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Proofs in proof assistants like Rocq can be brittle, breaking easily in response to changes. To address this, recent work introduced an algorithm and tool in Rocq to automatically repair broken proofs in response to changes that correspond…
Formal proof checkers such as Coq are capable of validating proofs of correction of algorithms for finite field arithmetics but they require extensive training from potential users. The delayed solution of a triangular system over a finite…
Mechanized verification of liveness properties for infinite programs with effects and nondeterminism is challenging. Existing temporal reasoning frameworks operate at the level of models such as traces and automata. Reasoning happens at a…
Reified Input/Output (I/O) logic[21] has been recently proposed to model real-world norms in terms of the logic in [11]. This is massively grounded on the notion of reification, and it has specifically designed to model meaning of natural…
In this article, we give a geometric description for any invertible operator on a finite dimensional inner--product space. With the aid of such a description, we are able to decompose any given conformal transformation as a product of…
Gentzen's 1936 proof of the consistency of Peano Arithmetic was a significant result in the foundations of mathematics. We provide here a modified version of the proof, based on G\"{o}del's reformulation, and including additional details…
A finite number of rational functions are compatible if they satisfy the compatibility conditions of a first-order linear functional system involving differential, shift and q-shift operators. We present a theorem that describes the…
We introduce a realisability semantics for infinitary intuitionistic set theory that is based on Ordinal Turing Machines (OTMs). We show that our notion of OTM-realisability is sound with respect to certain systems of infinitary…
We introduce a formal framework for analyzing trades in financial markets. These days, all big exchanges use computer algorithms to match buy and sell requests and these algorithms must abide by certain regulatory guidelines. For example,…
We describe several views of the semantics of a simple programming language as formal documents in the calculus of inductive constructions that can be verified by the Coq proof system. Covered aspects are natural semantics, denotational…
Given a static vertex-selection problem (e.g. independent set, dominating set) on a graph, we can define a corresponding temporally satisfying reconfiguration problem on a temporal graph which asks for a sequence of solutions to the…
We present a formal study of semantics for the relational programming language miniKanren. First, we formulate a denotational semantics which corresponds to the minimal Herbrand model for definite logic programs. Second, we present…
In this paper it is proved that if a finitely presented group acts properly discontinuously, cocompactly and by isometries on a simply connected Riemannian manifold, then the two Dehn functions, of the group and the manifold, respectively,…
Generating code from natural-language requirements has become a primary route for LLM-assisted software development. Although LLMs can successfully complete small programming tasks, generating an entire complex project remains unreliable…
We study the positive-definite completion problem for kernels on a variety of domains and prove results concerning the existence, uniqueness, and characterization of solutions. In particular, we study a special solution called the canonical…
Precise and autonomous clocks are of fundamental interest and central importance to both foundational studies and practical applications. Here, we construct a blueprint for a quantum clock governed by time-independent interactions. By…
This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic…
Composition technologies improve reuse in the development of large-scale complex systems. Safety critical systems require intensive validation and verification activities. These activities should be compositional in order to reduce the…
Formal verification of neuro-symbolic cyber-physical systems, such as drones, medical devices and robots, is complicated. Neural components must be trained to be optimal with respect to the available data as well as the safety…
Reasoning about real number expressions in a proof assistant is challenging. Several problems in theorem proving can be solved by using exact real number computation. I have implemented a library for reasoning and computing with complete…