# Interactive Realizability for second-order Heyting arithmetic with EM1 and SK1

@article{Aschieri2013InteractiveRF, title={Interactive Realizability for second-order Heyting arithmetic with EM1 and SK1}, author={Federico Aschieri}, journal={Mathematical Structures in Computer Science}, year={2013}, volume={24} }

We introduce a realizability semantics based on interactive learning for full second-order Heyting arithmetic with excluded middle and Skolem axioms over Σ10-formulas. Realizers are written in a classical version of Girard's System $\mathsf{F}$ and can be viewed as programs that learn by interacting with the environment. We show that the realizers of any Π20-formula represent terminating learning processes whose outcomes are numerical witnesses for the existential quantifier of the formula.

## 6 Citations

### Eliminating Skolem Functions in Peano Arithmetic with Interactive Realizability

- Mathematics
- 2012

We present a new syntactical proof that first-order Peano Arithmetic with Skolem axioms is conservative over Peano Arithmetic alone for arithmetical formulas. This result -- which shows that the…

### Interactive Realizability and the elimination of Skolem functions in Peano Arithmetic

- MathematicsCL&C
- 2012

A new syntactical proof is presented that first-order Peano Arithmetic with Skolem axioms is conservative over PeanoArithmetic alone for arithmetical formulas, and uses Interactive Realizability, a computational semantics for Peanos Arithmetic which extends Kreisel's modified realizability to the classical case.

### Interactive Realizability for Classical Peano Arithmetic with Skolem Axioms

- Computer ScienceCSL
- 2012

This work is to extend Interactive realizability to a system which includes classical first-order Peano Arithmetic with Skolem axioms, and realizers of atomic formulas will be update procedures in the sense of Avigad and thus will be understood as stratified-learning algorithms.

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

- MathematicsCSL
- 2013

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

### On Natural Deduction for Herbrand Constructive Logics I: Curry-Howard Correspondence for Dummett's Logic LC

- PhilosophyLog. Methods Comput. Sci.
- 2016

This work adds to the lambda calculus an operator which represents, from the viewpoint of programming, a mechanism for representing parallel computations and communication between them, and from the point of view logic, Dummett's axiom.

### On natural deduction in classical first-order logic: Curry-Howard correspondence, strong normalization and Herbrand's theorem

- MathematicsTheor. Comput. Sci.
- 2016

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