Related papers: Controlling unfolding in type theory
We show normalisation and decidability of convertibility for a type theory with a hierarchy of universes and a proof irrelevant type of propositions, close to the type system used in the proof assistant Lean. Contrary to previous arguments,…
We describe a prototype theorem prover, UTP2, developed to match the style of hand-written proof work in the Unifying Theories of Programming semantical framework. This is based on alphabetised predicates in a 2nd-order logic, with a strong…
PyLog is a minimal experimental proof assistant based on linearised natural deduction for intuitionistic and classical first-order logic extended with a comprehension operator. PyLog is interesting as a tool to be used in conjunction with…
Despite significant developments in Proof Theory, surprisingly little attention has been devoted to the concept of proof verifier. In particular, the mathematical community may be interested in studying different types of proof verifiers…
Congruence closure procedures are used extensively in automated reasoning and are a core component of most satisfiability modulo theories solvers. However, no known congruence closure algorithms can support any of the expressive logics…
In this chapter, we propose some future directions of work, potentially beneficial to Mathematics and its foundations, based on the recent import of methodology from the theory of programming languages into proof theory. This scientific…
Researchers in explainable artificial intelligence have developed numerous methods for helping users understand the predictions of complex supervised learning models. By contrast, explaining the $\textit{uncertainty}$ of model outputs has…
Large language models often struggle to recognize their knowledge limits in closed-book question answering, leading to confident hallucinations. While decomposed prompting is typically used to improve accuracy, we investigate its impact on…
Providing explanations along with predictions is crucial in some text processing tasks. Therefore, we propose a new self-interpretable model that performs output prediction and simultaneously provides an explanation in terms of the presence…
To solve Math Word Problems, human students leverage diverse reasoning logic that reaches different possible equation solutions. However, the mainstream sequence-to-sequence approach of automatic solvers aims to decode a fixed solution…
We consider the problem of formalizing the familiar notion of widening in abstract interpretation in higher-order logic. It turns out that many axioms of widening (e.g. widening sequences are ascending) are not useful for proving…
Adaptive robust optimization (ARO) extends static robust optimization by allowing decisions to depend on the realized uncertainty - weakly dominating static solutions within the modeled uncertainty set. However, ARO makes previous…
Harnessing the power of dependently typed languages can be difficult. Programmers must manually construct proofs to produce well-typed programs, which is not an easy task. In particular, migrating code to these languages is challenging.…
For controller design for systems on manifolds embedded in Euclidean space, it is convenient to utilize a theory that requires a single global coordinate system on the ambient Euclidean space rather than multiple local charts on the…
We propose a new language feature for ML-family languages, the ability to selectively unbox certain data constructors, so that their runtime representation gets compiled away to just the identity on their argument. Unboxing must be…
We show "free theorems" in the style of Wadler for polymorphic functions in homotopy type theory as consequences of the abstraction theorem. As an application, it follows that every space defined as a higher inductive type has the same…
Type checking algorithms and theorem provers rely on unification algorithms. In presence of type families or higher-order logic, higher-order (pre)unification (HOU) is required. Many HOU algorithms are expressed in terms of…
Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and co-induction. These proof principles…
This paper proposes a new adaptation methodology to find the control inputs for a class of nonlinear systems with time-varying bounded uncertainties. The proposed method does not require any prior knowledge of the uncertainties including…
A new notion of independence relation is given and associated to it, the class of flat theories, a subclass of strong stable theories including the superstable ones is introduced. More precisely, after introducing this independence…