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These notes present the essentials of first- and second-order monadic logics on strings with introductory purposes. We discuss Monadic First-Order logic and show that it is strictly less expressive than Finite-State Automata, in that it…

We develop a unified categorical theory of substructural abstract syntax with variable binding and single-variable (capture-avoiding) substitution. This is done for the gamut of context structural rules given by exchange (linear theory)…

In quantum information and computation research, symbolic methods have been widely used for human specification and reasoning about quantum states and operations. At the same time, they are essential for ensuring the scalability and…

Emerging computational paradigms, such as probabilistic and hybrid programming, introduce new primitive operations that often need to be combined with classic programming constructs. However, it still remains a challenge to provide a…

We develop the theory of strong and commutative monads in the 2-dimensional setting of bicategories. This provides a framework for the analysis of effects in many recent models which form bicategories and not categories, such as those based…

We extend the {\lambda}-calculus with constructs suitable for relational and functional-logic programming: non-deterministic choice, fresh variable introduction, and unification of expressions. In order to be able to unify…

Many interesting and useful symbolic computation algorithms manipulate mathematical expressions in mathematically meaningful ways. Although these algorithms are commonplace in computer algebra systems, they can be surprisingly difficult to…

We introduce a high-level language with Python-like syntax for string-to-string, polyregular, first-order definable transductions. This language features function calls, boolean variables, and nested for-loops. We devise and implement a…

We define sound and adequate denotational and operational semantics for the stochastic lambda calculus. These two semantic approaches build on previous work that used similar techniques to reason about higher-order probabilistic programs,…

We propose a hybrid-dynamic first-order logic as a formal foundation for specifying and reasoning about reconfigurable systems. As the name suggests, the formalism we develop extends (many-sorted) first-order logic with features that are…

The paper is devoted to showing how to systematically design a programming language in 'reverse order', i.e. from denotations to syntax. This construction is developed in an algebraic framework consisting of three many-sorted algebras: of…

We describe several views of the semantics of a simple programming language as formal documents in the calculus of inductive constructions that can be verified by the Coq proof system. Covered aspects are natural semantics, denotational…

Hybrid computation combines discrete and continuous dynamics in the form of an entangled mixture inherently present both in various natural phenomena, and in applications ranging from control theory to microbiology. The emergent behaviours…

We define and investigate source-modality monitoring -- the ability of multimodal models to track and communicate the input source from which pieces of information originate. We consider source-modality monitoring as an instance of the more…

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…

This paper proposes a general semantic framework for verifying programs with arbitrary monadic side-effects using Dijkstra monads, which we define as monad-like structures indexed by a specification monad. We prove that any monad morphism…

We provide a computational definition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higher-order computation and linear algebra. This language extends the Lambda-calculus…

First order formulas in a relational signature can be considered as operations on the relations of an underlying set, giving rise to multisorted algebras we call first order algebras. We present universal axioms so that an algebra satisfies…

Permutations can be viewed as pairs of linear orders, or more formally as models over a signature consisting of two binary relation symbols. This approach was adopted by Albert, Bouvel and F\'eray, who studied the expressibility of…

Monads can be interpreted as encoding formal expressions, or formal operations in the sense of universal algebra. We give a construction which formalizes the idea of "evaluating an expression partially": for example, "2+3" can be obtained…