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In this dissertation we collect some results about "interactive realizability", a realizability semantics that extends the Brouwer-Heyting-Kolmogorov interpretation to (sub-)classical logic, more precisely to first-order intuitionistic…
We propose a realizability interpretation of a system for quantifier free arithmetic which is equivalent to the fragment of classical arithmetic without "nested" quantifiers, called here EM1-arithmetic. We interpret classical proofs as…
We apply to the semantics of Arithmetic the idea of ``finite approximation'' used to provide computational interpretations of Herbrand's Theorem, and we interpret classical proofs as constructive proofs (with constructive rules for $\vee,…
We consider interactive learning in the realizable setting and develop a general framework to handle problems ranging from best arm identification to active classification. We begin our investigation with the observation that agnostic…
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…
The theory of classical realizability is a framework in which we can develop the proof-program correspondence. Using this framework, we show how to transform into programs the proofs in classical analysis with dependent choice and the…
In this dissertation we provide mathematical evidence that the concept of learning can be used to give a new and intuitive computational semantics of classical proofs in various fragments of Predicative Arithmetic. First, we extend Kreisel…
In this article, we investigate the arithmetical hierarchy from the perspective of realizability theory. An experimental observation in classical computability theory is that the notion of degrees of unsolvability for natural arithmetical…
Arithmetic automata recognize infinite words of digits denoting decompositions of real and integer vectors. These automata are known expressive and efficient enough to represent the whole set of solutions of complex linear constraints…
We treat uncertain linear programming problems by utilizing the notion of weighted analytic centers and notions from the area of multi-criteria decision making. After introducing our approach, we develop interactive cutting-plane algorithms…
We describe a realizability framework for classical first-order logic in which realizers live in (a model of) typed {\lambda}{\mu}-calculus. This allows a direct interpretation of classical proofs, avoiding the usual negative translation to…
We prove that interactive learning based classical realizability (introduced by Aschieri and Berardi for first order arithmetic) is sound with respect to Coquand game semantics. In particular, any realizer of an…
In an impressive series of papers, Krivine showed at the edge of the last decade how classical realizability provides a surprising technique to build models for classical theories. In particular, he proved that classical realizability…
We give a number of formal proofs of theorems from the field of computable analysis. Many of our results specify executable algorithms that work on infinite inputs by means of operating on finite approximations and are proven correct in the…
We develop a correspondence between the theory of sequential algorithms and classical reasoning, via Kreisel's no-counterexample interpretation. Our framework views realizers of the no-counterexample interpretation as dynamic processes…
Active learning is a paradigm of machine learning which aims at reducing the amount of labeled data needed to train a classifier. Its overall principle is to sequentially select the most informative data points, which amounts to determining…
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…
Predictive models are being increasingly used to support consequential decision making at the individual level in contexts such as pretrial bail and loan approval. As a result, there is increasing social and legal pressure to provide…
We present a new type system with support for proofs of programs in a call-by-value language with control operators. The proof mechanism relies on observational equivalence of (untyped) programs. It appears in two type constructors, which…
In the same sense as classical logic is a formal theory of truth, the recently initiated approach called computability logic is a formal theory of computability. It understands (interactive) computational problems as games played by a…