Tristan Crolard

A Formulae-as-Types Interpretation of Subtractive Logic

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 restricted form of first-class continuations).

Extending the loop language with higher-order procedural variables

We extend Meyer and Ritchies Loop language with higher-order procedures and procedural variables and we show that the resulting programming language (called Loop^ω) is a natural imperative counterpart of Gödel System T. The argument is two-fold: (1) we define a translation of the Loop^ω language into System T and we prove that this translation actually provides a lock-step simulation, (2) using a converse translation, we show that Loop^ω is expressive enough to encode any term of System T. Moreover, we define the “iteration rank” of a Loop^ω program, which corresponds to the classical notion of “recursion rank” in System T, and we show that both translations preserve ranks. Two applications of these results in the area of implicit complexity are described.

Deriving a Floyd–Hoare logic for non-local jumps from a formulæ-as-types notion of control

Abstract We derive a Floyd–Hoare logic for non-local jumps and mutable higher-order procedural variables from a formulae-as-types notion of control for classical logic. A key contribution of this work is the design of an imperative dependent type system for Hoare triples, which corresponds to classical logic, but where the famous consequence rule is admissible. Moreover, we prove that this system is complete for a reasonable notion of validity for Hoare judgments.

A type theory which is complete for Kreisel's modified realizability

Abstract We define a type theory with a strong elimination rule for existential quantification. As in Martin-Lofs type theory, the “axiom of choice” is thus derivable. Proofs are also annotated by realizers which are simply typed λ-terms. A new rule called “type extraction” which extracts the type of a realizer allows us to derive the so-called “independance of premisses” schema. Consequently, any formula which is realizable in HAω, according to Kreisels modified realizability, is derivable in this type theory.