Tseitin-formulas are systems of parity constraints whose structure is described by a graph. These formulas have been studied extensively in proof complexity as hard instances in many proof systems. In this paper, we prove that a class of unsatisfiable Tseitin-formulas of bounded degree has regular resolution refutations of polynomial length if and only if the treewidth of all underlying graphs G for that class is in O(log |V (G)|). It follows that unsatisfiable Tseitin-formulas with polynomial length of regular resolution refutations are completely determined by the treewidth of the underlying graphs when these graphs have bounded degree. To prove this, we show that any regular resolution refutation of an unsatisfiable Tseitin-formula with graph G of bounded degree has length 2Ω(tw(G))/|V (G)|, thus essentially matching the known 2O(tw(G))poly(|V (G)|) upper bound. Our proof first connects the length of regular resolution refutations of unsatisfiable Tseitin-formulas to the size of representations of satisfiable Tseitin-formulas in decomposable negation normal form (DNNF). Then we prove that for every graph G of bounded degree, every DNNF-representation of every satisfiable Tseitin-formula with graph G must have size 2Ω(tw(G)) which yields our lower bound for regular resolution.

Two major considerations when encoding pseudo-Boolean (PB) constraints into SAT are the size of the encoding and its propagation strength, that is, the guarantee that it has a good behaviour under unit propagation. Several encodings with propagation strength guarantees rely upon prior compilation of the constraints into DNNF (decomposable negation normal form), BDD (binary decision diagram), or some other sub-variants. However it has been shown that there exist PB-constraints whose ordered BDD (OBDD) representations, and thus the inferred CNF encodings, all have exponential size. Since DNNFs are more succinct than OBDDs, preferring encodings via DNNF to avoid size explosion seems a legitimate choice. Yet in this paper, we prove the existence of PB-constraints whose DNNFs all require exponential size.

A key point in successfully applying deep neural models is the construction of architecture families that contain inductive bias relevant to the application domain. Architectures such as CNNs and RNNs have become the preeminent favorites for modeling images and sequential data, respectively. For example, the inductive bias of CNNs favors locality, as well as translation and scale invariances. With these properties, CNNs work extremely well on image data, and are capable of generating problem-dependent representations that almost completely overcome the need for expert knowledge. Similarly, the inductive bias promoted by RNNs and LSTMs (and more recent models such as transformers) favors both locality and temporal stationarity. When considering tabular data, however, neural networks are not the hypothesis class of choice. Most often, the winning class in learning problems involving tabular data is decision forests. In Kaggle competitions, for example, gradient boosting of decision trees (GBDTs) [6, 9, 19, 14] are generally the superior model. While it is quite practical to use GBDTs for medium size datasets, it is extremely hard to scale these methods to very large datasets (e.g., Google or Facebook scale).

We consider bounded width CNF-formulas where the width is measured by popular graph width measures on graphs associated to CNF-formulas. Such restricted graph classes, in particular those of bounded treewidth, have been extensively studied for their uses in the design of algorithms for various computational problems on CNF-formulas. Here we consider the expressivity of these formulas in the model of clausal encodings with auxiliary variables. We first show that bounding the width for many of the measures from the literature leads to a dramatic loss of expressivity, restricting the formulas to those of low communication complexity. We then show that the width of optimal encodings with respect to different measures is strongly linked: there are two classes of width measures, one containing primal treewidth and the other incidence cliquewidth, such that in each class the width of optimal encodings only differs by constant factors. Moreover, between the two classes the width differs at most by a factor logarithmic in the number of variables. Both these results are in stark contrast to the setting without auxiliary variables where all width measures we consider here differ by more than constant factors and in many cases even by linear factors.

Structured $d$-DNNFs and SDDs are restricted negation normal form circuits used in knowledge compilation as target languages into which propositional theories are compiled. Structuredness is imposed by so-called vtrees. By definition SDDs are restricted structured $d$-DNNFs. Beame and Liew (2015) as well as Bova and Szeider (2017) mentioned the question whether structured $d$-DNNFs are really more general than SDDs w.r.t. polynomial-size representations (w.r.t. the number of Boolean variables the represented functions are defined on.) The main result in the paper is the proof that a function can be represented by SDDs of polynomial size if the function and its complement have polynomial-size structured $d$-DNNFs that respect the same vtree.

We investigate conjunctive normal form (CNF) encodings of a function represented with a smooth decomposable negation normal form (DNNF). Several encodings of DNNFs and decision diagrams were considered by (Abio et al. 2016). The authors differentiate between encodings which implement consistency or domain consistency from encodings which implement unit refutation completeness or propagation completeness (in both cases implements means by unit propagation). The difference is that in the former case we do not care about properties of the encoding with respect to the auxiliary variables while in the latter case we treat all variables (the input ones and the auxiliary ones) in the same way. The latter case is useful if a DNNF is a part of a problem containing also other constraints and a SAT solver is used to test satisfiability. The currently known encodings of smooth DNNF theories implement domain consistency. Building on this and the result of (Abio et al. 2016) on an encoding of decision diagrams which implements propagation completeness, we present a new encoding of a smooth DNNF which implements propagation completeness. This closes the gap left open in the literature on encodings of DNNFs.

We consider bounded width CNF-formulas where the width is measured by popular graph width measures on graphs associated to CNF-formulas. Such restricted graph classes, in particular those of bounded treewidth, have been extensively studied for their uses in the design of algorithms for various computational problems on CNF-formulas. Here we consider the expressivity of these formulas in the model of clausal encodings with auxiliary variables. We first show that bounding the width for many of the measures from the literature leads to a dramatic loss of expressivity, restricting the formulas to such of low communication complexity. We then show that the width of optimal encodings with respect to different measures is strongly linked: there are two classes of width measures, one containing primal treewidth and the other incidence cliquewidth, such that in each class the width of optimal encodings only differs by constant factors. Moreover, between the two classes the width differs at most by a factor logarithmic in the number of variables. Both these results are in stark contrast to the setting without auxiliary variables where all width measures we consider here differ by more than constant factors and in many cases even by linear factors.

In this paper we prove a space lower bound of $n^{\Omega(k)}$ for non-deterministic (syntactic) read-once branching programs ({\sc nrobp}s) on functions expressible as {\sc cnf}s with treewidth at most $k$ of their primal graphs. This lower bound rules out the possibility of fixed-parameter space complexity of {\sc nrobp}s parameterized by $k$. We use lower bound for {\sc nrobp}s to obtain a quasi-polynomial separation between Free Binary Decision Diagrams and Decision Decomposable Negation Normal Forms, essentially matching the existing upper bound introduced by Beame et al. and thus proving the tightness of the latter.

We consider in this paper the size of a Decomposable Negation Normal Form (DNNF) that respects a given vtree (known as structured DNNF). This representation of propositional knowledge bases was introduced recently and shown to include OBDD as a special case (an OBDD variable ordering is a special type of vtree). We provide a lower bound on the size of any structured DNNF and discuss three particular instances of this bound, which correspond to three distinct subsets of structured DNNF (including OBDD). We show that our lower bound subsumes the influential Sieling and Wegener’s lower bound for OBDDs, which is typically used for identifying variable orderings that will cause a blowup in the OBDD size. We show that our lower bound allows for similar usage but with respect to vtrees, which provide structure for DNNFs in the same way that variable orderings provide structure for OBDDs. We finally discuss some of the theoretical and practical implications of our lower bound.

The goal of testing is to discriminate between multiple hypotheses about a system - for example, different fault diagnoses - by applying input patterns and verifying or falsifying the hypotheses from the observed outputs. Definitely discriminating tests (DDTs) are those input patterns that are guaranteed to discriminate between different hypotheses of non-deterministic systems. Finding DDTs is important in practice, but can be very expensive. Even more challenging is the problem of finding a DDT that minimizes the cost of the testing process, i.e., an input pattern that can be most cheaply enforced and that is a DDT. This paper addresses both problems. We show how we can transform a given problem into a Boolean structure in decomposable negation normal form (DNNF), and extract from it a Boolean formula whose models correspond to DDTs. This allows us to harness recent advances in both knowledge compilation and satisfiability for efficient and scalable DDT computation in practice. Furthermore, we show how we can generate a DNNF structure compactly encoding all DDTs of the problem and use it to obtain a cost-optimal DDT in time linear in the size of the structure. Experimental results from a real-world application show that our method can compute DDTs in less than 1 second for instances that were previously intractable, and cost-optimal DDTs in less than 20 seconds where previous approaches could not even compute an arbitrary DDT.