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Large language models (LLMs) have shown an impressive ability to perform tasks believed to require thought processes. When the model does not document an explicit thought process, it becomes difficult to understand the processes occurring…
The notion of orbit finite data monoid was recently introduced by Bojanczyk as an algebraic object for defining recognizable languages of data words. Following Buchi's approach, we introduce a variant of monadic second-order logic with data…
One of the main claims of the paper is that Dirac's calculus and broader theories of physics can be treated as theories written in the language of Continuous Logic. Establishing its true interpretation (model) is a model theory problem. The…
The main goal of this paper is to settle a conceptual framework for cooperative game theory in which the notion of composition/aggregation of games is the defining structure. This is done via the mathematical theory of algebraic operads: we…
We continue to develop a research line initiated in \cite{wollic22}, studying I/O logic from an algebraic approach based on subordination algebras. We introduce the classes of slanted (co-)Heyting algebras as equivalent presentations of…
The present work presents some results about the categorial relation between logics and its categories of structures. A (propositional, finitary) logic is a pair given by a signature and Tarskian consequence relation on its formula algebra.…
Continuous chain-of-thought has been shown to be effective in saving reasoning tokens for large language models. By reasoning with continuous latent thought tokens, continuous CoT is able to perform implicit reasoning in a compact manner.…
Our understanding about things is conceptual. By stating that we reason about objects, it is in fact not the objects but concepts referring to them that we manipulate. Now, so long just as we acknowledge infinitely extending notions such as…
This paper concerns the relation between process algebra and Hoare logic. We investigate the question whether and how a Hoare logic can be used for reasoning about how data change in the course of a process when reasoning equationally about…
There is a wide range of modal logics whose semantics goes beyond relational structures, and instead involves, e.g., probabilities, multi-player games, weights, or neighbourhood structures. Coalgebraic logic serves as a unifying semantic…
A topos theoretic generalisation of the category of sets allows for modelling spaces which vary according to time intervals. Persistent homology, or more generally, persistence is a central tool in topological data analysis, which examines…
A series of recent papers by Bergfalk, Lupini and Panagiotopoulus developed the foundations of a field known as `definable algebraic topology,' in which classical cohomological invariants are enriched by viewing them as groups with a Polish…
A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left-, right- and bicovariance. A corresponding framework has been developed by Woronowicz, more…
We present algebraic semantics for the classical logic of proofs based on Boolean algebras. We also extend the language of the logic of proofs in order to have a Boolean structure on justification terms and equality predicate on terms. In…
This paper studies the homotopy theory of algebras and homotopy algebras over an operad. It provides an exhaustive description of their higher homotopical properties using the more general notion of morphisms called infinity-morphisms. The…
Combining abstract, symbolic reasoning with continuous neural reasoning is a grand challenge of representation learning. As a step in this direction, we propose a new architecture, called neural equivalence networks, for the problem of…
In this paper we show several similarities among logic systems that deal simultaneously with deductive and quantitative inference. We claim it is appropriate to call the tasks those systems perform as Quantitative Logic Reasoning. Analogous…
Tractable Boolean and arithmetic circuits have been studied extensively in AI for over two decades now. These circuits were initially proposed as "compiled objects," meant to facilitate logical and probabilistic reasoning, as they permit…
Computability logic (CL) (see http://www.cis.upenn.edu/~giorgi/cl.html) is a semantical platform and research program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth which it has more…
We use modal logic as a framework for coalgebraic trace semantics, and show the flexibility of the approach with concrete examples such as the language semantics of weighted, alternating and tree automata, and the trace semantics of…