Related papers: Facilitating Meta-Theory Reasoning (Invited Paper)
When performing complex multi-step reasoning tasks, the ability of Large Language Models (LLMs) to derive structured intermediate proof steps is important for ensuring that the models truly perform the desired reasoning and for improving…
Humans are black boxes -- we cannot observe their neural processes, yet society functions by evaluating verifiable arguments. AI explainability should follow this principle: stakeholders need verifiable reasoning chains, not mechanistic…
The formal analysis of automated systems is an important and growing industry. This activity routinely requires new verification frameworks to be developed to tackle new programming features, or new considerations (bugs of interest). Often,…
A number of flexible tactic-based logical frameworks are nowadays available that can implement a wide range of mathematical theories using a common higher-order metalanguage. Used as proof assistants, one of the advantages of such powerful…
Mathematical proofs should be paired with formal proofs, whenever feasible.
Convincing someone of the truth value of a premise requires understanding and articulating the core logical structure of the argument which proves or disproves the premise. Understanding the logical structure of an argument refers to…
A theory graph is a network of axiomatic theories connected with meaning-preserving mappings called theory morphisms. Theory graphs are well suited for organizing large bodies of mathematical knowledge. Traditional and formal proofs do not…
Mathematical proofs are a cornerstone of control theory, and it is important to get them right. Deduction systems can help with this by mechanically checking the proofs. However, the structure and level of detail at which a proof is…
Meta-learning is a framework for learning learning algorithms through repeated interactions with an environment as opposed to designing them by hand. In recent years, this framework has established itself as a promising tool for building…
Several formal systems, such as resolution and minimal model semantics, provide a framework for logic programming. In this paper, we will survey the use of structural proof theory as an alternative foundation. Researchers have been using…
Predicate intuitionistic logic is a well established fragment of dependent types. According to the Curry-Howard isomorphism proof construction in the logic corresponds well to synthesis of a program the type of which is a given formula. We…
A logic is presented for reasoning on iterated sequences of formulae over some given base language. The considered sequences, or "schemata", are defined inductively, on some algebraic structure (for instance the natural numbers, the lists,…
Large language models and other highly capable AI systems ease the burdens of deciding what to say or do, but this very ease can undermine the effectiveness of our actions in social contexts. We explain this apparent tension by introducing…
In this paper we study the logical foundations of automated inductive theorem proving. To that aim we first develop a theoretical model that is centered around the difficulty of finding induction axioms which are sufficient for proving a…
Symbolic reasoning, rule-based symbol manipulation, is a hallmark of human intelligence. However, rule-based systems have had limited success competing with learning-based systems outside formalized domains such as automated theorem…
We introduce a formal meta-language for probabilistic programming, capable of expressing both programs and the type systems in which they are embedded. We are motivated here by the desire to allow an AGI to learn not only relevant knowledge…
This paper concerns the development of metatheory for extensible languages. It uses as its starting point a view that programming languages tailored to specific application domains are to be constructed by composing components from an open…
The term {\em meta-programming} refers to the ability of writing programs that have other programs as data and exploit their semantics. The aim of this paper is presenting a methodology allowing us to perform a correct termination analysis…
Algorithmic meta-theorems are general algorithmic results applying to a whole range of problems, rather than just to a single problem alone. They often have a "logical" and a "structural" component, that is they are results of the form:…
Matching logic is a formalism for specifying, and reasoning about, mathematical structures, using patterns and pattern matching. Growing in popularity, it has been used to define many logical systems such as separation logic with recursive…