Related papers: Rewriting and Well-Definedness within a Proof Syst…
Large language models (LLMs) have been used to generate formal proofs of mathematical theorems in proofs assistants such as Lean. However, we often want to optimize a formal proof with respect to various criteria, depending on its…
A plausible definition of "reasoning" could be "algebraically manipulating previously acquired knowledge in order to answer a new question". This definition covers first-order logical inference or probabilistic inference. It also includes…
Multiset rewriting systems provide a formalism particularly suitable for the description of biological systems. We present an extension of this formalism with additional controls on the derivations as a tool for reducing possible…
Rewriting Induction (RI) is a principle to prove that an equation over terms is an inductive theorem of a rewrite system, i.e., that any ground instance of the equation is a theorem of the rewrite system. RI has been adapted to several…
In this article we investigate how a subterm pattern matching algorithm can be exploited to implement efficient term rewriting procedures. From the left-hand sides of the rewrite system we construct a set automaton, which can be used to…
A rewriting system is a set of equations over a given set of terms called rules that characterize a system of computation and is a powerful general method for providing decision procedures of equational theories, based upon the principle of…
Deduction systems and graph rewriting systems are compared within a common categorical framework. This leads to an improved deduction method in diagrammatic logics.
We generalize the notion of proof term to the realm of transfinite reduction. Proof terms represent reductions in the first-order term format, thereby facilitating their formal analysis. We show that any transfinite reduction can be…
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.…
Widely used complex code refactoring tools lack a solid reasoning about the correctness of the transformations they implement, whilst interest in proven correct refactoring is ever increasing as only formal verification can provide true…
This paper considers quasi-reductivity - essentially, the property that an evaluation cannot get "stuck" due to a missing case in pattern matching - in the context of term rewriting with logical constraints.
We describe techniques for synthesis and verification of recursive functional programs over unbounded domains. Our techniques build on top of an algorithm for satisfiability modulo recursive functions, a framework for deductive synthesis,…
Logically constrained term rewriting is a relatively new rewriting formalism that naturally supports built-in data structures, such as integers and bit vectors. In the analysis of logically constrained term rewrite systems (LCTRSs),…
A coverage type generalizes refinement types found in many functional languages with support for must-style underapproximate reasoning. Property-based testing frameworks are one particularly useful domain where such capabilities are useful…
In this note we give a simple unifying proof of the undecidability of several diagrammatic properties of term rewriting systems that include: local confluence, strong confluence, diamond property, subcommutative property, and the existence…
While deep reasoning with long chain-of-thought has dramatically improved large language models in verifiable domains like mathematics, its effectiveness for open-ended tasks such as writing remains unexplored. In this paper, we conduct a…
We propose a procedure for automated implicit inductive theorem proving for equational specifications made of rewrite rules with conditions and constraints. The constraints are interpreted over constructor terms (representing data values),…
We describe a "top down" approach for automated theorem proving (ATP). Researchers might usefully investigate the forms of the theorems mathematicians use in practice, carefully examine how they differ and are proved in practice, and code…
Theorem proving serves as a major testbed for evaluating complex reasoning abilities in large language models (LLMs). However, traditional automated theorem proving (ATP) approaches rely heavily on formal proof systems that poorly align…
Using multisets, we develop novel techniques for mechanizing the proofs of the synthesis conjectures for list-sorting algorithms, and we demonstrate them in the Theorema system. We use the classical principle of extracting the algorithm as…