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Stefan Mengel

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21 papers

2 author rows

AAAI Conference 2025 Conference Paper

Sum of Squares Circuits

Lorenzo Loconte
Stefan Mengel
Antonio Vergari

Designing expressive generative models that support exact and efficient inference is a core question in probabilistic ML. Probabilistic circuits (PCs) offer a framework where this tractability-vs-expressiveness trade-off can be analyzed theoretically. Recently, squared PCs encoding subtractive mixtures via negative parameters have emerged as tractable models that can be exponentially more expressive than monotonic PCs, i.e., PCs with positive parameters only. In this paper we provide a more precise theoretical characterization of the expressiveness relationships among these models. First, we prove that squared PCs can be less expressive than monotonic ones. Second, we formalize a novel class of PCs – sum of squares PCs – that can be exponentially more expressive than both squared and monotonic PCs. Around sum of squares PCs, we build an expressiveness hierarchy that allows us to precisely unify and separate different tractable model classes such as Born Machines and PSD models, and other recently introduced tractable probabilistic models by using complex parameters. Finally, we empirically show the effectiveness of sum of squares circuits in performing distribution estimation.

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ICLR Conference 2024 Conference Paper

Subtractive Mixture Models via Squaring: Representation and Learning

Lorenzo Loconte
Aleksanteri M. Sladek
Stefan Mengel
Martin Trapp 0001
Arno Solin
Nicolas Gillis
Antonio Vergari

Mixture models are traditionally represented and learned by adding several distributions as components. Allowing mixtures to subtract probability mass or density can drastically reduce the number of components needed to model complex distributions. However, learning such subtractive mixtures while ensuring they still encode a non-negative function is challenging. We investigate how to learn and perform inference on deep subtractive mixtures by squaring them. We do this in the framework of probabilistic circuits, which enable us to represent tensorized mixtures and generalize several other subtractive models. We theoretically prove that the class of squared circuits allowing subtractions can be exponentially more expressive than traditional additive mixtures; and, we empirically show this increased expressiveness on a series of real-world distribution estimation tasks.

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SAT Conference 2023 Conference Paper

Bounds on BDD-Based Bucket Elimination

Stefan Mengel

We study BDD-based bucket elimination, an approach to satisfiability testing using variable elimination which has seen several practical implementations in the past. We prove that it allows solving the standard pigeonhole principle formulas efficiently, when allowing different orders for variable elimination and BDD-representations, a variant of bucket elimination that was recently introduced. Furthermore, we show that this upper bound is somewhat brittle as for formulas which we get from the pigeonhole principle by restriction, i. e. , fixing some of the variables, the same approach with the same variable orders has exponential runtime. We also show that the more common implementation of bucket elimination using the same order for variable elimination and the BDDs has exponential runtime for the pigeonhole principle when using either of the two orders from our upper bound, which suggests that the combination of both is the key to efficiency in the setting.

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JAIR Journal 2023 Journal Article

Characterizing Tseitin-Formulas with Short Regular Resolution Refutations

Alexis de Colnet
Stefan Mengel

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.

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SAT Conference 2022 Conference Paper

Changing Partitions in Rectangle Decision Lists

Stefan Mengel

Rectangle decision lists are a form of decision lists that were recently shown to have applications in the proof complexity of certain OBDD-based QBF-solvers. We consider a version of rectangle decision lists with changing partitions, which corresponds to QBF-solvers that may change the variable order of the OBDDs they produce. We show that even allowing one single partition change generally leads to exponentially more succinct decision lists. More generally, we show that there is a succinctness hierarchy: for every k ∈ ℕ, when going from k partition changes to k+1, there are functions that can be represented exponentially more succinctly. As an application, we show a similar hierarchy for OBDD-based QBF-solvers.

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AAAI Conference 2022 Conference Paper

Lower Bounds on Intermediate Results in Bottom-Up Knowledge Compilation

Alexis de Colnet
Stefan Mengel

Bottom-up knowledge compilation is a paradigm for generating representations of functions by iteratively conjoining constraints using a so-called apply function. When the input is not efficiently compilable into a language – generally a class of circuits – because optimal compiled representations are provably large, the problem is not the compilation algorithm as much as the choice of a language too restrictive for the input. In contrast, in this paper, we look at CNF formulas for which very small circuits exists and look at the efficiency of their bottom-up compilation in one of the most general languages, namely that of structured decomposable negation normal forms (str-DNNF). We prove that, while the inputs have constant size representations as str-DNNF, any bottomup compilation in the general setting where conjunction and structure modification are allowed takes exponential time and space, since large intermediate results have to be produced. This unconditionally proves that the inefficiency of bottomup compilation resides in the bottom-up paradigm itself.

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KR Conference 2021 Conference Paper

A Compilation of Succinctness Results for Arithmetic Circuits

Alexis de Colnet
Stefan Mengel

Arithmetic circuits (AC) are circuits over the real numbers with 0/1-valued input variables whose gates compute the sum or the product of their inputs. Positive AC – that is, AC representing non-negative functions – subsume many interesting probabilistic models such as probabilistic sentential decision diagram (PSDD) or sum-product network (SPN) on indicator variables. Efficient algorithms for many operations useful in probabilistic reasoning on these models critically depend on imposing structural restrictions to the underlying AC. Generally, adding structural restrictions yields new tractable operations but increases the size of the AC. In this paper we study the relative succinctness of classes of AC with different combinations of common restrictions. Building on existing results for Boolean circuits, we derive an unconditional succinctness map for classes of monotone AC – that is, AC whose constant labels are non-negative reals – respecting relevant combinations of the restrictions we consider. We extend a small part of the map to classes of positive AC. Those are known to generally be exponentially more succinct than their monotone counterparts, but we observe here that for so-called deterministic circuits there is no difference between the monotone and the positive setting which allows us to lift some of our results. We end the paper with some insights on the relative succinctness of positive AC by showing exponential lower bounds on the representations of certain functions in positive AC respecting structured decomposability.

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SAT Conference 2021 Conference Paper

Characterizing Tseitin-Formulas with Short Regular Resolution Refutations

Alexis de Colnet
Stefan Mengel

Abstract 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 \(O(\log |V(G)|)\). To do so, we show that any regular resolution refutation of an unsatisfiable Tseitin-formula with graph G of bounded degree has length \(2^{\varOmega (tw(G))}/|V(G)|\), thus essentially matching the known \(2^{O(tw(G))}\text {poly}(|V(G)|)\) upper bound up. 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 \(2^{\varOmega (tw(G))}\) which yields our lower bound for regular resolution.

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SAT Conference 2021 Conference Paper

Proof Complexity of Symbolic QBF Reasoning

Stefan Mengel
Friedrich Slivovsky

Abstract We introduce and investigate symbolic proof systems for Quantified Boolean Formulas (QBF) operating on Ordered Binary Decision Diagrams (OBDDs). These systems capture QBF solvers that perform symbolic quantifier elimination, and as such admit short proofs of formulas of bounded path-width and quantifier complexity. As a consequence, we obtain exponential separations from standard clausal proof systems, specifically (long-distance) QU-Resolution and IR-Calc. We further develop a lower bound technique for symbolic QBF proof systems based on strategy extraction that lifts known lower bounds from communication complexity. This allows us to derive strong lower bounds against symbolic QBF proof systems that are independent of the variable ordering of the underlying OBDDs, and that hold even if the proof system is allowed access to an NP-oracle.

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JAIR Journal 2020 Journal Article

Graph Width Measures for CNF-Encodings with Auxiliary Variables

Stefan Mengel
Romain Wallon

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.

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IJCAI Conference 2020 Conference Paper

Lower Bounds for Approximate Knowledge Compilation

Alexis de Colnet
Stefan Mengel

Knowledge compilation studies the trade-off between succinctness and efficiency of different representation languages. For many languages, there are known strong lower bounds on the representation size, but recent work shows that, for some languages, one can bypass these bounds using approximate compilation. The idea is to compile an approximation of the knowledge for which the number of errors can be controlled. We focus on circuits in deterministic decomposable negation normal form (d-DNNF), a compilation language suitable in contexts such as probabilistic reasoning, as it supports efficient model counting and probabilistic inference. Moreover, there are known size lower bounds for d-DNNF which by relaxing to approximation one might be able to avoid. In this paper we formalize two notions of approximation: weak approximation which has been studied before in the decision diagram literature and strong approximation which has been used in recent algorithmic results. We then show lower bounds for approximation by d-DNNF, complementing the positive results from the literature.

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IJCAI Conference 2020 Conference Paper

On Irrelevant Literals in Pseudo-Boolean Constraint Learning

Daniel Le Berre
Pierre Marquis
Stefan Mengel
Romain Wallon

Learning pseudo-Boolean (PB) constraints in PB solvers exploiting cutting planes based inference is not as well understood as clause learning in conflict-driven clause learning solvers. In this paper, we show that PB constraints derived using cutting planes may contain irrelevant literals, i. e. , literals whose assigned values (whatever they are) never change the truth value of the constraint. Such literals may lead to infer constraints that are weaker than they should be, impacting the size of the proof built by the solver, and thus also affecting its performance. This suggests that current implementations of PB solvers based on cutting planes should be reconsidered to prevent the generation of irrelevant literals. Indeed, detecting and removing irrelevant literals is too expensive in practice to be considered as an option (the associated problem is NP-hard).

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SAT Conference 2019 Conference Paper

Revisiting Graph Width Measures for CNF-Encodings

Stefan Mengel
Romain Wallon

Abstract 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.

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IJCAI Conference 2018 Conference Paper

Pseudo-Boolean Constraints from a Knowledge Representation Perspective

Daniel Le Berre
Pierre Marquis
Stefan Mengel
Romain Wallon

We study pseudo-Boolean constraints (PBC) and their special case cardinality constraints (CARD) from the perspective of knowledge representation. To this end, the succinctness of PBC and CARD is compared to that of many standard propositional languages. Moreover, we determine which queries and transformations are feasible in polynomial time when knowledge is represented by PBC or CARD, and which are not (unconditionally or unless P = NP). In particular, the advantages and disadvantages compared to CNF are discussed.

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SAT Conference 2018 Conference Paper

QBF as an Alternative to Courcelle's Theorem

Michael Lampis
Stefan Mengel
Valia Mitsou

Abstract We propose reductions to quantified Boolean formulas (QBF) as a new approach to showing fixed-parameter linear algorithms for problems parameterized by treewidth. We demonstrate the feasibility of this approach by giving new algorithms for several well-known problems from artificial intelligence that are in general complete for the second level of the polynomial hierarchy. By reduction from QBF we show that all resulting algorithms are essentially optimal in their dependence on the treewidth. Most of the problems that we consider were already known to be fixed-parameter linear by using Courcelle’s Theorem or dynamic programming, but we argue that our approach has clear advantages over these techniques: on the one hand, in contrast to Courcelle’s Theorem, we get concrete and tight guarantees for the runtime dependence on the treewidth. On the other hand, we avoid tedious dynamic programming and, after showing some normalization results for CNF-formulas, our upper bounds often boil down to a few lines.

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Highlights Conference 2016 Conference Abstract

A Dual to a Classical Graph Homomorphism Theorem of Lovász

Hubie Chen
Stefan Mengel

Homomorphism between finite structures are a classical object of study considered in many areas including structural graph theory, constraint satisfaction and database theory. A classical result by Lovász states that every structure B is uniquely characterized (up to isomorphism) by the number of homomorphisms from other structures to B. In this talk I will I will give a dual version of this result that roughly says that every structure is uniquely described by the number of homomorphisms it has to other graphs.

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IJCAI Conference 2016 Conference Paper

Knowledge Compilation Meets Communication Complexity

Simone Bova
Florent Capelli
Stefan Mengel
Friedrich Slivovsky

Choosing a language for knowledge representation and reasoning involves a trade-off between two competing desiderata: succinctness (the encoding should be small) and tractability (the language should support efficient reasoning algorithms). The area of knowledge compilation is devoted to the systematic study of representation languages along these two dimensions - in particular, it aims to determine the relative succinctness of languages. Showing that one language is more succinct than another typically involves proving a nontrivial lower bound on the encoding size of a carefully chosen function, and the corresponding arguments increase in difficulty with the succinctness of the target language. In this paper, we introduce a general technique for obtaining lower bounds on Decomposable Negation Normal Form (DNNFs), one of the most widely studied and succinct representation languages, by relating the size of DNNFs to multi-partition communication complexity. This allows us to directly translate lower bounds from the communication complexity literature into lower bounds on the size of DNNF representations. We use this approach to prove exponential separations of DNNFs from deterministic DNNFs and of CNF formulas from DNNFs.

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SAT Conference 2016 Conference Paper

Parameterized Compilation Lower Bounds for Restricted CNF-Formulas

Stefan Mengel

Abstract We show unconditional parameterized lower bounds in the area of knowledge compilation, more specifically on the size of circuits in decomposable negation normal form (DNNF) that encode CNF-formulas restricted by several graph width measures. In particular, we show that there are CNF formulas of size n and modular incidence treewidth k whose smallest DNNF-encoding has size \(n^{\varOmega (k)}\), and there are CNF formulas of size n and incidence neighborhood diversity k whose smallest DNNF-encoding has size \(n^{\varOmega (\sqrt{k})}\). These results complement recent upper bounds for compiling CNF into DNNF and strengthen—quantitatively and qualitatively—known conditional lower bounds for cliquewidth. Moreover, they show that, unlike for many graph problems, the parameters considered here behave significantly differently from treewidth.

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SAT Conference 2015 Conference Paper

On Compiling CNFs into Structured Deterministic DNNFs

Simone Bova
Florent Capelli
Stefan Mengel
Friedrich Slivovsky

Abstract We show that the traces of recently introduced dynamic programming algorithms for #SAT can be used to construct structured deterministic DNNF (decomposable negation normal form) representations of propositional formulas in CNF (conjunctive normal form). This allows us prove new upper bounds on the complexity of compiling CNF formulas into structured deterministic DNNFs in terms of parameters such as the treewidth and the clique-width of the incidence graph.

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SAT Conference 2014 Conference Paper

Hypergraph Acyclicity and Propositional Model Counting

Florent Capelli
Arnaud Durand 0001
Stefan Mengel

Abstract We show that the propositional model counting problem #SAT for CNF-formulas with hypergraphs that allow a disjoint branches decomposition can be solved in polynomial time. We show that this class of hypergraphs is incomparable to hypergraphs of bounded incidence cliquewidth which were the biggest class of hypergraphs for which #SAT was known to be solvable in polynomial time so far. Furthermore, we present a polynomial time algorithm that computes a disjoint branches decomposition of a given hypergraph if it exists and rejects otherwise. Finally, we show that some slight extensions of the class of hypergraphs with disjoint branches decompositions lead to intractable #SAT, leaving open how to generalize the counting result of this paper.

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MFCS Conference 2013 Conference Paper

Arithmetic Branching Programs with Memory

Stefan Mengel

Abstract We extend the well known characterization of the arithmetic circuit class VP ws as the class of polynomials computed by polynomial size arithmetic branching programs to other complexity classes. In order to do so we add additional memory to the computation of branching programs to make them more expressive. We show that allowing different types of memory in branching programs increases the computational power even for constant width programs. In particular, this leads to very natural and robust characterizations of VP and VNP by branching programs with memory.

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