions, coupled with the model translation function, preserves satisfiability of logical formulae.

Random models and solvable Skolem classes

A Skolem class is a class of formulas of pure quantification theory in Skolem normal form: closed, prenex formulas with prefixes ∀…∀∃…∃. (Pure quantification theory contains quantifiers, truth-functions, and predicate letters, but neither the identity sign nor function letters.) The Gödel Class, in which the number of universal quantifiers is limited to two, was shown effectively solvable (for satisfiability) sixty years ago [G1]. Various solvable Skolem classes that extend the Gödel Class can be obtained by allowing more universal quantifiers but restricting the combinations of variables that may occur together in atomic subformulas [DG, Chapter 2]. The Gödel Class and these extensions are also finitely controllable, that is, every satisfiable formula in them has a finite model. In this paper we isolate a model-theoretic property that connects the usual solvability proofs for these classes and their finite controllability. For formulas in the solvable Skolem classes, the property is necessary and sufficient for satisfiability. The solvability proofs implicitly relied on this fact. Moreover, for any formula in Skolem normal form, the property implies the existence of a finite model.The proof of the latter implication uses the random models technique introduced in [GS] for the Gödel Class and modified and applied in [Go] to the Maslov Class. The proof thus substantiates the claim made in [Go] that random models can be adapted to the Skolem classes of [DG, Chapter 2]. As a whole, the results of this paper provide a more general, systematic approach to finite controllability than previous methods.

BREAKUP: a preprocessing algorithm for satisfiability testing of CNF formulas.

An algorithm called BREAKUP, which processes CNF formulas by separating them into components, is introduced. BREAKUP is then used to speed up the testing of some first-order formulas for satisfi- ability using Iwama's IS Algorithm. The complexity of this algorithm is shown to be on the order of O(nc-nv), where nc is the number of clauses and nv is the number of variables. 1 Introduction Connectedness is an attribute of graphs that can be useful in deciding the satisfiability of propositional formulas in conjunctive normal form (CNF). Preprocessing to separate a formula into components is generally fast and easy, and can significantly speed up testing of the formula for satisfiability. BREAKUP, the algorithm introduced here, will be used in con- junction with Iwama's IS Algorithm (4),(5) and will be shown to significantly speed up (as compared to using IS alone) the processing of some formulas from Gilmore (3) and Davis and Putnam (2). A CNF formula is a conjunction of clauses where each clause is a disjunc- tion of literals. A literal is a propositional variable or the negation of a propo- sitional variable. Given a pair of literals {p9-*p}9 each will be called the mate of the other (cf. Davis (1)). Clearly, we can assume that no clause contains both a literal and its mate. There are two ways to think of clauses in a formula as con- nected. First, two clauses c,d, are connected if there exists a chain of clauses starting with c and ending with d such that for any two adjacent clauses x,y, either y contains a literal that occurs in x or y contains the mate of a literal that occurs in x. Alternatively, c and d can be said to be connected if there is a chain of clauses starting with c and ending with d such that if x and y are adjacent then y contains the mate of a literal that occurs in x. In either case, a set of clauses is said to be connected if every pair in the set is connected and a component is a maximally connected set of the clauses of the formula.

A hypergraph model for constraint logic programming and applications to bus drivers' scheduling

In a previous paper, a hypergraph model for the satisfiability of Datalog formulas was proposed. Here, we extend that approach in order to deal with a class ofconstraint logic programming (CLP) formulas, that is, Datalog formulas in the presence of constraints. A CLP formula is represented by means of a weighted hypergraph and the problem of evaluating this formula is reduced to a sequence of shortest path computations on hypergraphs. To evaluate the performance of this approach, the bus drivers' scheduling problem is formulated as the problem of checking the satisfiability of a CLP formula and it is solved by means of the hypergraph-based algorithm embedded within a local search procedure. Preliminary experimental results are quite encouraging and suggest that the proposed approach may provide an efficient way to tackle hard real-life combinatorial problems.

An action-based framework for veryfying logical and behavioural properties of concurrent systems

A system is described which supports proving both behavioural and logical properties of concurrent systems, these are specified by means of a process algebra and its associated logic. The logic is an action based version of the branching time logic CTL, which we call ACTL. It is interpreted over transition labelled structured while CTL is interpreted over state labelled ones. The core of the system are two existing tools, AUTO and EMC. The first builds the labelled transition system corresponding to a term of a process algebra and permits proof of equivalence and simplification of terms, while the second checks the validity of CTL logical formulae. The integration is realized by means of two translation functions from the action based branching time logic ACTL to CTL and from transition-labelled to state-labelled structures. The correctness of the integration is guaranteed by the proof that the two translation functions when coupled preserve satisfiability of logical formulae.

Quantitative temporal reasoning

A substantially large class of programs operate in distributed and real-time environments, and an integral part of their correctness specification requires the expression of time-critical properties that relate the occurrence of events of the system. We focus on the formal specification and reasoning about the correctness of such programs. We propose a system of temporal logic, RTCTL (Real-Time Computation Tree Logic), that allows the melding of qualitative temporal assertions together with real-time constraints to permit specification and reasoning at the twin levels of abstraction: qualitative and quantitative. We argue that many practically useful correctness properties of temporal systems, which need to express timing as an essential part of their functionality requirements, can be expressed in RTCTL. We develop a model-checking algorithm for RTCTL whose complexity is linear in the size of the RTCTL specification formula and in the size of the structure. We also present an essentially optimal, exponential time tableau-based decision procedure for the satisfiability of RTCTL formulae. Finally, we consider several variants and extensions of RTCTL for real-time reasoning.

A model parametric real-time logic

TRIO is a formal notation for the logic-based specification of real-time systems. In this paper the language and its straightforward model-theoretic semantics are briefly summarized. Then the need for assigning a consistent meaning to TRIO specifications is discussed, with reference to a variety of underlying time structures such as infinite-time structures (both dense and discrete) and finite-time structures. The main motivation is the ability to validate formal specifications. A solution to this problem is presented, which gives a new, model-parametric semantics to the language. An algorithm for constructively verifying the satisfiability of formulas in the decidable cases is defined, and several important temporal properties of specifications are characterized.

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)).

Satisfiability of formulae with one ∀ is decidable in exponential time

In first order logic without equality, but with arbitrary relations and functions the ∃*∀∃* class is the unique maximal solvable prefix class. We show that the satisfiability problem for this class is decidable in deterministic exponential time The result is established by a structural analysis of a particular infinite subset of the Herbrand universe and by a polynomial space bounded alternating procedure.

A hierarchy of propositional Horn formulas

A hierarchy of propositional Horn formulas is introduced. The levels σ H k and ∏ H k of the hierarchy are defined by way of the number of alternations between players in a certain game related to the satisfiability of Horn formulas. The satisfiability problems for formulas from a given level of the hierarchy are shown to be complete in NSPACE(log n). A certain relationship between the hierarchy and the bounded-depth circuits is exhibited. Using it we show that for some σ H k and ∏ H k formulas the equivalent formulas in the lower levels of the hierarchy must be exponentially longer.

Algorithms for testing the satisfiability of propositional formulae

The best-known algorithm for the satisfiability problem in the case of propositional formulae ( SAT) is the implicit enumeration version of the Davis–Putnam algorithm, as described in Loveland's book. We review this algorithm, which we call DPL, and some recent variants. As it is often the case with enumerative algorithms for decision type problems, neither DPL nor its variants incorporate any effective device to prune the search tree. We investigate the idea of using the solution of related subproblems as such a pruning device. It is well known that HORN-SAT, the satisfiability problem in the case of Horn clauses, is easy: in fact an algorithm, DG, has been proposed by Dowling and Gallier, which solves HORN-SAT in linear time. Here we show by means of a set of experiments that the efficiency of DG is not only theoretical but practical as well. In fact we show that on a set of randomly generated problems the complexity of DG grows almost linearly with the problem size, while, for instance, the complexity of the unit resolution approach grows almost quadratically. Then we propose two relaxation schemes which map instances of SAT into instances of HORN-SAT; such relaxation schemes are used to derive two new enumerative algorithms for SAT: HORN1 and HORN2. These algorithms have been compared experimentally with DPL and with its variants. Our results show that HORN2 outperforms the other algorithms; in particular it runs several times faster than DPL on almost all the test problems we have generated.

Random models and the Maslov class

In [GS] Gurevich and Shelah introduce a novel method for proving that every satisfiable formula in the Gödel class has a finite model (the Gödel class is the class of prenex formulas of pure quantification theory with prefixes ∀∀∃ … ∃). They dub their method “random models”: it proceeds by delineating, given any F in the Gödel class and any integer p, a set of structures for F with universe {1, …, p} that can be treated as a finite probability space S. They then show how to calculate an upper bound on the probability that a structure chosen at random from S makes F false; from this bound they are able to infer that if p is sufficiently large, that probability will be less than one, so that there will exist a structure in S that is a model for F. The Gurevich-Shelah proof is somewhat simpler than those known heretofore. In particular, there is no need for the combinatorial partitionings of finite universes that play a central role in the earlier proofs (see [G] and [DG, p. 86]). To be sure, Gurevich and Shelah obtain a larger bound on the size of the finite models, but this is relatively unimportant, since searching for finite models is not the most efficient method to decide satisfiability.Gurevich and Shelah note that the random model method can be used to treat the Gödel class extended by initial existential quantifiers, that is, the prefix-class ∀…∀∃…∃; but they do not investigate further its range of applicability to syntactically specified classes.

A logic to reason about likelihood

We present a logic LL which uses a modal operator L to help capture the notion of being likely. Despite the fact that likelihood is not assigned quantitative values through probabilities, LL captures many of the properties of likelihood in an intuitively appealing way. We give a possible-worlds style semantics to LL, and, using standard techniques of modal logic, we give a complete axiomatization for LL and show that satisfiability of LL formulas can be decided in exponential time. We discuss how the logic might be used in areas such as medical diagnosis, where decision making in the face of uncertainties is crucial. We conclude by using LL to give a formal proof of correctness of some aspects of a protocol for exchanging secrets.

Probabilistic propositional temporal logics

We present two (closely-related) propositional probabilistic temporal logics based on temporal logics of branching time as introduced by Ben-Ari, Pnueli, and Manna ( Acta Inform. 20 (1983), 207–226), Emerson and Halpern (“Proceedings, 14th ACM Sympos. Theory of Comput.,” 1982, pp. 169–179, and Emerson and Clarke ( Sci. Comput. Program. 2 (1982), 241–266). The first logic, PTL f , is interpreted over finite models, while the second logic, PTL b , which is an extension of the first one, is interpreted over infinite models with transition probabilities bounded away from 0. The logic PTL f allows us to reason about finite-state sequential probabilistic programs, and the logic PTL b allows us to reason about (finite-state) concurrent probabilistic programs, without any explicit reference to the actual values of their state-transition probabilities. A generalization of the tableau method yields deterministic single-exponential time decision procedures for our logics, and complete axiomatizations of them are given. Several meta-results, including the absence of a finite-model property for PTL b , and the connection between satisfiable formulae of PTL b and finite state concurrent probabilistic programs, are also discussed.

The unsolvability of the Gödel class with identity

The Gödel class with identity (GCI) is the class of closed, prenex quantificational formulas whose prefixes have the form ∀∀∃ … ∃ and whose matrices contain arbitrary predicate letters and the identity sign “ = ”, but do not contain function signs or individual constants. The ∀∀∃ … ∃ class without identity was shown solvable over fifty years ago ([4], [12], [17]); slightly later, that class was shown to possess the stronger property of finite controllability ([5], [18]). (A class of formulas is solvable iff it is decidable for satisfiability; it is finitely controllable iff every satisfiable formula in it has a finite model.) At the end of [5], Gödel claims that the finite controllability of the GCI can be shown “by the same method” as he employed to show this for the class without identity. This claim has been questioned for nearly twenty years; in §1 below we give a brief history of investigations into it. The major result of this paper shows Gödel to have been mistaken: the GCI is unsolvable. §2 contains the basic construction, which yields a satisfiable formula in the GCI that lacks finite models. This formula may easily be exploited to encode undecidable problems into the GCI.

Complete problems in the first-order predicate calculus

Some problems concerning the satisfiability of first-order predicate calculus formulae in Schonfinkel-Bernays form provide a natural hierarchy of complete problems for various complexity classes. Also, problems concerning the existence of resolution proofs from sets of clauses not necessarily in Schonfinkel-Bernays form provide another such hierarchy. In this way we obtain problems complete for P, NP, PSPACE, deterministic and nondeterministic exponential, deterministic and nondeterministic double exponential time, and exponential space. The results concerning resolution proofs may have practical implications for the design of resolution theorem proving programs. Also, these results enable us to make precise statements about the relative difficulty of various resolution strategies. Some connections with temporal logic and alternating Turing machines are discussed.