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On Natural Deduction for Herbrand Constructive Logics II: Curry-Howard Correspondence for Markov's Principle in First-Order Logic and Arithmetic

2 3 4 5 6 7 8 9 10 13 Federico Aschieri . Strong normalization for HA + EM1 by non-deterministic choice . In Proceedings First Workshop on Control Operators and their Semantics , COS 2013 , Eindhoven ... , The Netherlands , June 24-25, 2013 ., pages 1 - 14 , 2013 . doi: 10 .4204/EPTCS.127.1. Federico Aschieri , Stefano Berardi, and Giovanni Birolo . Realizability and strong normalization for a Curry

A "Game Semantical" Intuitionistic Realizability Validating Markov's Principle

We propose a very simple modification of Kreisel's modified realizability in order to computationally realize Markov's Principle in the context of Heyting Arithmetic. Intuitively, realizers correspond to arbitrary strategies in Hintikka-Tarski games, while in Kreisel's realizability they can only represent winning strategies. Our definition, however, does not employ directly game...

Interactive Realizability for Classical Peano Arithmetic with Skolem Axioms

Interactive realizability is a computational semantics of classical Arithmetic. It is based on interactive learning and was originally designed to interpret excluded middle and Skolem axioms for simple existential formulas. A realizer represents a proof/construction depending on some state, which is an approximation of some Skolem functions. The realizer interacts with the...

Realizability and Strong Normalization for a Curry-Howard Interpretation of HA + EM1

We present a new Curry-Howard correspondence for HA + EM_1, constructive Heyting Arithmetic with the excluded middle on \Sigma^0_1-formulas. We add to the lambda calculus an operator ||_a which represents, from the viewpoint of programming, an exception operator with a delimited scope, and from the viewpoint of logic, a restricted version of the excluded middle. We motivate the...

Transfinite Update Procedures for Predicative Systems of Analysis

We present a simple-to-state, abstract computational problem, whose solution implies the 1-consistency of various systems of predicative Analysis and offers a way of extracting witnesses from classical proofs. In order to state the problem, we formulate the concept of transfinite update procedure, which extends Avigad's notion of update procedure to the transfinite and can be...