Related papers: Formalized Lambek Calculus in Higher Order Logic (…
Relational program verification is a variant of program verification where one can reason about two programs and as a special case about two executions of a single program on different inputs. Relational program verification can be used for…
We present a translation of the Lambek calculus with brackets and the unit constant, $\mathbf{Lb}^{\boldsymbol{*}}_{\mathbf{1}}$, into the Lambek calculus with brackets allowing empty antecedents, but without the unit constant,…
The "Concrete Semantics" book gives an introduction to imperative programming languages accompanied by an Isabelle/HOL formalization. In this paper we discuss a re-formalization of the book using the Coq proof assistant. In order to achieve…
Ordered logics and type systems have been used in a variety of applications including computational linguistics, memory allocation, stream processing, logical frameworks, parametricity, and enforcing security protocols. In most…
We formally verify an abstract machine for a call-by-value lambda-calculus with de Bruijn terms, simple substitution, and small-step semantics. We follow a stepwise refinement approach starting with a naive stack machine with substitution.…
We enrich the Lambek calculus with the cyclic shift operation, which is expected to model the closure operator of formal languages with respect to cyclic shifts. We introduce a Gentzen-style calculus and prove cut elimination. Secondly, we…
Modern large language models (LLMs) show promising progress in formalizing informal mathematics into machine-verifiable theorems. However, these methods still face bottlenecks due to the limited quantity and quality of multilingual parallel…
This paper presents a new system of logic, LF, that is intended to be used as the foundation of the formalization of science. That is, deductive validity according to LF is to be used as the criterion for assessing what follows from the…
G\"odel's ontological proof has been analysed for the first-time with an unprecedent degree of detail and formality with the help of higher-order theorem provers. The following has been done (and in this order): A detailed natural deduction…
Autoformalization is the process of automatically translating from natural language mathematics to formal specifications and proofs. A successful autoformalization system could advance the fields of formal verification, program synthesis,…
This comprehensive survey examines Lean 4, a state-of-the-art interactive theorem prover and functional programming language. We analyze its architectural design, type system, metaprogramming capabilities, and practical applications in…
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,…
Autoformalization - automatically translating natural language mathematical texts into formal proof language such as Lean4 - can help accelerate AI-assisted mathematical research, be it via proof verification or proof search. I fine-tune…
Informal mathematics has been central to modern large language model (LLM) reasoning, offering flexibility and enabling efficient construction of arguments. However, purely informal reasoning is prone to logical gaps and subtle errors that…
We advocates here the use of (mathematical) logic for systems biology, as a unified framework well suited for both modeling the dynamic behaviour of biological systems, expressing properties of them, and verifying these properties. The…
We argue that the implementation and verification of compilers for functional programming languages are greatly simplified by employing a higher-order representation of syntax known as Higher-Order Abstract Syntax or HOAS. The underlying…
The Lambek calculus provides a foundation for categorial grammar in the form of a logic of concatenation. But natural language is characterized by dependencies which may also be discontinuous. In this paper we introduce the displacement…
Formalization of real analysis offers a chance to rebuild traditional proofs of important theorems as unambiguous theories that can be interactively explored. This paper provides a comprehensive overview of the Lebesgue Differentiation…
Automating the translation of natural language to first-order logic (FOL) is crucial for knowledge representation and formal methods, yet remains challenging. We present a systematic evaluation of fine-tuned LLMs for this task, comparing…
We present a formalization, in the theorem prover Lean, of the classification of solvable Lie algebras of dimension at most three over arbitrary fields. Lie algebras are algebraic objects which encode infinitesimal symmetries, and as such…