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Regular cost functions have been introduced recently as an extension to the notion of regular languages with counting capabilities, which retains strong closure, equivalence, and decidability properties. The specificity of cost functions is…

Characterisations theorems serve as important tools in model theory and can be used to assess and compare the expressive power of temporal languages used for the specification and verification of properties in formal methods. While complete…

We give an algebraic characterisation of first-order logic with the neighbour relation, on finite words. For this, we consider languages of finite words over alphabets with an involution on them. The natural algebras for such languages are…

In this paper we demonstrate how logic programming systems and Automated first-order logic Theorem Provers (ATPs) can improve the accuracy of Large Language Models (LLMs) for logical reasoning tasks where the baseline performance is given…

We study the positive logic FO+ on finite words, and its fragments, pursuing and refining the work initiated in [Kuperberg 2023]. First, we transpose notorious logic equivalences into positive first-order logic: FO+ is equivalent to LTL+ ,…

We study the Monadic Second Order (MSO) Hierarchy over colourings of the discrete plane, and draw links between classes of formula and classes of subshifts. We give a characterization of existential MSO in terms of projections of tilings,…

We present automated theorem provers for the first-order logic of here and there (HT). They are based on a native sequent calculus for the logic of HT and an axiomatic embedding of the logic of HT into intuitionistic logic. The analytic…

The overall goal of this paper is to investigate the theoretical foundations of algorithmic verification techniques for first order linear logic specifications. The fragment of linear logic we consider in this paper is based on the linear…

Large computer-understandable proofs consist of millions of intermediate logical steps. The vast majority of such steps originate from manually selected and manually guided heuristics applied to intermediate goals. So far, machine learning…

In this paper, we study whether transformer-based language models can extract predicate argument structure from simple sentences. We firstly show that language models sometimes confuse which predicates apply to which objects. To mitigate…

We present a novel automata-based approach to address linear temporal logic modulo theory (LTL-MT) as a specification language for data words. LTL-MT extends LTL_f by replacing atomic propositions with quantifier-free multi-sorted…

The connection between languages defined by computational models and logic for languages is well-studied. Monadic second-order logic and finite automata are shown to closely correspond to each-other for the languages of strings, trees, and…

The central open question in Descriptive Complexity is whether there is a logic that characterizes deterministic polynomial time (PTIME) on relational structures. Towards this goal, we define a logic that is obtained from first-order logic…

Large language models (LLMs) have recently attracted considerable interest for their ability to perform complex reasoning tasks, such as chain-of-thought (CoT) reasoning. However, most of the existing approaches to enhance this ability rely…

This paper relates the well-known Linear Temporal Logic with the logic of propositional schemata introduced by the authors. We prove that LTL is equivalent to a class of schemata in the sense that polynomial-time reductions exist from one…

Large Language Models (LLMs) have revolutionized natural language processing, yet they struggle with inconsistent reasoning, particularly in novel domains and complex logical sequences. This research introduces Proof of Thought, a framework…

Over finite words, languages of dot-depth one are expressively complete for alternation-free first-order logic. This fragment is also known as the Boolean closure of existential first-order logic. Here, the atomic formulas comprise order,…

Dependent type theory gives an expressive type system facilitating succinct formalizations of mathematical concepts. In practice, it is mainly used for interactive theorem proving with intensional type theories, with PVS being a notable…

Logical transductions provide a very useful tool to encode classes of structures inside other classes of structures. In this paper we study first-order (FO) transductions and the quasiorder they induce on infinite classes of finite graphs.…

Natural language understanding applications such as interactive planning and face-to-face translation require extensive inferencing. Many of these inferences are based on the meaning of particular open class words. Providing a…