Counting Quantifiers Research Articles

Lifted algorithms for symmetric weighted first-order model sampling

Weighted model counting (WMC) is the task of computing the weighted sum of all satisfying assignments (i.e., models) of a propositional formula. Similarly, weighted model sampling (WMS) aims to randomly generate models with probability proportional to their respective weights. Both WMC and WMS are hard to solve exactly, falling under the #P-hard complexity class. However, it is known that the counting problem may sometimes be tractable, if the propositional formula can be compactly represented and expressed in first-order logic. In such cases, model counting problems can be solved in time polynomial in the domain size, and are known as domain-liftable. The following question then arises: Is it also the case for WMS? This paper addresses this question and answers it affirmatively. Specifically, we prove the domain-liftability under sampling for the two-variables fragment of first-order logic with counting quantifiers in this paper, by devising an efficient sampling algorithm for this fragment that runs in time polynomial in the domain size. We then further show that this result continues to hold even in the presence of cardinality constraints. To empirically validate our approach, we conduct experiments over various first-order formulas designed for the uniform generation of combinatorial structures and sampling in statistical-relational models. The results demonstrate that our algorithm outperforms a state-of-the-art WMS sampler by a substantial margin, confirming the theoretical results.

Lifted inference with tree axioms

We consider the problem of weighted first-order model counting (WFOMC): given a first-order sentence ϕ and domain size n∈N, determine the weighted sum of models of ϕ over the domain {1,…,n}. Past work has shown that any sentence using at most two logical variables admits an algorithm for WFOMC that runs in time polynomial in the given domain size [1,2]. The same property was later also shown to hold for C2, the two-variable fragment with counting quantifiers [3]. In this paper, we further expand this result to any C2 sentence ϕ with the addition of a tree axiom, stating that some distinguished binary relation in ϕ forms a tree in the graph-theoretic sense.

On Presburger arithmetic extended with non-unary counting quantifiers

We consider a first-order logic for the integers with addition. This logic extends classical first-order logic by modulo-counting, threshold-counting and exact-counting quantifiers, all applied to tuples of variables (here, residues are given as terms while moduli and thresholds are given explicitly). Our main result shows that satisfaction for this logic is decidable in two-fold exponential space. If only threshold- and exact-counting quantifiers are allowed, we prove an upper bound of alternating two-fold exponential time with linearly many alternations. This latter result almost matches Berman's exact complexity of first-order logic without counting quantifiers. To obtain these results, we first translate threshold- and exact-counting quantifiers into classical first-order logic in polynomial time (which already proves the second result). To handle the remaining modulo-counting quantifiers for tuples, we first reduce them in doubly exponential time to modulo-counting quantifiers for single elements. For these quantifiers, we provide a quantifier elimination procedure similar to Reddy and Loveland's procedure for first-order logic and analyse the growth of coefficients, constants, and moduli appearing in this process. The bounds obtained this way allow to restrict quantification in the original formula to integers of bounded size which then implies the first result mentioned above. Our logic is incomparable with the logic considered by Chistikov et al. in 2022. They allow more general counting operations in quantifiers, but only unary quantifiers. The move from unary to non-unary quantifiers is non-trivial, since, e.g., the non-unary version of the H\"artig quantifier results in an undecidable theory.

Towards logical foundations for probabilistic computation

The overall purpose of the present work is to lay the foundations for a new approach to bridge logic and probabilistic computation. To this aim we introduce extensions of classical and intuitionistic propositional logic with counting quantifiers, that is, quantifiers that measure to which extent a formula is true. The resulting systems, called cCPL and iCPL, respectively, admit a natural semantics, based on the Borel σ-algebra of the Cantor space, together with a sound and complete proof system. Our main results consist in relating cCPL and iCPL with some central concepts in the study of probabilistic computation. On the one hand, the validity of cCPL-formulae in prenex form characterizes the corresponding level of Wagner's hierarchy of counting complexity classes, closely related to probabilistic complexity. On the other hand, proofs in iCPL correspond, in the sense of Curry and Howard, to typing derivations for a randomized extension of the λ-calculus, so that counting quantifiers reveal the probability of termination of the underlying probabilistic programs.

On counting propositional logic and Wagner's hierarchy

We introduce an extension of classical propositional logic with counting quantifiers. These forms of quantification make it possible to express that a formula is true in a certain portion of the set of all its interpretations. Beyond providing a sound and complete proof system for this logic, we show that validity problems for counting propositional logic can be used to capture counting complexity classes. More precisely, we show that the complexity of the decision problems for validity of prenex counting formulas perfectly matches the appropriate levels of Wagner's counting hierarchy.

Structure and Power: an Emerging Landscape

In this paper, we give an overview of some recent work on applying tools from category theory in finite model theory, descriptive complexity, constraint satisfaction, and combinatorics. The motivations for this work come from Computer Science, but there may also be something of interest for model theorists and other logicians. The basic setting involves studying the category of relational structures via a resource-indexed family of adjunctions with some process category - which unfolds relational structures into tree-like forms, allowing natural resource parameters to be assigned to these unfoldings. One basic instance of this scheme allows us to recover, in a purely structural, syntax-free way: the Ehrenfeucht-Fraïssé game; the quantifier rank fragments of first-order logic; the equivalences on structures induced by (i) the quantifier rank fragments, (ii) the restriction of this fragment to the existential positive part, and (iii) the extension with counting quantifiers; and the combinatorial parameter of tree-depth (Nesetril and Ossona de Mendez). Another instance recovers the k-pebble game, the finite-variable fragments, the corresponding equivalences, and the combinatorial parameter of treewidth. Other instances cover modal, guarded and hybrid fragments, generalized quantifiers, and a wide range of combinatorial parameters. This whole scheme has been axiomatized in a very general setting, of arboreal categories and arboreal covers. Beyond this basic level, a landscape is beginning to emerge, in which structural features of the resource categories, adjunctions and comonads are reflected in degrees of logical and computational tractability of the corresponding languages. Examples include semantic characterisation and preservation theorems, and Lovász-type results on counting homomorphisms.

Weighted Model Counting in FO2 with Cardinality Constraints and Counting Quantifiers: A Closed Form Formula

Weighted First-Order Model Counting (WFOMC) computes the weighted sum of the models of a first-order logic theory on a given finite domain. First-Order Logic theories that admit polynomial-time WFOMC w.r.t domain cardinality are called domain liftable. We introduce the concept of lifted interpretations as a tool for formulating closed forms for WFOMC. Using lifted interpretations, we reconstruct the closed-form formula for polynomial-time FOMC in the universally quantified fragment of FO2, earlier proposed by Beame et al. We then expand this closed-form to incorporate cardinality constraints, existential quantifiers, and counting quantifiers (a.k.a C2) without losing domain-liftability. Finally, we show that the obtained closed-form motivates a natural definition of a family of weight functions strictly larger than symmetric weight functions.

Alternating complexity of counting first-order logic for the subword order

This paper considers the structure consisting of the set of all words over a given alphabet together with the subword relation, regular predicates, and constants for every word. We are interested in the counting extension of first-order logic by threshold counting quantifiers. The main result shows that the two-variable fragment of this logic can be decided in twofold exponential alternating time with linearly many alternations (and therefore in particular in twofold exponential space as announced in the conference version (Kuske and Schwarz, in: MFCS’20, Leibniz International Proceedings in Informatics (LIPIcs) vol. 170, pp 56:1–56:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020) of this paper) provided the regular predicates are restricted to piecewise testable ones. This result improves prior insights by Karandikar and Schnoebelen by extending the logic and saving one exponent in the space bound. Its proof consists of two main parts: First, we provide a quantifier elimination procedure that results in a formula with constants of bounded length (this generalises the procedure by Karandikar and Schnoebelen for first-order logic). From this, it follows that quantification in formulas can be restricted to words of bounded length, i.e., the second part of the proof is an adaptation of the method by Ferrante and Rackoff to counting logic and deviates significantly from the path of reasoning by Karandikar and Schnoebelen.

Weighted First-Order Model Counting in the Two-Variable Fragment With Counting Quantifiers

It is known due to the work of Van den Broeck, Meert and Darwiche that weighted first-order model counting (WFOMC) in the two-variable fragment of first-order logic can be solved in time polynomial in the number of domain elements. In this paper we extend this result to the two-variable fragment with counting quantifiers.

Non-finitely axiomatisable modal product logics with infinite canonical axiomatisations

Our concern is the axiomatisation problem for modal and algebraic logics that correspond to various fragments of two-variable first-order logic with counting quantifiers. In particular, we consider modal products with Diff, the propositional unimodal logic of the difference operator. We show that the two-dimensional product logic Diff×Diff is non-finitely axiomatisable, but can be axiomatised by infinitely many Sahlqvist axioms. We also show that its ‘square’ version (the modal counterpart of the substitution and equality free fragment of two-variable first-order logic with counting to two) is non-finitely axiomatisable over Diff×Diff, but can be axiomatised by adding infinitely many Sahlqvist axioms. These are the first examples of products of finitely axiomatisable modal logics that are not finitely axiomatisable, but axiomatisable by explicit infinite sets of canonical axioms.

Graph pattern matching with counting quantifiers and label-repetition constraints

In recent years, we have witnessed an increasing use of graph pattern matching in a wide variety of applications such as social networks analysis, knowledge discovery, software plagiarism detection and many more. It is typically defined in terms of subgraph isomorphism, an NP-Complete problem. To overcome this cost, many extensions of graph simulation have been proposed that allow graph pattern matching to be conducted in cubic-time. However, in emerging applications, more expressive patterns are needed, notably ones with counting quantifiers (CQs) which are not considered by simulation-based approaches. In this article, we propose a simulation-based graph pattern matching approach that supports CQs on edges of graph patterns. We first consider CQs that express numeric aggregates only. We show that our approach is in ptime as earlier extensions of graph simulation by providing a cubic-time quantified matching algorithm, i.e., an algorithm for matching graph patterns that contain CQs. In the second part, we discuss the problem of Label-Repetition Constraints (LRCs). We define a necessary and sufficient condition for the satisfaction of LRCs. Based on this condition, we give an extension of our quantified matching algorithm to deal with LRCs in ptime, together with an optimization technique. Finally, we show that our quantified graph pattern matching approach retains the same complexity bounds when dealing with ratio aggregates. To our knowledge, this is the first effort to deal with numeric aggregates, ratio aggregates, and LRCs on graph patterns in ptime.

Reasoning about Social Choice and Games in Monadic Fixed-Point Logic

Whether it be in normal form games, or in fair allocations, or in voter preferences in voting systems, a certain pattern of reasoning is common. From a particular profile, an agent or a group of agents may have an incentive to shift to a new one. This induces a natural graph structure that we call the improvement graph on the strategy space of these systems. We suggest that the monadic fixed-point logic with counting, an extension of monadic first-order logic on graphs with fixed-point and counting quantifiers, is a natural specification language on improvement graphs, and thus for a class of properties that can be interpreted across these domains. The logic has an efficient model checking algorithm (in the size of the improvement graph).

Descriptive complexity of graph spectra

Two graphs are cospectral if their respective adjacency matrices have the same multi-set of eigenvalues. A graph is said to be determined by its spectrum if all graphs that are cospectral with it are isomorphic to it. We consider these properties in relation to logical definability. We show that any pair of graphs that are elementarily equivalent with respect to the three-variable counting first-order logic C3 are cospectral, and this is not the case with C2, nor with any number of variables if we exclude counting quantifiers. We also show that the class of graphs that are determined by their spectra is definable in partial fixed-point logic with counting. We relate these properties to other algebraic and combinatorial problems.

Achieving New Upper Bounds for the Hypergraph Duality Problem through Logic

The hypergraph duality problem DUAL is defined as follows: given two simple hypergraphs $\mathcal{G}$ and $\mathcal{H}$, decide whether $\mathcal{H}$ consists precisely of all minimal transversals of $\mathcal{G}$ (in which case we say that $\mathcal{G}$ is the dual of $\mathcal{H}$). This problem is equivalent to deciding whether two given non-redundant monotone DNFs are dual. It is known that non-DUAL, the complementary problem to DUAL, is in $\mathrm{GC}(\log^2 n,\mathrm{PTIME})$, where $\mathrm{GC}(f(n),\mathcal{C})$ denotes the complexity class of all problems that after a nondeterministic guess of $O(f(n))$ bits can be decided (checked) within complexity class $\mathcal{C}$. It was conjectured that non-DUAL is in $\mathrm{GC}(\log^2 n,\mathrm{LOGSPACE})$. In this paper we prove this conjecture and actually place the non-DUAL problem into the complexity class $\mathrm{GC}(\log^2 n,\mathrm{TC}^0)$ which is a subclass of $\mathrm{GC}(\log^2 n,\mathrm{LOGSPACE})$. We here refer to the logtime-uniform version of $\mathrm{TC}^0$, which corresponds to $\mathrm{FO(COUNT)}$, i.e., first order logic augmented by counting quantifiers. We achieve the latter bound in two steps. First, based on existing problem decomposition methods, we develop a new nondeterministic algorithm for non-DUAL that requires to guess $O(\log^2 n)$ bits. We then proceed by a logical analysis of this algorithm, allowing us to formulate its deterministic part in $\mathrm{FO(COUNT)}$. From this result, by the well known inclusion $\mathrm{TC}^0\subseteq\mathrm{LOGSPACE}$, it follows that DUAL belongs also to $\mathrm{DSPACE}[\log^2 n]$. Finally, by exploiting the principles on which the proposed nondeterministic algorithm is based, we devise a deterministic algorithm that, given two hypergraphs $\mathcal{G}$ and $\mathcal{H}$, computes in quadratic logspace a transversal of $\mathcal{G}$ missing in $\mathcal{H}$.

Managing Change in Graph-Structured Data Using Description Logics

In this article, we consider the setting of graph-structured data that evolves as a result of operations carried out by users or applications. We study different reasoning problems, which range from deciding whether a given sequence of actions preserves the satisfaction of a given set of integrity constraints, for every possible initial data instance, to deciding the (non)existence of a sequence of actions that would take the data to an (un)desirable state, starting either from a specific data instance or from an incomplete description of it. For describing states of the data instances and expressing integrity constraints on them, we use description logics (DLs) closely related to the two-variable fragment of first-order logic with counting quantifiers. The updates are defined as actions in a simple yet flexible language, as finite sequences of conditional insertions and deletions, which allow one to use complex DL formulas to select the (pairs of) nodes for which (node or arc) labels are added or deleted. We formalize the preceding data management problems as a static verification problem and several planning problems and show that, due to the adequate choice of formalisms for describing actions and states of the data, most of these data management problems can be effectively reduced to the (un)satisfiability of suitable formulas in decidable logical formalisms. Leveraging this, we provide algorithms and tight complexity bounds for the formalized problems, both for expressive DLs and for a variant of the popular DL-Lite, advocated for data management in recent years.

Uniform proofs of ACC representations

We give a uniform proof of the theorems of Yao and Beigel–Tarui representing ACC predicates as constant depth circuits with $$\hbox {MOD}_{m}$$ gates and a symmetric gate. The proof is based on a relativized, generalized form of Toda’s theorem expressed in terms of closure properties of formulas under bounded universal, existential and modular counting quantifiers. This allows the main proofs to be expressed in terms of formula classes instead of Boolean circuits. The uniform version of the Beigel–Tarui theorem is then obtained automatically via the Furst–Saxe–Sipser and Paris–Wilkie translations. As a special case, we obtain a uniform version of Razborov and Smolensky’s representation of $$\hbox {AC}^{0}[p]$$ circuits. The paper is partly expository, but is also motivated by the desire to recast Toda’s theorem, the Beigel–Tarui theorem, and their proofs into the language of bounded arithmetic. However, no knowledge of bounded arithmetic is needed.

On the locality of arb-invariant first-order formulas with modulo counting quantifiers

We study Gaifman locality and Hanf locality of an extension of first-order logic with modulo p counting quantifiers (FO+MOD_p, for short) with arbitrary numerical predicates. We require that the validity of formulas is independent of the particular interpretation of the numerical predicates and refer to such formulas as arb-invariant formulas. This paper gives a detailed picture of locality and non-locality properties of arb-invariant FO+MOD_p. For example, on the class of all finite structures, for any p >= 2, arb-invariant FO+MOD_p is neither Hanf nor Gaifman local with respect to a sublinear locality radius. However, in case that p is an odd prime power, it is weakly Gaifman local with a polylogarithmic locality radius. And when restricting attention to the class of string structures, for odd prime powers p, arb-invariant FO+MOD_p is both Hanf and Gaifman local with a polylogarithmic locality radius. Our negative results build on examples of order-invariant FO+MOD_p formulas presented in Niemist\"o's PhD thesis. Our positive results make use of the close connection between FO+MOD_p and Boolean circuits built from NOT-gates and AND-, OR-, and MOD_p- gates of arbitrary fan-in.

Cardinality constraints for arrays (decidability results and applications)

Enriching logic formalisms with counting capabilities is an important task in view of the needs of many application areas, ranging from database theory to formal verification. In this paper, we consider a very expressive language obtained by enriching linear integer arithmetic with free function symbols and cardinality constraints for interpreted sets. We obtain positive results for a flat fragment via a reduction to decidability of Presburger arithmetic with unary counting quantifiers (Schweikhart in Arithmetic, first-order logic, and counting quantifiers, ACM TOCL, New York, 2004). We isolate also an easier simple flat subfragment, whose satisfiability is in NP, and we show that this subfragment is adequate to formalize problems arising in the area of the verification of fault-tolerant distributed algorithms. We finally discuss our first implementation, the related experimental results, as well as further algorithmic problems suggested by model-checking applications.

Algorithms for Deciding Counting Quantifiers over Unary Predicates

We study algorithms for fragments of first order logic ex- tended with counting quantifiers, which are known to be highly complex in general. We propose a fragment over unary predicates that is NP-complete and for which there is a nor- mal form where Counting Quantification sentences have a single Unary predicate, thus call it the CQU fragment. We provide an algebraic formulation of the CQU satisfiability problem in terms of Integer Linear Programming based on which two algorithms are proposed, a direct reduction to SAT instances and an Integer Linear Programming version extended with a column generation mechanism. The latter is shown to lead to a viable implementation and experiments shows this algorithm presents a phase transition behavior.

The complexity of counting quantifiers on equality languages

An equality language is a relational structure with infinite domain whose relations are first-order definable in equality. We classify the extensions of the quantified constraint satisfaction problem over equality languages in which the native existential and universal quantifiers are augmented by some subset of counting quantifiers. In doing this, we find ourselves in various worlds in which dichotomies or trichotomies subsist.