# A Formulae-as-Types Interpretation of Subtractive Logic

@article{Crolard2004AFI, title={A Formulae-as-Types Interpretation of Subtractive Logic}, author={Tristan Crolard}, journal={J. Log. Comput.}, year={2004}, volume={14}, pages={529-570} }

We present a formulae-as-types interpretation of Subtractive Logic (i.e. bi-intuitionistic logic). This presentation is two-fold: we first define a very natural restriction of the λμ-calculus which is closed under reduction and whose type system is a constructive restriction of the Classical Natural Deduction. Then we extend this deduction system conservatively to Subtractive Logic. From a computational standpoint, the resulting calculus provides a type system for first-class coroutines (a…

## 59 Citations

### Relating Sequent Calculi for Bi-intuitionistic Propositional Logic

- PhilosophyCL&C
- 2010

Three sequent calculi for bi-intuitionistic propositional logic are compared and a basic standard-style sequent calculus that restricts the pre mises of implication-right and exclusion-left inferences to be single-conclusion resp.

### A TERM ASSIGNMENT FOR DUAL INTUITIONISTIC LOGIC

- Mathematics
- 2006

We study the proof-theory of co-Heyting algebras and present a calculus of continuations typed in the disjunctive–subtractive fragment of dual intuitionistic logic. We give a single-assumption…

### Classical Logic with Mendler Induction - A Dual Calculus and Its Strong Normalization

- Mathematics, PhilosophyLFCS
- 2016

An extension of the Dual Calculus with a Mendler-style (co-)iterator that remains strongly normalizing under head reduction is introduced, and is proved using a non-constructive realizability argument.

### Sequent calculi for bi-intuitionistic propositional logic

- Computer Science
- 2010

This talk compares three sequent calculi for bi-intuitionistic propositional logic and shows that these calculi can be translated into each other and discuss the ineliminable cuts of the standard-style sequent calculus.

### A PARIGOT-STYLE LINEAR -CALCULUS FOR FULL INTUITIONISTIC LINEAR LOGIC

- Philosophy
- 2006

This paper describes a natural deduction formulation for Full Intuitionistic Linear Logic (FILL), an intriguing variation of multiplicative linear logic, due to Hyland and de Paiva. The system FILL…

### Dualized Simple Type Theory

- PhilosophyLog. Methods Comput. Sci.
- 2016

DTT is a simple type theory with perfect intuitionistic duality, and corresponds to a single-sided polarized sequent calculus, which gives a direct proof of consistency, but proves completeness by reduction to L.

### Categorical Proof Theory of Co-Intuitionistic Linear Logic

- PhilosophyLog. Methods Comput. Sci.
- 2014

This work builds models of co-intuitionistic logic in symmetric monoidal left-closed categories with additional structure using a variant of Crolard's term assignment to co-intsubstantial logic in the construction of a free category.

### Dualized Type Theory

- Computer Science
- 2014

DTT is a type theory with perfect intuitionistic duality, and corresponds to a single-sided polarized sequent calculus, and it is proved strongly normalizing, and type preservation is proved.

### A type-theoretic foundation of continuations and prompts

- GeologyICFP '04
- 2004

The addition of prompts corresponds to the addition of a single dynamically-scoped variable modelling the special top-level continuation of classical logic with first-class continuations.

## References

SHOWING 1-10 OF 55 REFERENCES

### Strong Normalization of Classical Natural Deduction with Disjunction

- MathematicsTLCA
- 2001

An extension of Parigot's λµ-calculus where disjunction is taken as a primitive is introduced, and the associated reduction relation is Church-Rosser, strongly normalizing, and such that the normal deductions satisfy the subformula property.

### Proof-terms for classical and intuitionistic resolution

- PhilosophyJ. Log. Comput.
- 2000

We exploit a system of realizers for classical logic, and a translation from resolution into the sequent calculus, to assess the intuitionistic force of classical resolution for a fragment of…

### A formulae-as-type notion of control

- Computer SciencePOPL '90
- 1989

It is proved that all evaluations of typed terms in Idealized Scheme are finite, and the existence of computationally interesting “classical programs” is illustrated by the definition of conjunctively, disjunctive, and existential types using standard classical definitions.

### Dual Intuitionistic Logic Revisited

- Computer ScienceTABLEAUX
- 2000

It is shown that a previously reported generalised display framework does deliver the required cut-free display calculus and the structural rule necessary to turn this display calculus into one for classical logic is pinpointed.

### Lambda-Mu-Calculus: An Algorithmic Interpretation of Classical Natural Deduction

- Computer ScienceLPAR
- 1992

This paper presents a way of extending the paradigm "proofs as programs" to classical proofs, which can be seen as a simple extension of intuitionistic natural deduction, whose algorithmic interpretation is very well known.

### A confluent λ-calculus with a catch/throw mechanism

- MathematicsJournal of Functional Programming
- 1999

The subject reduction property for the λ-calculus is obtained, as well as the strong normalization for λct-terms typable in the second order classical natural deduction.

### Declarative Continuations: an Investigation of Duality in Programming Language Semantics

- Computer ScienceCategory Theory and Computer Science
- 1989

A symmetric extension of the typed λ-calculus is introduced, where values and continuations play dual roles, permitting mirror-image syntax for dual categorical concepts like products and coproducts.

### A Simple Calculus of Exception Handling

- Computer ScienceTLCA
- 1995

A simply-typed λ-calculus featuring an ML-like exception handling mechanism, whose type system corresponds to classical logic through the Curry-Howard isomorphism satifies several interesting properties: among other, Church-Rosser, subject reduction, and strong-normalisation.