The Complexity of the Boolean Formula Value Problem

Annals of Mathematics and Artificial Intelligence, 1996

In Artificial Intelligence there has been a great deal of interest in the tradeoff between expressiveness and tractability for various areas of symbolic reasoning. Here we present several complexity theory results for two areas, wherein we restrict the application of negation. First, we consider the problem of determining a minimum satisfying assignment for a (restricted) propositional sentence. We show that the problem of determining a minimum satisfying assignment for a sentence in negation-free CNF, even with no more than two disjuncts per clause, is NP-complete. We also show that unless P = NP, no polynomial time approximation scheme can exist for this problem. However, the problem is in polynomial time if either each clause contains exactly one negative and one positive literal or we use exclusive-OR in the clauses instead of the more standard inclusive-OR. Second, the problem of determining logical implication between sentences composed solely of conjunctions and disjunctions is shown to be as difficult as that between arbitrary sentences. We also study this problem when the sentences are restricted to being in CNF or DNE Determining whether a CNF sentence logically implies a DNF sentence is co-NP-complete, but in all other cases this problem is polynomial time. We argue that these results are relevant, first to areas where a least solution (in some fashion) is desired, and second, to limited deductive systems.

It has been proved that almost all n-bit Boolean functions have exact classical query complexity n. However, the situation seemed to be very different when we deal with exact quantum query complexity. In this paper, we prove that almost all n-bit Boolean functions can be computed by an exact quantum algorithm with less than n queries. More exactly, we prove that AND n is the only n-bit Boolean function, up to isomorphism, that requires n queries.

This short note deals with several classes of Boolean formulae which have the property, that satisfiabilty can be tested for them in polynomial time with respect to the length of the formula. The studied classes known ffom the literature build uP on the $\mathrm{w}\mathrm{e}\mathrm{U}$-known classes of quadratic on Horn formulae. We prove several interesting properties of these classes and show their mutual positions with respect to inclusion, aproblem which was not previously studied. 1Introduction The class of Horn formulae is avery important and extensively studied subclass of general Boolean formulae. The principal reason for their importance is the fact, that the $\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{i}\mathrm{a}\mathrm{b}\underline{\mathrm{i}\mathrm{l}\mathrm{i}}\mathrm{t}\mathrm{y}$ problem (SAT), which is well-known to be $\mathrm{N}\mathrm{P}$-complete for general Boolean formulae, can be solved efficiently (in linear time with respect to the length of the formula) for Horn formulae [11, 15, 17]. This has significant practical implications. Many real-life problems require for their solution to solve SAT as asubproblem, and hence are in general intractable; however, they become tractable if the underlying Boolean formula in the problem is Horn. Such problems arise in several application areas, among others in artificial inteUigence [9, 13, 14] and database design $[10, 16]$. The limiting factor in using Horn formulae is their expressing power. Not every real-life problem can be formulated in such away, that the underlying formula is Horn. For the above reasons it is obvious, that finding broader classes of formulae, which preserve the property that satisfiability is decidable for them in polynomial time, is highly desirable. Several attempts in this direction were successfully made. The &st natural generalzation that was considered is the class of hidden Horn formulae, which are in literature sometimes also called renameable or disguised Horn formulae. This class consists of formulae, which can be obtained from Horn formulae by so called "variable complementing" (also known as "vaiable renaming" or "variable switching"), i.e. by replacing some Boolean variables by their complements. Aspvall showed in [1] that recognizing whether agiven Boolean formula is hidden Horn can be done in linear time. Moreover, the recognition algorithm combined with the linear time SAT algorithm for Horn formulae [11, 15, 17] directly yields alinear time SAT algorithm for the class of hidden Horn formulae. Yamasaki and Doshita [19] defined adifferent generalzation of Horn formulae, called their class $S_{0}$ , and developed acubic time SAT algorithm for formulae in So. This was later improved to quadratic time by Arvind and Biswas [3]. Moreover, recognizing whether agiven formula belongs to $S_{0}$ can be decided also in quadratic time by astraightforward algorithm which uses in asimple way the definition of the class. The class $S_{0}$ was further generalized by Gallo and Scuteu\'a [12] who came up with arecursively

Lecture Notes in Computer Science, 2008

We consider the logical system of boolean expressions built on the single connector of implication and on positive literals. Assuming all expressions of a given size to be equally likely, we prove that we can define a probability distribution on the set of boolean functions expressible in this system. We then show how to approximate the probability of a function f when the number of variables grows to infinity, and that this asymptotic probability has a simple expression in terms of the complexity of f . We also prove that most expressions computing any given function in this system are "simple", in a sense that we make precise.

Discrete Mathematics & Theoretical Computer Science

Any attempt to find connections between mathematical properties and complexity has a strong relevance to the field of Complexity Theory. This is due to the lack of mathematical techniques to prove lower bounds for general models of computation.\par This work represents a step in this direction: we define a combinatorial property that makes Boolean functions ''\emphhard'' to compute in constant depth and show how the harmonic analysis on the hypercube can be applied to derive new lower bounds on the size complexity of previously unclassified Boolean functions.