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A consecutive pattern in a permutation $\pi$ is another permutation $\sigma$ determined by the relative order of a subsequence of contiguous entries of $\pi$. Traditional notions such as descents, runs and peaks can be viewed as particular…
The goal is to construct three related "prismatization" functors from the category of p-adic formal schemes to that of formal stacks. This should provide a good category of coefficients for prismatic cohomology in the spirit of F-gauges. In…
Finite Automata (FAs) are fundamental components in the domains of programming languages. For instance, regular expressions, which are pivotal in languages such as JavaScript and Python, are frequently implemented using FAs. Finite…
Stepwise refinement of algebraic specifications is a well known formal methodology for program development. However, traditional notions of refinement based on signature morphisms are often too rigid to capture a number of relevant…
We describe a new notation for finite transformations. This attractor-cycle notation extends the orbit-cycle notation for permutations and builds upon existing transformation notations. How the basins of attraction of a finite…
We show how to compute efficiently with nominal sets over the total order symmetry, by developing a direct representation of such nominal sets and basic constructions thereon. In contrast to previous approaches, we work directly at the…
A study of assisted problem solving formalized via decompositions of deterministic finite automata is initiated. The landscape of new types of decompositions of finite automata this study uncovered is presented. Languages with various…
The goal of this paper is threefold. First, we describe the notion of dissociation for closed subgroups of the group of permutations on a countably infinite set and explain its numerous consequences on unitary representations…
We present the formalization of a theory of syntax with bindings that has been developed and refined over the last decade to support several large formalization efforts. Terms are defined for an arbitrary number of constructors of varying…
Recent research suggests that systematic generalization in natural language understanding remains a challenge for state-of-the-art neural models such as Transformers and Graph Neural Networks. To tackle this challenge, we propose Edge…
The contribution of this paper is the development of the syntax and semantics of multi-sorted nominal abstract binding trees (abts), an extension of second order universal algebra to support symbol-indexed families of operators. Nominal…
Sets with atoms serve as an alternative to ZFC foundations for mathematics, where some infinite, though highly symmetric sets, behave in a finitistic way. Therefore, one can try to carry over analysis of the classical algorithms from finite…
We present an easily defined countable family of permutations of the natural numbers for which explicit rearrangements (i.e., the sums induced by the permutations) can be computed. The digamma function proves to be the key tool for the…
Superposition is an established decision procedure for a variety of first-order logic theories represented by sets of clauses. A satisfiable theory, saturated by superposition, implicitly defines a minimal term-generated model for the…
To a finite group $G$, one can associate several notions of dimensions (or degrees). In this survey, we attempt to bring together some of the notions of dimensions or degrees defined using representations of the group in General Linear…
The theory of finitely supported algebraic structures represents a reformulation of Zermelo-Fraenkel set theory in which every construction is finitely supported according to the action of a group of permutations of some basic elements…
Presheaves and nominal sets provide alternative abstract models of sets of syntactic objects with free and bound variables, such as lambda-terms. One distinguishing feature of the presheaf-based perspective is its elegant syntax-free…
Nominal logic is a variant of first-order logic that provides support for reasoning about bound names in abstract syntax. A key feature of nominal logic is the new-quantifier, which quantifies over fresh names (names not appearing in any…
We present three ordinal notation systems representing ordinals below $\varepsilon_0$ in type theory, using recent type-theoretical innovations such as mutual inductive-inductive definitions and higher inductive types. We show how ordinal…
We give a semantics for the lambda-calculus based on a topological duality theorem in nominal sets. A novel interpretation of lambda is given in terms of adjoints, and lambda-terms are interpreted absolutely as sets (no valuation is…