CNF Representations Research Articles

Matroid Horn functions

Hypergraph Horn functions were introduced as a subclass of Horn functions that can be represented by a collection of circular implication rules. These functions possess distinguished structural and computational properties. In particular, their characterizations in terms of implicate-duality and the closure operator provide extensions of matroid duality and the Mac Lane–Steinitz exchange property of matroid closure, respectively.In the present paper, we introduce a subclass of hypergraph Horn functions that we call matroid Horn functions. We provide multiple characterizations of matroid Horn functions in terms of their canonical and complete CNF representations. We also study the Boolean minimization problem for this class, where the goal is to find a minimum size representation of a matroid Horn function given by a CNF representation. While there are various ways to measure the size of a CNF, we focus on the number of circuits and circuit clauses. We determine the size of an optimal representation for binary matroids, and give lower and upper bounds in the uniform case. For uniform matroids, we show a strong connection between our problem and Turán systems that might be of independent combinatorial interest.

A Separator Theorem for Hypergraphs and a CSP-SAT Algorithm

We show that for every $r \ge 2$ there exists $\epsilon_r > 0$ such that any $r$-uniform hypergraph with $m$ edges and maximum vertex degree $o(\sqrt{m})$ contains a set of at most $(\frac{1}{2} - \epsilon_r)m$ edges the removal of which breaks the hypergraph into connected components with at most $m/2$ edges. We use this to give an algorithm running in time $d^{(1 - \epsilon_r)m}$ that decides satisfiability of $m$-variable $(d, k)$-CSPs in which every variable appears in at most $r$ constraints, where $\epsilon_r$ depends only on $r$ and $k\in o(\sqrt{m})$. Furthermore our algorithm solves the corresponding #CSP-SAT and Max-CSP-SAT of these CSPs. We also show that CNF representations of unsatisfiable $(2, k)$-CSPs with variable frequency $r$ can be refuted in tree-like resolution in size $2^{(1 - \epsilon_r)m}$. Furthermore for Tseitin formulas on graphs with degree at most $k$ (which are $(2, k)$-CSPs) we give a deterministic algorithm finding such a refutation.

A Novel SAT-Based Approach to Model Based Diagnosis

This paper introduces a novel encoding of Model Based Diagnosis (MBD) to Boolean Satisfaction (SAT) focusing on minimal cardinality diagnosis. The encoding is based on a combination of sophisticated MBD preprocessing algorithms and the application of a SAT compiler which optimizes the encoding to provide more succinct CNF representations than obtained with previous works. Experimental evidence indicates that our approach is superior to all published algorithms for minimal cardinality MBD. In particular, we can determine, for the first time, minimal cardinality diagnoses for the entire standard ISCAS-85 and 74XXX benchmarks. Our results open the way to improve the state-of-the-art on a range of similar MBD problems.

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 $2^{\log^{1-o(1)} n}$ . This is the case even when the inputs are restricted to pure Horn 3-CNFs with O(n 1+? ) clauses, for some small positive constant ?. 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.

Boolean functions with a simple certificate for CNF complexity

In this paper we study relationships between CNF representations of a given Boolean function f and essential sets of implicates of f. It is known that every CNF representation and every essential set must intersect. Therefore the maximum number of pairwise disjoint essential sets of f provides a lower bound on the size of any CNF representation of f. We are interested in functions, for which this lower bound is tight, and call such functions coverable. We prove that for every coverable function there exists a polynomially verifiable certificate (witness) for its minimum CNF size. On the other hand, we show that not all functions are coverable, and construct examples of non-coverable functions. Moreover, we prove that computing the lower bound, i.e. the maximum number of pairwise disjoint essential sets, is NP-hard under various restrictions on the function and on its input representation.

Model Counting in Product Configuration

We describe how to use propositional model counting for a quantitative analysis of product configuration data. Our approach computes valuable meta information such as the total number of valid configurations or the relative frequency of components. This information can be used to assess the severity of documentation errors or to measure documentation quality. As an application example we show how we apply these methods to product documentation formulas of the Mercedes-Benz line of vehicles. In order to process these large formulas we developed and implemented a new model counter for non-CNF formulas. Our model counter can process formulas, whose CNF representations could not be processed up till now.

Exclusive and essential sets of implicates of Boolean functions

In this paper we study relationships between CNF representations of a given Boolean function f and certain sets of implicates of f . We introduce two definitions of sets of implicates which are both based on the properties of resolution. The first type of sets, called exclusive sets of implicates, is shown to have a functional property useful for decompositions. The second type of sets, called essential sets of implicates, is proved to possess an orthogonality property, which implies that every CNF representation and every essential set must intersect. The latter property then leads to an interesting question, to which we give an affirmative answer for some special subclasses of Horn Boolean functions.

Composition of Post classes and normal forms of Boolean functions

The class composition C ∘ K of Boolean clones, being the set of composite functions f ( g 1 , … , g n ) with f ∈ C , g 1 , … , g n ∈ K , is investigated. This composition C ∘ K is either the join C ∨ K in the Post Lattice or it is not a clone, and all pairs of clones C , K are classified accordingly. Factorizations of the clone Ω of all Boolean functions as a composition of minimal clones are described and seen to correspond to normal form representations of Boolean functions. The median normal form, arising from the factorization of Ω with the clone SM of self-dual monotone functions as the leftmost composition factor, is compared in terms of complexity with the well-known DNF, CNF, and Zhegalkin (Reed–Muller) polynomial representations, and it is shown to provide a more efficient normal form representation.

On converting CNF to DNF

We study how big the blow-up in size can be when one switches between the CNF and DNF representations of Boolean functions. For a function f : { 0 , 1 } n → { 0 , 1 } , cnfsize ( f ) denotes the minimum number of clauses in a CNF for f; similarly, dnfsize ( f ) denotes the minimum number of terms in a DNF for f. For 0 ⩽ m ⩽ 2 n - 1 , let dnfsize ( m , n ) be the maximum dnfsize ( f ) for a function f : { 0 , 1 } n → { 0 , 1 } with cnfsize ( f ) ⩽ m . We show that there are constants c 1 , c 2 ⩾ 1 and ε > 0 , such that for all large n and all m ∈ [ 1 ε n , 2 ε n ] , we have 2 n - c 1 ( n / log ( m / n ) ) ⩽ dnfsize ( m , n ) ⩽ 2 n - c 2 ( n / log ( m / n ) ) . In particular, when m is the polynomial n c , we get dnfsize ( n c , n ) = 2 n - θ ( c - 1 ( n / log n ) ) .

On converting CNF to DNF

We study how big the blow-up in size can be when one switches between the CNF and DNF representations of boolean functions. For a function f : {0,1}^n -> {0,1}, cnfsize(f) denotes the minimum number of clauses in a CNF for f; similarly, dnfsize(f) denotes the minimum number of terms in a DNF for f. For 0 <= m <= 2^{n-1} , let dnfsize(m,n) be the maximum dnfsize(f) for a function f : {0,1}^n -> {0,1} with cnfsize(f) <= m. We show that there are constants c_1, c_2 >= 1 and epsilon > 0 , such that for all large n and all m in [ 1/epsilon n , 2^{epsilon n}], we have <br />2^{(n - c_1) n / log(m/n)} <= dnfsize(m,n) <= 2^{(n - c_2) n / log(m/n)}.<br /> In particular, when m is the polynomial n^c, we get dnfsize(n^c,n) = 2^{(n - theta) (c^{-1} n / log n)}.

The monotone theory for the PAC-model

In this paper we extend the Monotone Theory to the PAC-learning Model with membership queries. Using this extention we show that a DNF formula that has at least one “1/ poly -heavy” clause in one of its CNF representation (a clause that is not satisfied with probability 1/ poly ( n , s ) where n is the number of variables and s is the number of terms in f ) with respect to a distribution D is weakly learnable under this distribution. So DNF that are not weakly learnable under the distribution D do not have any “1/ poly -heavy” clauses in any of their CNF representations. A DNF f is called τ -CDNF if there is τ ′ > τ and a CNF representation of f that contains poly ( n , s ) clauses that τ ′ -approximates f according to a distribution D . We show that the class of all τ -CDNF is weakly ( τ + ϵ )-PAC-learnable with membership queries under the distribution D . We then show how to change our algorithm to a parallel algorithm that runs in polylogarithmic time with a polynomial number of processors. In particular, decision trees are (strongly) PAC-learnable with membership queries under any distribution in parallel in polylogarithmic time with a polynomial number of processors. Finally we show that no efficient parallel exact learning algorithm exists for decision trees.

Almost all monotone Boolean functions are polynomially learnable using membership queries

We consider exact learning or identification of monotone Boolean functions by only using membership queries. It is shown that almost all monotone Boolean functions are polynomially identifiable in the input number of variables as well as the output being the sum of the sizes of the CNF and DNF representations.

The learnability of exclusive-or expansions based on monotone DNF formulas

The learnability of the class of exclusive-or expansions based on monotone DNF formulas is investigated. The class consists of the formulas of the form f=f 1⊕⋯⊕f d , where f 1>⋯>f d are monotone DNF formulas. It is shown that any Boolean function can be represented as a formula in this class, and that the representation in the simplest form is unique. Learning algorithms that learn such formulas using various queries are presented: An algorithm with subset and superset queries and one with membership and equivalence queries are given. The former can learn any formula in the class, while the latter is proved to learn formulas of constant depth, i.e., formulas represented as exclusive-or of a constant number of monotone DNF formulas. In spite of the seemingly strong restriction of the depth being constant, the class of formulas of constant depth includes functions with very high complexity in terms of DNF and CNF representations, so the latter algorithm could learn Boolean functions significantly complex otherwise represented.

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 of f 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 NPdhard. 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.