Related papers: Realizability algebras: a program to well order R
Order types are a well known abstraction of combinatorial properties of a point set. By Mn\"ev's universality theorem for each semi-algebraic set $V$ there is an order type with a realization space that is \emph{stably equivalent} to $V$.…
Linearizability is the commonly accepted notion of correctness for concurrent data structures. It requires that any execution of the data structure is justified by a linearization --- a linear order on operations satisfying the data…
We show that verification of object-oriented programs by means of the assertional method can be achieved in a simple way by exploiting a syntax-directed transformation from object-oriented programs to recursive programs. This transformation…
When the inverse of an algorithm is well-defined -- that is, when its output can be deterministically transformed into the input producing it -- we say that the algorithm is invertible. While one can describe an invertible algorithm using a…
Using the proof-program (Curry-Howard) correspondence, we give a new method to obtain models of ZF and relative consistency results in set theory. We show the relative consistency of ZF + DC + there exists a sequence of subsets of R the…
In considering the reliability of numerical programs, it is normal to "limit our study to the semantics dealing with numerical precision" (Martel, 2005). On the other hand, there is a great deal of work on the reliability of programs that…
We investigate replicable learning algorithms. Ideally, we would like to design algorithms that output the same canonical model over multiple runs, even when different runs observe a different set of samples from the unknown data…
The provability logic of a theory T is the set of modal formulas, which under any arithmetical realization are provable in T . We slightly modify this notion by requiring the arithmetical realizations to come from a specified set $\Gamma$.…
In this vision paper, we explore the challenges and opportunities of a form of computation that employs an empirical (rather than a formal) approach, where the solution of a computational problem is returned as empirically most likely…
We present an extension to the $\mathtt{mathlib}$ library of the Lean theorem prover formalizing the foundations of computability theory. We use primitive recursive functions and partial recursive functions as the main objects of study, and…
The Curry-Howard correspondence is about a relationship between types and programs on the one hand and propositions and proofs on the other. The implications for programming language design and program verification is an active field of…
The point of this work is to explore axiomatisations of concurrent computation using the technology of proof theory and realizability. To deal with this problem, we redefine the Concurrent Realizability of Beffara using as realizers a…
Formal explainability guarantees the rigor of computed explanations, and so it is paramount in domains where rigor is critical, including those deemed high-risk. Unfortunately, since its inception formal explainability has been hampered by…
Realizability, introduced by Kleene, can be understood as a concretization of the Brouwer-Heyting-Kolmogorov (BHK) interpretation of proofs, providing a framework to interpret mathematical statements and proofs in terms of their…
This paper explores epistemic realizability, a form of realizability in which the property that a piece of data constitutes evidence for a logical proposition is semi-decidable. In this framework, each proposition A is assigned a verifier}…
Argumentation theory is a powerful paradigm that formalizes a type of commonsense reasoning that aims to simulate the human ability to resolve a specific problem in an intelligent manner. A classical argumentation process takes into account…
Runtime efficiency and termination are crucial properties in the studies of program verification. Instead of dealing with these issues in an ad hoc manner, it would be useful to develop a robust framework in which such properties are…
In this paper, we introduce a semantics of realisability for the classical propositional natural deduction and we prove a correctness theorem. This allows to characterize the operational behaviour of some typed terms.
We give a new presentation of interactive realizability with a more explicit syntax. Interactive realizability is a realizability semantics that extends the Curry-Howard correspondence to (sub-)classical logic, more precisely to first-order…
We present tools for analysing ordinals in realizability models of classical set theory built using Krivine's technique for realizability. This method uses a conservative extension of $ZF$ known as $ZF_{\varepsilon}$, where two membership…