Related papers: Natural Deduction as Higher-Order Resolution
Superdeduction is a method specially designed to ease the use of first-order theories in predicate logic. The theory is used to enrich the deduction system with new deduction rules in a systematic, correct and complete way. A proof-term…
We propose an automated deduction method which allows us to produce proofs close to the human intuition and practice. This method is based on tableaux, which generate more natural proofs than similar methods relying on clausal forms, and…
An inductive inference system for proving validity of formulas in the initial algebra $T_{\mathcal{E}}$ of an order-sorted equational theory $\mathcal{E}$ is presented. It has 20 inference rules, but only 9 of them require user interaction;…
Modern machine learning pipelines are built on numerical algorithms. Reliable numerical methods are thus a prerequisite for trustworthy machine learning and cyber-physical systems. Therefore, we contribute a framework for verified numerical…
How difficult are interactive theorem provers to use? We respond by reviewing the formalization of Hilbert's tenth problem in Isabelle/HOL carried out by an undergraduate research group at Jacobs University Bremen. We argue that, as…
Resolution is the rule of inference at the basis of most procedures for automated reasoning. In these procedures, the input formula is first translated into an equisatisfiable formula in conjunctive normal form (CNF) and then represented as…
Translating formal language into natural language is a foundational challenge in NLP, driving various downstream applications in semantic parsing, theorem validation, and question answering. In this study, we introduce First-Order Logic to…
Higher order derivatives of functions are structured high dimensional objects which lend themselves to many alternative representations, with the most popular being multi-index, matrix and tensor representations. The choice between them…
Using Isabelle/HOL, we verify a union-find data structure with an explain operation due to Nieuwenhuis and Oliveras. We devise a simpler, more naive version of the explain operation whose soundness and completeness is easy to verify. Then,…
We propose a novel logic, called Frame Logic (FL), that extends first-order logic (with recursive definitions) using a construct Sp(.) that captures the implicit supports of formulas -- the precise subset of the universe upon which their…
A key feature of inductive logic programming (ILP) is its ability to learn first-order programs, which are intrinsically more expressive than propositional programs. In this paper, we introduce techniques to learn higher-order programs.…
Transformers have been shown to emulate logical deduction over natural language theories (logical rules expressed in natural language), reliably assigning true/false labels to candidate implications. However, their ability to generate…
Undergraduate students of artificial intelligence often struggle with representing knowledge as logical sentences. This is a skill that seems to require extensive practice to obtain, suggesting a teaching strategy that involves the…
In this paper, we propose the use of interactive theorem proving for explainable machine learning. After presenting our proposition, we illustrate it on the dedicated application of explaining security attacks using the Isabelle…
We develop a system for solving logical deduction one-dimensional ordering problems by transforming natural language premises and candidate statements into first-order logic. Building on Heim and Kratzer's syntax-based compositional…
Formal programming language semantics are imperative when trying to verify properties of programs in an automated manner. Using a new approach, Din et al. strengthen the ability of reasoning about concurrent programs by proposing a modular…
The Isabelle proof assistant includes a small functional language, which allows users to write and reason about programs. So far, these programs could be extracted into a number of functional languages: Standard ML, OCaml, Scala, and…
A number of first-order calculi employ an explicit model representation formalism for automated reasoning and for detecting satisfiability. Many of these formalisms can represent infinite Herbrand models. The first-order fragment of…
Formal verification via interactive theorem proving is increasingly used to ensure the correctness of critical systems, yet constructing large proof scripts remains highly manual and limits scalability. Advances in large language models…
Logical reasoning is a core challenge in natural language understanding and a fundamental capability of artificial intelligence, underpinning scientific discovery, mathematical theorem proving, and complex decision-making. Despite the…