Horn minimization by iterative decomposition

Given a Horn CNF representing a Boolean function f, the problem of Horn minimization consists in constructing a CNF representation off which has a minimum possible number of clauses. This problem is the formalization of the problem of knowledge compression for speeding up queries to propositional Horn expert systems, and it is known to be NP-hard. In this paper we present a linear time algorithm which takes a Horn CNF as an input, and through a series of decompositions reduces the minimization of the input CNF to the minimization problem on a“shorter” CNF. The correctness of this decomposition algorithm rests on several interesting properties of Horn functions which, as we prove here, turn out to be independent of the particular CNF representations. This revised version was published online in June 2006 with corrections to the Cover Date.     

Recognition and dualization of disguised bidual Horn functions

We consider the problem of dualizing a Boolean function f given by CNF, i.e., computing a CNF for its dual f^d. While this problem is not solvable in quasi-polynomial total time in general (unless SAT is solvable in quasi-polynomial time), it is so in case the input belongs to special classes, e.g., the class of bidual Horn CNF ? [Discrete Appl. Math. 96-97 (1999) 55-88] (i.e., both ? and its dual ?^d represent Horn functions). In this paper, we show that a disguised bidual Horn CNF ? (i.e., ? becomes a bidual Horn CNF after renaming of variables) can be recognized in polynomial time, and its dualization can be done in quasi-polynomial total time. We also establish a similar result for dualization of prime CNFs.     

Disjoint essential sets of implicates of a CQ Horn function

In this paper we study a class of CQ Horn functions introduced in Boros et al. (Ann Math Artif Intell 57(3–4):249–291, 2010). We prove that given a CQ Horn function f, the maximal number of pairwise disjoint essential sets of implicates of f equals the minimum number of clauses in a CNF representing f. In other words, we prove that the maximum number of pairwise disjoint essential sets of implicates of f constitutes a tight lower bound on the size (the number of clauses) of any CNF representation of f.     

Double Horn Functions

In this paper, we define double Horn functions, which are the Boolean functionsfsuch that bothfand its complement (i.e., negation)fare Horn, and investigate their semantical and computational properties. Double Horn functions embody a balanced treatment of positive and negative information in the course of the extension problem of partially defined Boolean functions (pdBfs), where a pdBf is a pair (T, F) of disjoint setsT, F⊆{0, 1}ⁿof true and false vectors, respectively, and an extension of (T, F) is a Boolean functionfthat is compatible withTandF. We derive syntactic and semantic characterizations of double Horn functions, and determine the number of such functions. The characterizations are then exploited to give polynomial time algorithms (i) that recognize double Horn functions from Horn DNFs (disjunctive normal forms), and (ii) that compute the prime DNF from an arbitrary formula, as well as its complement and its dual. Furthermore, we consider the problem of determining a double Horn extension of a given pdBf. We describe a polynomial time algorithm for this problem and moreover an algorithm that enumerates all double Horn extensions of a pdBf with polynomial delay. However, finding a shortest double Horn extension (in terms of the size of a formula?representing it) is shown to be intractable.     

Polynomial certificates for propositional classes

This paper studies the complexity of learning classes of expressions in propositional logic from equivalence queries and membership queries. In particular, we focus on bounding the number of queries that are required to learn the class ignoring computational complexity. This quantity is known to be captured by a combinatorial measure of concept classes known as the certificate complexity. The paper gives new constructions of polynomial size certificates for monotone expressions in conjunctive normal form (CNF), for unate CNF functions where each variable affects the function either positively or negatively but not both ways, and for Horn CNF functions. Lower bounds on certificate size for these classes are derived showing that for some parameter settings the new certificate constructions are optimal. Finally, the paper gives an exponential lower bound on the certificate size for a natural generalization of these classes known as renamable Horn CNF functions, thus implying that the class is not learnable from a polynomial number of queries.     

Hardness results for approximate pure Horn CNF formulae minimization

We study the hardness of approximation of clause minimum and literal minimum representations of pure Horn functions in n Boolean variables. We show that unless P=NP, it is not possible to approximate in polynomial time the minimum number of clauses and the minimum number of literals of pure Horn CNF representations to within a factor of \(2^{\log^{1-o(1)} n}\) . This is the case even when the inputs are restricted to pure Horn 3-CNFs with O(n ^1+ε ) clauses, for some small positive constant ε. Furthermore, we show that even allowing sub-exponential time computation, it is still not possible to obtain constant factor approximations for such problems unless the Exponential Time Hypothesis turns out to be false.     

Equivalence of Polynomial Identity Testing and Polynomial Factorization

In this paper, we show that the problem of deterministically factoring multivariate polynomials reduces to the problem of deterministic polynomial identity testing. Specifically, we show that given an arithmetic circuit (either explicitly or via black-box access) that computes a multivariate polynomial f, the task of computing arithmetic circuits for the factors of f can be solved deterministically, given a deterministic algorithm for the polynomial identity testing problem (we require either a white-box or a black-box algorithm, depending on the representation of f).Together with the easy observation that deterministic factoring implies a deterministic algorithm for polynomial identity testing, this establishes an equivalence between these two central derandomization problems of arithmetic complexity.Previously, such an equivalence was known only for multilinear circuits (Shpilka & Volkovich, 2010).     

Compact DSOP and Partial DSOP Forms

Given a Boolean function f on n variables, a Disjoint Sum-of-Products (DSOP) of f is a set of products (ANDs) of subsets of literals whose sum (OR) equals f, such that no two products cover the same minterm of f. DSOP forms are a special instance of partial DSOPs, i.e. the general case where a subset of minterms must be covered exactly once and the other minterms (typically corresponding to don’t care conditions of f) can be covered any number of times. We discuss finding DSOPs and partial DSOPs with a minimal number of products, a problem theoretically connected with various properties of Boolean functions and practically relevant in the synthesis of digital circuits. Finding an absolute minimum is hard, in fact we prove that the problem of absolute minimization of partial DSOPs is NP-hard. Therefore it is crucial to devise a polynomial time heuristic that compares favorably with the known minimization tools. To this end we develop a further piece of theory starting from the definition of the weight of a cube c as a functions of the number of fragments induced on other cubes by the selection of c, and show how cube weights can be exploited for building a class of minimization heuristics for DSOP and partial DSOP synthesis. A set of experiments conducted on major benchmark functions show that our method, with a family of variants, always generates better results than the ones of previous heuristics, including the method based on a BDD representation of f.     

A Case of Depth-3 Identity Testing, Sparse Factorization and Duality

The polynomial identity testing (PIT) problem is known to be challenging even for constant depth arithmetic circuits. In this work, we study the complexity of two special but natural cases of identity testing—first is a case of depth-3 PIT, the other of depth-4 PIT. Our first problem is a vast generalization of verifying whether a bounded top fan-in depth-3 circuit equals a sparse polynomial (given as a sum of monomial terms). Formally, given a depth-3 circuit C, having constant many general product gates and arbitrarily many semidiagonal product gates test whether the output of C is identically zero. A semidiagonal product gate in C computes a product of the form ${m \cdot \prod^b_{i=1}l^{e_i}_i}$ , where m is a monomial, l _i is a linear polynomial, and b is a constant. We give a deterministic polynomial time test, along with the computation of leading monomials of semidiagonal circuits over local rings. The second problem is on verifying a given sparse polynomial factorization, which is a classical question (von zur Gathen, FOCS 1983): Given multivariate sparse polynomials f, g1, . . . , gt explicitly check whether ${f = \prod^t_{i=1} {g_i}}$ . For the special case when every gi is a sum of univariate polynomials, we give a deterministic polynomial time test. We characterize the factors of such g _i ’s and even show how to test the divisibility of f by the powers of such polynomials. The common tools used are Chinese remaindering and dual representation. The dual representation of polynomials (Saxena, ICALP 2008) is a technique to express a product-of-sums of univariates as a sum-ofproducts of univariates. We generalize this technique by combining it with a generalized Chinese remaindering to solve these two problems (over any field).     

Inferring Minimal Functional Dependencies in Horn and q-Horn Theories

In this paper, we study various problems related to the inference of minimal functional dependencies in Horn and q-Horn theories. We show that if a Horn theory is represented by a Horn CNF, then there exists an incrementally polynomial algorithm for inferring all minimal functional dependencies. On the other hand, if a Horn theory is represented as the Horn envelope of a given set of models, then there exists a polynomial total time algorithm for this inference problem if and only if there exists such an algorithm for dualizing a positive CNF. Finally, we generalize our results to the case of q-Horn theories, and show that all the considered problems can be reduced in polynomial time to the corresponding problems for Horn theories.     

Random walks for selected boolean implication and equivalence problems

This paper is concerned with the design and analysis of a random walk algorithm for the 2CNF implication problem (2CNFI). In 2CNFI, we are given two 2CNF formulas f₁{\phi_{1}} and f₂{\phi_{2}} and the goal is to determine whether every assignment that satisfies f₁{\phi_{1}} , also satisfies f₂{\phi_{2}} . The implication problem is clearly coNP-complete for instances of kCNF, k ≥ 3; however, it can be solved in polynomial time, when k ≤ 2. The goal of this paper is to provide a Monte Carlo algorithm for 2CNFI with a bounded probability of error. The technique developed for 2CNFI is then extended to derive a randomized, polynomial time algorithm for the problem of checking whether a given 2CNF formula Nae-implies another 2CNF formula.     

An efficient algorithm for Horn description

We give a new algorithm for computing a prepositional Horn CNF formula given the set of its models. Its running time is O(|R|n(|R|+n)), where |R| is the number of models and n that of variables, and the computed CNF contains at most |R|n clauses. This algorithm also uses the well-known closure property of Horn relations in a new manner.     

Error-Free and Best-Fit Extensions of Partially Defined Boolean Functions

In this paper, we address a fundamental problem related to the induction of Boolean logic: Given a set of data, represented as a set of binary “truen-vectors” (or “positive examples”) and a set of “falsen-vectors” (or “negative examples”), we establish a Boolean function (or an extension)f, so thatfis true (resp., false) in every given true (resp., false) vector. We shall further require that such an extension belongs to a certain specified class of functions, e.g., class of positive functions, class of Horn functions, and so on. The class of functions represents our a priori knowledge or hypothesis about the extensionf, which may be obtained from experience or from the analysis of mechanisms that may or may not cause the phenomena under consideration. The real-world data may contain errors, e.g., measurement and classification errors might come in when obtaining data, or there may be some other influential factors not represented as variables in the vectors. In such situations, we have to give up the goal of establishing an extension that is perfectly consistent with the given data, and we are satisfied with an extensionfhaving the minimum number of misclassifications. Both problems, i.e., the problem of finding an extension within a specified class of Boolean functions and the problem of finding a minimum error extension in that class, will be extensively studied in this paper. For certain classes we shall provide polynomial algorithms, and for other cases we prove their NP-hardness.     

On P versus NP $ \cap $ co-NP for decision trees and read-once branching programs

It is known that if a Boolean function f in n variables has a DNF and a CNF of size then f also has a (deterministic) decision tree of size exp(O(log n log² N)). We show that this simulation cannot be made polynomial: we exhibit explicit Boolean functions f that require deterministic trees of size exp where N is the total number of monomials in minimal DNFs for f and ?f. Moreover, we exhibit new examples of explicit Boolean functions that require deterministic read-once branching programs of exponential size whereas both the functions and their negations have small nondeterministic read-once branching programs. One example results from the Bruen—Blokhuis bound on the size of nontrivial blocking sets in projective planes: it is remarkably simple and combinatorially clear. Other examples have the additional property that f is in AC⁰. Received: June 5 1997.     

Strong computational lower bounds via parameterized complexity

We develop new techniques for deriving strong computational lower bounds for a class of well-known NP-hard problems. This class includes weighted satisfiability, dominating set, hitting set, set cover, clique, and independent set. For example, although a trivial enumeration can easily test in time O(nk) if a given graph of n vertices has a clique of size k, we prove that unless an unlikely collapse occurs in parameterized complexity theory, the problem is not solvable in time f(k)no(k) for any function f, even if we restrict the parameter values to be bounded by an arbitrarily small function of n. Under the same assumption, we prove that even if we restrict the parameter values k to be of the order Θ(μ(n)) for any reasonable function μ, no algorithm of running time no(k) can test if a graph of n vertices has a clique of size k. Similar strong lower bounds on the computational complexity are also derived for other NP-hard problems in the above class. Our techniques can be further extended to derive computational lower bounds on polynomial time approximation schemes for NP-hard optimization problems. For example, we prove that the NP-hard distinguishing substring selection problem, for which a polynomial time approximation scheme has been recently developed, has no polynomial time approximation schemes of running time f(1/?)no(1/?) for any function f unless an unlikely collapse occurs in parameterized complexity theory.     

Learning Read-Constant Polynomials of Constant Degree Modulo Composites

Boolean functions that have constant degree polynomial representation over a fixed finite ring form a natural and strict subclass of the complexity class ACC⁰. They are also precisely the functions computable efficiently by programs over fixed and finite nilpotent groups. This class is not known to be learnable in any reasonable learning model. In this paper, we provide a deterministic polynomial time algorithm for learning Boolean functions represented by polynomials of constant degree over arbitrary finite rings from membership queries, with the additional constraint that each variable in the target polynomial appears in a constant number of monomials. Our algorithm extends to superconstant but low degree polynomials and still runs in quasipolynomial time.     

Conjunctions of Unate DNF Formulas: Learning and Structure

A central topic in query learning is to determine which classes of Boolean formulas are efficiently learnable with membership and equivalence queries. We consider the class ^kconsisting of conjunctions ofkunate DNF formulas. This class generalizes the class ofk-clause CNF formulas and the class of unate DNF formulas, both of which are known to be learnable in polynomial time with membership and equivalence queries. We prove that ²can be properly learned with a polynomial number of polynomial-size membership and equivalence queries, but can be properly learned in polynomial time with such queries if and only if P=NP. Thus the barrier to properly learning ²with membership and equivalence queries is computational rather than informational. Few results of this type are known. In our proofs, we use recent results of Hellersteinet al.(1997,J. Assoc. Comput. Mach.43(5), 840–862), characterizing the classes that are polynomial-query learnable, together with work of Bshouty on the monotone dimension of Boolean functions. We extend some of our results to ^kand pose open questions on learning DNF formulas of small monotone dimension. We also prove structural results for ^k. We construct, for any fixedk2, a class of functionsfthat cannot be represented by any formula in ^k, but which cannot be “easily” shown to have this property. More precisely, for any functionfonnvariables in the class, the value offon any polynomial-size set of points in its domain is not a witness thatfcannot be represented by a formula in ^k. Our construction is based on BCH codes.     

Quasi-acyclic propositional Horn knowledge bases: optimalcompression

Horn knowledge bases are widely used in many applications. The paper is concerned with the optimal compression of propositional Horn production rule bases-one of the most important knowledge bases used in practice. The problem of knowledge compression is interpreted as a problem of Boolean function minimization. It was proved by P.L. Hammer and A. Kogan (1993) that the minimization of Horn functions, i.e., Boolean functions associated with Horn knowledge bases, is NP complete. The paper deals with the minimization of quasi acyclic Horn functions, the class of which properly includes the two practically significant classes of quadratic and of acyclic functions. A procedure is developed for recognizing in quadratic time the quasi acyclicity of a function given by a Horn CNF, and a graph based algorithm is proposed for the quadratic time minimization of quasi acyclic Horn functions     

On Learning Visual Concepts and DNF Formulae

We consider the problem of learning DNF formulae in the mistake-bound and the PAC models. We develop a new approach, which is called polynomial explainability, that is shown to be useful for learning some new subclasses of DNF (and CNF) formulae that were not known to be learnable before. Unlike previous learnability results for DNF (and CNF) formulae, these subclasses are not limited in the number of terms or in the number of variables per term; yet, they contain the subclasses of k-DNF and k-term-DNF (and the corresponding classes of CNF) as special cases. We apply our DNF results to the problem of learning visual concepts and obtain learning algorithms for several natural subclasses of visual concepts that appear to have no natural boolean counterpart. On the other hand, we show that learning some other natural subclasses of visual concepts is as hard as learning the class of all DNF formulae. We also consider the robustness of these results under various types of noise.An earlier version of this paper appeared in the Proceedings of the Sixth Annual ACM Workshop on Computational Learning Theory, COLT93.Current address: Department of Computer Science, Technion, Israel. e-mail:eyalk@cs.technion.ac.il     

Random arithmetic formulas can be reconstructed efficiently

Informally stated, we present here a randomized algorithm that given black-box access to the polynomial f computed by an unknown/hidden arithmetic formula ? reconstructs, on the average, an equivalent or smaller formula \({\hat{\phi}}\) in time polynomial in the size of its output \({\hat{\phi}}\) . Specifically, we consider arithmetic formulas wherein the underlying tree is a complete binary tree, the leaf nodes are labeled by affine forms (i.e., degree one polynomials) over the input variables and where the internal nodes consist of alternating layers of addition and multiplication gates. We call these alternating normal form (ANF) formulas. If a polynomial f can be computed by an arithmetic formula μ of size s, it can also be computed by an ANF formula ?, possibly of slightly larger size s ^O(1). Our algorithm gets as input black-box access to the output polynomial f (i.e., for any point x in the domain, it can query the black box and obtain f(x) in one step) of a random ANF formula ? of size s (wherein the coefficients of the affine forms in the leaf nodes of ? are chosen independently and uniformly at random from a large enough subset of the underlying field). With high probability (over the choice of coefficients in the leaf nodes), the algorithm efficiently (i.e., in time s ^O(1)) computes an ANF formula \({\hat{\phi}}\) of size s computing f. This then is the strongest model of arithmetic computation for which a reconstruction algorithm is presently known, albeit efficient in a distributional sense rather than in the worst case.