Related papers: Extending the Applicability Condition in the Forma…
We present $\cal L$, an extension of Parigot's $\lambda\mu$-calculus by adding negation as a type constructor, together with syntactic constructs that represent negation introduction and elimination. We will define a notion of reduction…
While large language models (LLMs) have shown progress in mathematical reasoning, they still face challenges in formalizing theorems that arise from instantiating abstract structures in concrete settings. With the goal of auto-formalizing…
We propose a validity preserving translation from a subset of epistemic Alternating-time Temporal Logic (ATL) to epistemic Computation Tree Logic (CTL). The considered subset of epistemic ATL is known to have the finite model property and…
The lambda-Pi-calculus modulo theory is a logical framework in which many type systems can be expressed as theories. We present such a theory, the theory U, where proofs of several logical systems can be expressed. Moreover, we identify a…
To take advantage of Large Language Model in theorem formalization and proof, we propose a reinforcement learning framework to iteratively optimize the pretrained LLM by rolling out next tactics and comparing them with the expected ones.…
We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. The use of guarded recursive types ensures the productivity of well-typed programs. Guarded recursive…
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…
In reductive proof search, proofs are naturally generalized by solutions, comprising all possibly infinite structures generated by locally correct, bottom-up application of inference rules. We propose an extension of the Curry-Howard…
We describe a realizability framework for classical first-order logic in which realizers live in (a model of) typed {\lambda}{\mu}-calculus. This allows a direct interpretation of classical proofs, avoiding the usual negative translation to…
The Abella interactive theorem prover has proven to be an effective vehicle for reasoning about relational specifications. However, the system has a limitation that arises from the fact that it is based on a simply typed logic:…
Just as the $\lambda$-calculus uses three primitives (abstraction, application, variable) as the foundation of functional programming, inheritance-calculus uses three primitives (record, definition, inheritance) as the foundation of…
We give arithmetical proofs of the strong normalization of two symmetric $\lambda$-calculi corresponding to classical logic. The first one is the $\bar{\lambda}\mu\tilde{\mu}$-calculus introduced by Curien & Herbelin. It is derived via the…
Tabular data have been playing a vital role in diverse real-world fields, including healthcare, finance, etc. With the recent success of Large Language Models (LLMs), early explorations of extending LLMs to the domain of tabular data have…
Since the very beginning of the theory of linear logic it is known how to represent the $\lambda$-calculus as linear logic proof nets. The two systems however have different granularities, in particular proof nets have an explicit notion of…
We present a general and user-extensible equality checking algorithm that is applicable to a large class of type theories. The algorithm has a type-directed phase for applying extensionality rules and a normalization phase based on…
Computation Tree Logic (CTL) and its extensions CTL* and CTL+ are widely used in automated verification as a basis for common model checking tools. But while they can express many properties of interest like reachability, even simple…
Cut-elimination is the bedrock of proof theory with a multitude of applications from computational interpretations to proof analysis. It is also the starting point for important meta-theoretical investigations including decidability,…
Denotational models of type theory, such as set-theoretic, domain-theoretic, or category-theoretic models use (actual) infinite sets of objects in one way or another. The potential infinite, seen as an extensible finite, requires a dynamic…
Herbrand's theorem is one of the most fundamental insights in logic. From the syntactic point of view it suggests a compact representation of proofs in classical first- and higher-order logic by recording the information which instances…
We compare results from $\delta$--expansion, in simple theories, with self--consistent calculations as well as calculations involving the principle of minimal sensitivity. We show that the latter methods give relatively more accurate…