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}
}
  • Federico Aschieri
  • Published 25 October 2013
  • Computer Science
  • Mathematical Structures in Computer Science
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. 

Eliminating Skolem Functions in Peano Arithmetic with Interactive Realizability

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

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

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

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

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.

References

SHOWING 1-10 OF 45 REFERENCES

A Calculus of Realizers for EM1-Arithmetic

We propose a realizability interpretation of a system for quantifier free arithmetic which is equivalent to the fragment of classical arithmetic without nested quantifiers, which we call

A Calculus of Realizers for EM1 Arithmetic (Extended Abstract)

A realizability interpretation of a system for quantifier free arithmetic which is equivalent to the fragment of classical arithmetic without nested quantifiers, which is called EM 1 -arithmetic, is proposed, and any two quantifier-free formulas provably equivalent in classical arithmetic have the same realizer.

Learning, realizability and games in classical arithmetic

There is a one-to-one correspondence between realizers and recursive winning strategies in the 1-Backtracking version of Tarski games and a new realizability semantics is introduced, called "Interactive Learning-Based Realizability".

Interactive Learning-Based Realizability for Heyting Arithmetic with EM1

The idea of ``finite approximation'' used to provide computational interpretations of Herbrand's Theorem is applied to the semantics of Arithmetic, and a new Realizability semantics is introduced, which interprets atomic realizers as ``learning agents''.

Relating Classical Realizability and Negative Translation for Existential Witness Extraction

This paper shows how to achieve the same goal efficiently using Krivine realizability with primitive numerals, and proves that the corresponding program is but the direct-style equivalent (using call-cc) of the CPS-style program underlying Friedman's method.

On Krivine's Realizability Interpretation of Classical Second-Order Arithmetic

Krivine's realizability interpretation of classical second-order arithmetic and its recent extension handling countable choice is investigated and a twostep interpretation is presented which first eliminates classical logic via a negative translation and then applies standard realizable interpretation.

Split-2 bisimilarity has a finite axiomatization over CCS with Hennessy's merge

This note shows that split-2 bisimulation equivalence affords a finite equational axiomatization over the process algebra obtained by adding an auxiliary operation proposed by Hennessy in 1981 to the recursion free fragment of Milner's Calculus of Communicating Systems.

A Realizability Interpretation for Classical Arithmetic

Summary. A constructive realizablity interpretation for classical arithmetic is presented, enabling one to extract witnessing terms from proofs of � 1 sentences. The interpretation is shown to

Eliminating definitions and Skolem functions in first-order logic

  • J. Avigad
  • Mathematics, Computer Science
    Proceedings 16th Annual IEEE Symposium on Logic in Computer Science
  • 2001
The author considers how in any classical first-order theory strong enough to code finite functions, including sequential theories, one can also eliminate Skolem functions with a polynomial bound on the increase in proof length.