We characterise the computational power of recurrent graph neural networks (GNNs) in terms of arithmetic circuits over the real numbers. Our networks are not restricted to aggregate-combine GNNs or other particular types. Generalising similar notions from the literature, we introduce the model of recurrent arithmetic circuits, which can be seen as arithmetic analogues of sequential or logical circuits. These circuits utilise so-called memory gates which are used to store data between iterations of the recurrent circuit. While (recurrent) GNNs work on labelled graphs, we construct arithmetic circuits that obtain encoded labelled graphs as real valued tuples and then compute the same function. For the other direction we construct recurrent GNNs which are able to simulate the computations of recurrent circuits. These GNNs are given the circuit-input as initial feature vectors and then, after the GNN-computation, have the circuit-output among the feature vectors of its nodes. In this way we establish an exact correspondence between the expressivity of recurrent GNNs and recurrent arithmetic circuits operating over real numbers. Our results both deepen our understanding of the capabilities of trained neural networks and open new approaches to study recurrent neural networks using the lens of circuit complexity theory.

Abstract In the eighties, Immerman showed that the class of languages definable by first-order formulae coincides with the class of languages decidable by unbounded fan-in Boolean circuits of constant depth and polynomial size. We show an analogous result for real-valued computation, i.e. we define circuits of unbounded fan-in operating over real numbers and show that families of such circuits of polynomial size and constant depth decide exactly those sets of vectors of reals that can be defined in first-order logic on real valued structures. Our characterization holds both non-uniformly as well as for many natural uniformity conditions.

We characterize the computational power of neural networks that follow the graph neural network (GNN) architecture, not restricted to aggregate-combine GNNs or other particular types. We establish an exact correspondence between the expressivity of GNNs using diverse activation functions and arithmetic circuits over real numbers. In our results the activation function of the network becomes a gate type in the circuit. Our result holds for families of constant depth circuits and networks, both uniformly and non-uniformly, for all common activation functions.

Abstract In this article, we study the complexity of weighted team definability for logics with team semantics. This problem is a natural analog of one of the most studied problems in parameterized complexity, the notion of weighted Fagin-definability, which is formulated in terms of satisfaction of first-order formulas with free relation variables. We focus on the parameterized complexity of weighted team definability for a fixed formula $\varphi$ of central team-based logics. Given a first-order structure $\mathcal{A}$ and the parameter value $k\in \mathbb N$ as input, the question is to determine whether $\mathcal{A},T\models \varphi$ for some team T of size k. We show several results on the complexity of this problem for dependence, independence, and inclusion logic formulas. Moreover, we also relate the complexity of weighted team definability to the complexity classes in the well-known W-hierarchy as well as paraNP.

We refine the complexity landscape for enumeration problems by introducing very low classes defined by using Boolean circuits as enumerators. We locate well-known enumeration problems, e.g., from graph theory, Gray code enumeration, and propositional satisfiability in our classes. In this way we obtain a framework to distinguish between the complexity of different problems known to be in $\mathbf{DelayP}$, for which a formal way of comparison was not possible to this day.

We investigate the computational complexity of the satisfiability problem of modal inclusion logic. We distinguish two variants of the problem: one for the strict and another one for the lax semantics. Both problems turn out to be EXPTIME-complete on general structures. Finally, we show how for a specific class of structures NEXPTIME-completeness for these problems under strict semantics can be achieved.

We study descriptive complexity of counting complexity classes in the range from #P to #*NP. A corollary of Fagin’s characterization of NP by existential second-order logic is that #P can be logically described as the class of functions counting satisfying assignments to free relation variables in first-order formulae. In this paper we extend this study to classes beyond #P and extensions of first-order logic with team semantics. These team-based logics are closely related to existential second-order logic and its fragments, hence our results also shed light on the complexity of counting for extensions of first-order logic in Tarski’s semantics. Our results show that the class #*NP can be logically characterized by independence logic and existential second-order logic, whereas dependence logic and inclusion logic give rise to subclasses of #*NP and #P, respectively. We also study the function class generated by inclusion logic and relate it to the complexity class TotP, which is a subclass of #P. Our main technical result shows that the problem of counting satisfying assignments for monotone Boolean Sigma_1-formulae is #*NP-complete with respect to Turing reductions as well as complete for the function class generated by dependence logic with respect to first-order reductions.

In this paper we give a characterization of both Boolean and arithmetic circuit classes of logarithmic depth in the vein of descriptive complexity theory, i.e., the Boolean classes NC1, SAC1 and AC1 as well as their arithmetic counterparts #NC1, #SAC1 and #AC1. We build on Immerman's characterization of constant-depth polynomial-size circuits by formulae of first-order logic, i.e., AC0 = FO, and augment the logical language with an operator for defining relations in an inductive way. Considering slight variations of the new operator, we obtain uniform characterizations of the three just mentioned Boolean classes. The arithmetic classes can then be characterized by functions counting winning strategies in semantic games for formulae characterizing languages in the corresponding Boolean class.

We classify the computational complexity of the satisfiability, validity, and model-checking problems for propositional independence, inclusion, and team logic. Our main result shows that the satisfiability and validity problems for propositional team logic are complete for alternating exponential-time with polynomially many alternations.

This article introduces modal independence logic MIL, a modal logic that can explicitly talk about independence among propositional variables. Formulas of MIL are not evaluated in worlds but in sets of worlds, so called teams. In this vein, MIL can be seen as a variant of Väänänen’s modal dependence logic MDL. We show that MIL embeds MDL and is strictly more expressive. However, on singleton teams, MIL is shown to be not more expressive than usual modal logic, but MIL is exponentially more succinct. Making use of a new form of bisimulation, we extend these expressivity results to modal logics extended by various generalized dependence atoms. We demonstrate the expressive power of MIL by giving a specification of the anonymity requirement of the dining cryptographers protocol in MIL. We also study complexity issues of MIL and show that, though it is more expressive, its satisfiability and model checking problem have the same complexity as for MDL.

We introduce a new framework for a descriptive complexity approach to arithmetic computations. We define a hierarchy of classes based on the idea of counting assignments to free function variables in first-order formulae. We completely determine the inclusion structure and show that #P and #AC^0 appear as classes of this hierarchy. In this way, we unconditionally place #AC^0 properly in a strict hierarchy of arithmetic classes within #P. We compare our classes with a hierarchy within #P defined in a model-theoretic way by Saluja et al. We argue that our approach is better suited to study arithmetic circuit classes such as #AC^0 which can be descriptively characterized as a class in our framework.

The famous van Benthem theorem states that modal logic corresponds exactly to the fragment of first-order logic that is invariant under bisimulation. In this article we prove an exact analogue of this theorem in the framework of modal dependence logic (MDL) and team semantics. We show that Modal Team Logic (MTL) extending MDL by classical negation captures exactly the FO-definable bisimulation invariant properties of Kripke structures and teams. We also compare the expressive power of MTL to most of the variants and extensions of MDL recently studied in the area.

We consider the weighted satisfiability problem for Boolean circuits and propositional formulæ, where the weight of an assignment is the number of variables set to true. We study the parameterized complexity of these problems and initiate a systematic study of the complexity of its fragments. Only the monotone fragment has been considered so far and proven to be of same complexity as the unrestricted problems. Here, we consider all fragments obtained by semantically restricting circuits or formulæ to contain only gates (connectives) from a fixed set B of Boolean functions. We obtain a dichotomy result by showing that for each such B, the weighted satisfiability problems are either W[P]-complete (for circuits) or W[SAT]-complete (for formulæ) or efficiently solvable. We also consider the related enumeration and counting problems.

In this work, we present LoCo, a fragment of classical first-order logic carefully tailored for expressing technical product configuration problems. The core feature of LoCo is that the number of components used in configurations does not have to be finitely bounded explicitly, but instead is bounded implicitly through the axioms. Computing configurations is equivalent to the task of model finding. We present the language, related algorithms, and complexity results as well as a prototypical implementation via answer set programming.

Default logic was introduced by Reiter in 1980. In 1992, Gottlob classified the complexity of the extension existence problem for propositional default logic as Σ2p-complete, and the complexity of the credulous and skeptical reasoning problem as Σ2p-complete, respectively Π2p-complete. Additionally, he investigated restrictions on the default rules, i.e. semi-normal default rules. Selman used in 1992 a similar approach with disjunction-free and unary default rules. In this article, we systematically restrict the set of allowed propositional connectives. We give a complete complexity classification for all sets of Boolean functions in the meaning of Post's lattice for all three common decision problems for propositional default logic. We show that the complexity is a hexachotomy (⁠Σ2p-, Δ2p-, NP-, P-, NL-complete, trivial) for the extension existence problem, while for the credulous and skeptical reasoning problem we obtain similar classifications without trivial cases.

Autoepistemic logic extends propositional logic by the modal operator L . A formula φ that is preceded by an L is said to be “believed.” The logic was introduced by Moore in 1985 for modeling an ideally rational agent’s behavior and reasoning about his own beliefs. In this article we analyze all Boolean fragments of autoepistemic logic with respect to the computational complexity of the three most common decision problems expansion existence, brave reasoning and cautious reasoning. As a second contribution we classify the computational complexity of checking that a given set of formulae characterizes a stable expansion and that of counting the number of stable expansions of a given knowledge base. We improve the best known Δ 2 p -upper bound on the former problem to completeness for the second level of the Boolean hierarchy. To the best of our knowledge, this is the first paper analyzing counting problem for autoepistemic logic.

We study the extension of dependence logic D by a majority quantifier M over finite structures. We show that the resulting logic is equi-expressive with the extension of second-order logic by second-order majority quantifiers of all arities. Our results imply that, from the point of view of descriptive complexity theory, D(M) captures the complexity class counting hierarchy.

In a seminal paper from 1985, Sistla and Clarke showed that the model-checking problem for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If in contrast, the set of propositional operators is restricted, the complexity may decrease. This article systematically studies the model-checking problem for LTL formulae over restricted sets of propositional and temporal operators. For almost all combinations of temporal and propositional operators, we determine whether the model-checking problem is tractable (in PTIME) or intractable (NP-hard). We then focus on the tractable cases, showing that they all are NL-complete or even logspace solvable. This leads to a surprising gap in complexity between tractable and intractable cases. It is worth noting that our analysis covers an infinite set of problems, since there are infinitely many sets of propositional operators.

Imposing an extensional uniformity condition on a nonuniform circuit complexity class $\mathcal{C}$ means simply intersecting $\mathcal{C}$ with a uniform class $\mathcal{L}$. By contrast, the usual intensional uniformity conditions require that a resource-bounded machine be able to exhibit the circuits in the circuit family defining $\mathcal{C}$. We say that $(\mathcal{C},\mathcal{L})$ has the uniformity duality property if the extensionally uniform class $\mathcal{C}\cap\mathcal{L}$ can be captured intensionally by means of adding so-called $\mathcal{L}$-numerical predicates to the first-order descriptive complexity apparatus describing the connection language of the circuit family defining $\mathcal{C}$. This paper exhibits positive instances and negative instances of the uniformity duality property.

The model checking problem for CTL is known to be P-complete (Clarke, Emerson, and Sistla (1986), see Schnoebelen (2002)). We consider fragments of CTL obtained by restricting the use of temporal modalities or the use of negations---restrictions already studied for LTL by Sistla and Clarke (1985) and Markey (2004).For all these fragments, except for the trivial case without any temporal operator, we systematically prove model checking to be either inherently sequential (P-complete) or very efficiently parallelizable (LOGCFL-complete). For most fragments, however, model checking for CTL is already P-complete. Hence our results indicate that in most applications, approaching CTL model checking by parallelism will not result in the desired speed up. We also completely determine the complexity of the model checking problem for all fragments of the extensions ECTL, CTL+, and ECTL+.

In a seminal paper from 1985, Sistla and Clarke showed that satisfiability for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrease. This paper undertakes a systematic study of satisfiability for LTL formulae over restricted sets of propositional and temporal operators. Since every propositional operator corresponds to a Boolean function, there exist infinitely many propositional operators. In order to systematically cover all possible sets of them, we use Post's lattice. With its help, we determine the computational complexity of LTL satisfiability for all combinations of temporal operators and all but two classes of propositional functions. Each of these infinitely many problems is shown to be either PSPACE-complete, NP-complete, or in P.

This paper addresses the problems of counting proof-trees (as introduced by Venkateswaran and Tompa) and counting proof-circuits, a related but seemingly more natural question. These problems lead to a common generalization of straight-line programs which we call polynomial replacement systems {PRSs}. We contribute a classification of these systems and we investigate their complexity. Diverse problems falling within the scope of this study include, for example, counting proof-circuits and evaluating $\{\cup,+\}$-circuits over the natural numbers. A number of complexity results are obtained, including a proof that counting proof-circuits is $\numP$-complete.

A binary sequence $A=A(0)A(1)\ldots$ is called infinitely often (i.o.)~Turing-au\-to\-re\-duc\-ible if A~is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either $A(x)$ or a don't-know symbol on any given input~x, and outputs $A(x)$ for infinitely many~x. If in addition the oracle Turing machine terminates on all inputs and oracles, A~is called i.o.~truth-table-autoreducible. We obtain the somewhat counterintuitive result that every Martin-L\"of random sequence, in fact even every rec-random or p-random sequence, is i.o.~truth-table-autoreducible. Furthermore, we investigate the question of how dense the set of guessed bits can be when i.o.~autoreducing a random sequence. We show that rec-random sequences are never i.o.~truth-table-autoreducible such that the set of guessed bits has positive constant density in the limit and that a similar assertion holds for Martin-L\"of random sequences and i.o.~Turing autoreducibility. On the other hand, we show that for any rational-valued computable function~r that goes nonascendingly to zero, any rec-random sequence is i.o.~truth-table-autoreducible such that on any prefix of length~m at least a fraction of~$r(m)$ of the m~bits in the prefix are guessed. We include a self-contained account of the hat problem, a puzzle that has received some attention outside of theoretical computer science. The hat problem asks for guessing bits of a finite sequence, thus illustrating the notion of i.o.~autoreducibility in a finite setting. The solution to the hat problem is then used as a module in the proofs of the positive results on i.o.~autoreducibility.

We discuss the use of operators of higher types in complexity theory. These are operators ranging over sets of words, i.e. over oracles. Depending on different oracle access mechanisms we consider two types of operators. In particular we examine existential, universal, and bounded-error probabilistic operators. We identify some of the emerging classes and we interpret recent results about interactive protocols in terms of these operators.

We define the counting classes NC/sup 1/, GapNC/sup 1/ PNC/sup 1/ and C/sub =/NC/sup 1/. We prove that Boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We observe that NC/sup 1//spl sube/L and that C/sub =/NC/sup 1//spl sube/L. Then we exploit our finite automaton model and extend the padding techniques used to investigate leaf languages. Finally, we draw some consequences from the resulting body of leaf language characterizations of complexity classes, including the unconditional separation of ACC/sup 0/ from MOD-PH as well as that of TC/sup 0/ from the counting hierarchy. Moreover we obtain that dlogtime-uniformity and logspace-uniformity for AC/sup 0/ coincide if and only if the polynomial time hierarchy equals PSPACE.

We study the class coNPMV of complements of NPMV functions. Though defined symmetrically to NPMV this class exhibits very different properties. We clarify the complexity of coNPMV by showing that it is essentially the same as that of NPMV/sup NP/ complete functions for coNPMV are exhibited and central complexity-theoretic properties of this class are studied. We show that computing maximum satisfying assignments can be done in coNPMV, which leads us to a comparison of NPMV and coNPMV with Krentel's classes Max P and Min P. The difference hierarchy for NPMV is related to the query hierarchy for coNPMV. Finally, we examine a functional analogue of Chang and Kadin's relationship between a collapse of the Boolean hierarchy over NP and a collapse of the polynomial time hierarchy.