arXiv subjects

Michael Benedikt

Publications and source records attributed to Michael Benedikt.

At least 19 recordsLinked to original sources

arXiv2026

Interpolation and Query Rewriting

We overview applications of Craig interpolation and Beth definability to simplifying logical expressions or database queries. From the perspective of the theory of interpolation and definability the results give a number of new angles. First, they give a different take on what it means to make definability or interpolation results effective, looking at algorithms that take a proof as input and return an interpolant or explicit definition as output. Secondly, they relate interpolation and definability to preservation theorems in model theory: interpolation and definability theorems are the basis for many "semantics-to-syntax" results, relating a semantic property of a formula to its equivalence with a certain syntactic form. Thirdly, they motivate new forms of interpolation and definability, focusing on syntactic forms that are of interest in databases.

cs.LO↗

arXiv2026

Using LLMs for Explainable, Data-Driven Insight Generation from Time Series

Time series forecasts are widely used in decision-critical domains, where they are rarely consumed without accompanying explanations. Producing such explanations is usually a manual and costly process, and attempts to automate it using large language models often suffer from hallucination when applied to temporal data. We propose a domain-agnostic framework for grounded natural language explanation generation for time series forecasts, illustrated in Figure 1. The framework consists of three components: (i) extraction of structured explanatory factors from historical analyst-written explanations, (ii) evidence-conditioned explanation generation, and (iii) scalable evaluation for readability, logical consistency, and persuasiveness. The design explicitly constrains generation to verifiable evidence, reducing unsupported claims. We evaluate the framework on a financial forecasting case study involving the NASDAQ-100 index and a freight pricing case study using data from Vortexa. Results show that generated explanations approached analyst-written explanations in terms of readability, consistency and persuasiveness. These findings demonstrate that grounded explanation generation for time series forecasting can be achieved at scale without domain-specific fine-tuning.

cs.AI↗

arXiv2026

How (and when) can you fit examples to logic-based hypothesis classes over infinite structures?

We study fitting problems, sometimes called ``training problems'', where we have a finite sample consisting of inputs and outputs, and we want to know whether there is a function in a certain class that could produce these outputs, exactly or approximately, on the given inputs. We focus on the computational and descriptive complexity of fitting for logically-defined classes in common decidable structures, like the real ordered field and Presburger arithmetic, and also for broader classes defined via combinatorial or model-theoretic properties. We isolate the complexity of these fitting problems, with particular attention to cases where we can use queries in a natural query language over the sample to determine whether a sample is fittable.

cs.LO↗

arXiv2026

A logical approach to concentration

Concentration results say that a sequence of random variables becomes progressively concentrated around the mean. Such results are common in the study of functions of random graphs. We introduce a real-valued logic with various aggregate operators on graphs, including summation, and prove that every term in the language, seen as a random variable on random graphs within the classical Erd\H{o}s-R\'{e}nyi random graph model, is concentrated. We prove this for dense and sparse variants of Erd\H{o}s-R\'{e}nyi graphs. On the one hand, our results extend the line of work originating with Fagin and Glebskii et al. on zero-one laws for dense random graphs, as well as the zero-one law of Shelah and Spencer for sparse random graphs. On the other hand, they can be seen as a meta-theorem for inferring concentration results on random graphs, and we give examples of such applications.

math.PR↗

arXiv2026

How Expressive Are Graph Neural Networks in the Presence of Node Identifiers?

Graph neural networks (GNNs) are a widely used class of machine learning models for graph-structured data, based on local aggregation over neighbors. GNNs have close connections to logic. In particular, their expressive power is linked to that of modal logics and bounded-variable logics with counting. In many practical scenarios, graphs processed by GNNs have node features that act as unique identifiers. In this work, we study how such identifiers affect the expressive power of GNNs. We initiate a study of the key-invariant expressive power of GNNs, inspired by the notion of order-invariant definability in finite model theory: which node queries that depend only on the underlying graph structure can GNNs express on graphs with unique node identifiers? We provide answers for various classes of GNNs with local max- or sum-aggregation.

cs.LO↗

arXiv2025

Robustness Verification of Graph Neural Networks Via Lightweight Satisfiability Testing

Graph neural networks (GNNs) are the predominant architecture for learning over graphs. As with any machine learning model, an important issue is the detection of attacks, where an adversary can change the output with a small perturbation of the input. Techniques for solving the adversarial robustness problem - determining whether an attack exists - were originally developed for image classification. In the case of graph learning, the attack model usually considers changes to the graph structure in addition to or instead of the numerical features of the input, and the state of the art techniques proceed via reduction to constraint solving, working on top of powerful solvers, e.g. for mixed integer programming. We show that it is possible to improve on the state of the art in structural robustness by replacing the use of powerful solvers by calls to efficient partial solvers, which run in polynomial time but may be incomplete. We evaluate our tool RobLight on a diverse set of GNN variants and datasets.

cs.LG↗

arXiv2025

Analysis of logics with arithmetic

We present new results on finite satisfiability of logics with counting and arithmetic. One result is a tight bound on the complexity of satisfiability of logics with so-called local Presburger quantifiers, which sum over neighbors of a node in a graph. A second contribution concerns computing a semilinear representation of the cardinalities associated with a formula in two variable logic extended with counting quantifiers. Such a representation allows you to get bounds not only on satisfiability for these logics, but for satisfiability in the presence of additional ``global cardinality constraints'': restrictions on cardinalities of unary formulas, expressed using arbitrary decidability logics over arithmetic. In the process, we provide simpler proofs of some key prior results on finite satisfiability and semi-linearity of the spectrum for these logics.

cs.LO↗

arXiv2025

Constraint-aware Learning of Probabilistic Sequential Models for Multi-Label Classification

We investigate multi-label classification involving large sets of labels, where the output labels may be known to satisfy some logical constraints. We look at an architecture in which classifiers for individual labels are fed into an expressive sequential model, which produces a joint distribution. One of the potential advantages for such an expressive model is its ability to modelling correlations, as can arise from constraints. We empirically demonstrate the ability of the architecture both to exploit constraints in training and to enforce constraints at inference time.

cs.LG↗

arXiv2025

Convergence Laws for Extensions of First-Order Logic with Averaging

For many standard models of random structure, first-order logic sentences exhibit a convergence phenomenon on random inputs. The most well-known example is for random graphs with constant edge probability, where the probabilities of first-order sentences converge to 0 or 1. In other cases, such as certain ``sparse random graph'' models, the probabilities of sentences converge, although not necessarily to 0 or 1. In this work we deal with extensions of first-order logic with aggregate operators, variations of averaging. These logics will consist of real-valued terms, and we allow arbitrary Lipschitz functions to be used as ``connectives''. We show that some of the well-known convergence laws extend to this setting.

cs.LO↗

arXiv2025

From learnable objects to learnable random objects

We consider the relationship between learnability of a "base class" of functions on a set $X$, and learnability of a class of statistical functions derived from the base class. For example, we refine results showing that learnability of a family $h_p: p \in Y$ of functions implies learnability of the family of functions $h_\mu=\lambda p: Y. E_\mu(h_p)$, where $E_\mu$ is the expectation with respect to $\mu$, and $\mu$ ranges over probability distributions on $X$. We will look at both Probably Approximately Correct (PAC) learning, where example inputs and outputs are chosen at random, and online learning, where the examples are chosen adversarily. For agnostic learning, we establish improved bounds on the sample complexity of learning for statistical classes, stated in terms of combinatorial dimensions of the base class. We connect these problems to techniques introduced in model theory for "randomizing a structure". We also provide counterexamples for realizable learning, in both the PAC and online settings.

cs.LO↗

arXiv2024

Almost Surely Asymptotically Constant Graph Neural Networks

We present a new angle on the expressive power of graph neural networks (GNNs) by studying how the predictions of real-valued GNN classifiers, such as those classifying graphs probabilistically, evolve as we apply them on larger graphs drawn from some random graph model. We show that the output converges to a constant function, which upper-bounds what these classifiers can uniformly express. This strong convergence phenomenon applies to a very wide class of GNNs, including state of the art models, with aggregates including mean and the attention-based mechanism of graph transformers. Our results apply to a broad class of random graph models, including sparse and dense variants of the Erdős-Rényi model, the stochastic block model, and the Barabási-Albert model. We empirically validate these findings, observing that the convergence phenomenon appears not only on random graphs but also on some real-world graphs.

cs.LG↗

arXiv2024

Monotone Rewritability and the Analysis of Queries, Views, and Rules

We study the interaction of views, queries, and background knowledge in the form of existential rules. The motivating questions concern monotonic determinacy of a query using views w.r.t. rules, which refers to the ability to recover the query answer from the views via a monotone function. We study the decidability of monotonic determinacy, and compare with variations that require the ``recovery function'' to be in a well-known monotone query language, such as conjunctive queries or Datalog. Surprisingly, we find that even in the presence of basic existential rules, the borderline between well-behaved and badly-behaved answerability differs radically from the unconstrained case. In order to understand this boundary, we require new results concerning entailment problems involving views and rules.

cs.DB↗

arXiv2024

Synthesizing nested relational queries from implicit specifications: via model theory and via proof theory

Derived datasets can be defined implicitly or explicitly. An implicit definition (of dataset O in terms of datasets I) is a logical specification involving two distinguished sets of relational symbols. One set of relations is for the "source data" I, and the other is for the "interface data" O. Such a specification is a valid definition of O in terms of I, if any two models of the specification agreeing on I agree on O. In contrast, an explicit definition is a transformation (or "query" below) that produces O from I. Variants of Beth's theorem state that one can convert implicit definitions to explicit ones. Further, this conversion can be done effectively given a proof witnessing implicit definability in a suitable proof system. We prove the analogous implicit-to-explicit result for nested relations: implicit definitions, given in the natural logic for nested relations, can be converted to explicit definitions in the nested relational calculus (NRC). We first provide a model-theoretic argument for this result, which makes some additional connections that may be of independent interest, between NRC queries, interpretations, a standard mechanism for defining structure-to-structure translation in logic, and between interpretations and implicit to definability "up to unique isomorphism". The latter connection uses a variation of a result of Gaifman concerning "relatively categorical" theories. We also provide a proof-theoretic result that provides an effective argument: from a proof witnessing implicit definability, we can efficiently produce an NRC definition. This will involve introducing the appropriate proof system for reasoning with nested sets, along with some auxiliary Beth-type results for this system. As a consequence, we can effectively extract rewritings of NRC queries in terms of NRC views, given a proof witnessing that the query is determined by the views.

cs.LO↗

arXiv2024

Revisiting the Expressiveness Landscape of Data Graph Queries

The study of graph queries in database theory has spanned more than three decades, resulting in a multitude of proposals for graph query languages. We can identify three main families of languages, with the canonical representatives being: (1) regular path queries, (2) walk logic, and (3) first-order logic with transitive closure operators. This paper provides a complete picture of the expressive power of these languages in the context of data graphs. Specifically, we consider a graph data model that supports querying over both data and topology. For example, ``Does there exist a path between two different persons in a social network with the same last name?''. We also show that an extension of (1) with regular path comparisons, augmented with transitive closure operators, can unify the expressivity of (1)--(3).

cs.DB↗

arXiv2024

Model Equivalences

We look at equivalence relations on the set of models of a theory -- MERs, for short -- such that the class of equivalent pairs is itself an elementary class, in a language appropriate for pairs of models. We provide many examples of definable MERs, along with the first steps of a classification theory for them. We characterize the special classes of definable MERs associated with preservation of formulas, either in classical first order logic or in continuous logic, and uncover an intrinsic role for the latter. We bring out a nontrivial relationship with interpretations (imaginary sorts), leading to a wider hierarchy of classes related to the preservation of reducts. We give results about the relationship between these classes, both for general theories and for theories satisfying additional model-theoretic properties, such as stability.

math.LO↗

arXiv2024

Embedded Finite Models Beyond Restricted Quantifier Collapse

We revisit evaluation of logical formulas that allow both uninterpreted relations, constrained to be finite, as well as an interpreted vocabulary over an infinite domain. This formalism was denoted embedded finite model theory in the past. It is clear that the expressiveness and evaluating complexity of formulas of this type depends heavily on the infinite structure. If we embed in a wild structure like the integers with additive and multiplicative arithmetic, logic is extremely expressive and formulas are impossible to evaluate. On the other hand, for some well-known decidable structures, the expressiveness and evaluating complexity are similar to the situation without the additional infrastructure. The latter phenomenon was formalized via the notion of ``Restricted Quantifier Collapse'': adding quantification over the infinite structure does not add expressiveness. Beyond these two extremes little was known. In this work we show that the possibilities for expressiveness and complexity are much wider. We show that we can get almost any possible complexity of evaluation while staying within a decidable structure. We also show that in some decidable structures, there is a disconnect between expressiveness of the logic and complexity, in that we cannot eliminate quantification over the structure, but this is not due to an ability to embed complex relational computation in the logic. We show failure of collapse for the theory of finite fields and the related theory of pseudo-finite fields, which will involve coding computation in the logic. As a by-product of this, we establish new lower-bounds for the complexity of decision procedures for several decidable theories of fields, including the theory of finite fields. In the process of investigating this landscape, we investigate several weakenings of collapse.

cs.LO↗

arXiv2024

Decidability of Graph Neural Networks via Logical Characterizations

We present results concerning the expressiveness and decidability of a popular graph learning formalism, graph neural networks (GNNs), exploiting connections with logic. We use a family of recently-discovered decidable logics involving "Presburger quantifiers". We show how to use these logics to measure the expressiveness of classes of GNNs, in some cases getting exact correspondences between the expressiveness of logics and GNNs. We also employ the logics, and the techniques used to analyze them, to obtain decision procedures for verification problems over GNNs. We complement this with undecidability results for static analysis problems involving the logics, as well as for GNN verification problems.

cs.LO↗

arXiv2024

Two variable logic with ultimately periodic counting

We consider the extension of two variable logic with quantifiers that state that the number of elements where a formula holds should belong to a given ultimately periodic set. We show that both satisfiability and finite satisfiability of the logic are decidable. We also show that the spectrum of any sentence is definable in Presburger arithmetic. In the process we present several refinements to the ``biregular graph method''. In this method, decidability issues concerning two-variable logics are reduced to questions about Presburger definability of integer vectors associated with partitioned graphs, where nodes in a partition satisfy certain constraints on their in- and out-degrees.

cs.LO↗