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This paper describes a categorical interpretation of the Wolfram Language and introduces a simple implementation of monadic types and the "do" notation. The monadic style of programming combined with the many built in functions of the…
In semantics and in programming practice, algebraic concepts such as monads or, essentially equivalently, (large) Lawvere theories are a well-established tool for modelling generic side-effects. An important issue in this context are…
Randomized controlled trials are a cornerstone of medicine and the social sciences as they enable reliable estimates of causal effects. However, they are costly and time-consuming to conduct, motivating interest in predicting causal effects…
In recent years, the interest in using proof assistants to formalise and reason about mathematics and programming languages has grown. Type-logical grammars, being closely related to type theories and systems used in functional programming,…
The motivation for this paper comes out of our experience with teaching natural deduction (ND) and with the way this formal system is implemented by the \textsc{Coq} proof assistant, namely by means of so-called tactics, which are…
Weighted monadic second-order logic is a weighted extension of monadic second-order logic that captures exactly the behaviour of weighted automata. Its semantics is parameterized with respect to a semiring on which the values that weighted…
Capitalizing on previous encodings and formal developments about nominal calculi and type systems, we propose a weak Higher-Order Abstract Syntax formalization of the type language of pure System F<: within Coq, a proof assistant based on…
Representation theorems for formal systems often take the form of an inductive translation that satisfies certain invariants, which are proved inductively. Theory morphisms and logical relations are common patterns of such inductive…
We define an extension of lambda-calculus with dependents types that enables us to encode transparent and opaque probabilistic programs and prove a strong normalisation result for it by a reducibility technique. While transparent…
For those of us who generally live in the world of syntax, semantic proof techniques such as reducibility, realizability or logical relations seem somewhat magical despite -- or perhaps due to -- their seemingly unreasonable effectiveness.…
ML5 is a programming language for spatially distributed computing, based on a Curry-Howard correspondence with the modal logic S5. Despite being designed by a correspondence with S5 modal logic, the ML5 programming language differs from the…
Rule-based reasoning, a fundamental type of legal reasoning, enables us to draw conclusions by accurately applying a rule to a set of facts. We explore causal language models as rule-based reasoners, specifically with respect to…
Mathematical theorems are human knowledge able to be accumulated in the form of symbolic representation, and proving theorems has been considered intelligent behavior. Based on the BHK interpretation and the Curry-Howard isomorphism, proof…
Bidirectional transformations (bx) have primarily been modeled as pure functions, and do not account for the possibility of the side-effects that are available in most programming languages. Recently several formulations of bx that use…
"Interaction trees" (ITrees) are a general-purpose data structure for representing the behaviors of recursive programs that interact with their environments. A coinductive variant of "free monads," ITrees are built out of uninterpreted…
In this paper, we take a pervasively effectful (in the style of ML) typed lambda calculus, and show how to extend it to permit capturing pure expressions with types. Our key observation is that, just as the pure simply-typed lambda calculus…
As deep neural models in NLP become more complex, and as a consequence opaque, the necessity to interpret them becomes greater. A burgeoning interest has emerged in rationalizing explanations to provide short and coherent justifications for…
Context-free language theory is a well-established area of mathematics, relevant to computer science foundations and technology. This paper presents the preliminary results of an ongoing formalization project using context-free grammars and…
Training large language models (LLMs) with chain-of-thought (CoT) supervision has proven effective for enhancing their reasoning abilities. However, obtaining reliable and accurate reasoning supervision remains a significant challenge. We…
Consequence-based reasoning can be used to construct proofs that explain entailments of description logic (DL) ontologies. In the literature, one can find multiple consequence-based calculi for reasoning in the $\mathcal{EL}$ family of DLs,…