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Text editors represent one of the fundamental tools that writers use - software developers, book authors, mathematicians. A text editor must work as intended in that it should allow the users to do their job. We start by introducing a small…
Reinforcement learning algorithms solve sequential decision-making problems in probabilistic environments by optimizing for long-term reward. The desire to use reinforcement learning in safety-critical settings inspires a recent line of…
Whereas proof assistants based on Higher-Order Logic benefit from external solvers' automation, those based on Type Theory resist automation and thus require more expertise. Indeed, the latter use a more expressive logic which is further…
Traditional concurrent-programming techniques require programmers to painstakingly write programs for each participant in a concurrent system. Choreographic programming, in contrast, allows a programmer to write one centralized program and…
In recent years, a number of lightweight programs have been deployed in critical domains, such as in smart contracts based on blockchain technology. Therefore, the security and reliability of such programs should be guaranteed by the most…
Formal verification using proof assistants, such as Coq, is an effective way of improving software quality, but requires significant effort and expertise. Machine learning can automatically synthesize proofs, but such tools are able to…
The calculus of constructions (CC) is a core theory for dependently typed programming and higher-order constructive logic. Originally introduced in Coquand's 1985 thesis, CC has inspired 25 years of research in programming languages and…
Real-life conjectures do not come with instructions saying whether they they should be proven or, instead, refuted. Yet, as we now know, in either case the final argument produced had better be not just convincing but actually verifiable in…
We exhibit a uniform method for obtaining (wellfounded and non-wellfounded) cut-free sequent-style proof systems that are sound and complete for various classes of action algebras, i.e., Kleene algebras enriched with meets and residuals.…
This technical report contains the full set of definitions and projection rules of the paper ``Verified Parameterized Choreographies'' by Rubbens et al. It also supplements the artefact.
The Coq Platform is a continuously developed distribution of the Coq proof assistant together with commonly used libraries, plugins, and external tools useful in Coq-based formal verification projects. The Coq Platform enables reproducing…
When writing programs involving matrices or tensors in general, it is desirable to rule out the inconsistency of tensor shapes (i.e., the generalization of matrix sizes) before actual computation. For this purpose, some languages provide…
In this article we describe the formalisation of the Bruhat-Tits tree - an important tool in modern number theory - in the Lean Theorem Prover. Motivated by the goal of connecting to ongoing research, we apply our formalisation to verify a…
The theorem of three circles in real algebraic geometry guarantees the termination and correctness of an algorithm of isolating real roots of a univariate polynomial. The main idea of its proof is to consider polynomials whose roots belong…
In this article, we establish the foundations of a computational field theory, which we term Topological Kleene Field Theory (TKFT), inspired by Stephen Kleene's seminal work on partial recursive functions and drawing parallels with…
Classical first-order logic is in many ways central to work in mathematics, linguistics, computer science and artificial intelligence, so it is worthwhile to define it in full detail. We present soundness and completeness proofs of a…
We report the results of the first experiments with learning proof dependencies from the formalizations done with the Coq system. We explain the process of obtaining the dependencies from the Coq proofs, the characterization of formulas…
We present a deep embedding of Bellantoni and Cook's syntactic characterization of polytime functions. We prove formally that it is correct and complete with respect to the original characterization by Cobham that required a bound to be…
This paper presents a Coq formalization of linear algebra over elementary divisor rings, that is, rings where every matrix is equivalent to a matrix in Smith normal form. The main results are the formalization that these rings support…
Kleene algebra with tests is an extension of Kleene algebra, the algebra of regular expressions, which can be used to reason about programs. We develop a coalgebraic theory of Kleene algebra with tests, along the lines of the coalgebraic…