Stefano Berardi

A Symmetric Lambda Calculus for Classical Program Extraction

We introduce a?-calculus with symmetric reduction rules and “classical” types, i.e., types corresponding to formulas of classical propositional logic. The strong normalization property is proved to hold for such a calculus, as well as for its extension to a system equivalent to Peano arithmetic. A theorem on the shape of terms in normal form is also proved, making it possible to get recursive functions out of proofs of?02formulas, i.e., those corresponding to program specifications.

On the computational content of the axiom of choice

We present a possible computational content of the negative translation of classical analysis with the Axiom of (countable) Choice. Interestingly, this interpretation uses a refinement of the realizability semantics of the absurdity proposition, which is not interpreted as the empty type here. We also show how to compute witnesses from proofs in classical analysis of ∃-statements and how to extract algorithms from proofs of ∀∃-statements. Our interpretation seems computationally more direct than the one based on Godels Dialectica interpretation.

Extracting Constructive Content from Classical Logic via Control-like Reductions

Recently there has been much interest in the problem of finding the computational content of classical reasoning. One of the most appealing directions for the computer scientist to tackle such a problem is the relation which has been established between classical logic and lambda calculi with control operators, like Felleisens control operator C. In this paper we introduce a typed lambda calculus with the C operator corresponding to Peano Arithmetic, and a set of reduction rules related to the ones of the usual control calculi with C. We show how these rules, which are proved to be strongly normalizing, can be used to extract witnesses from proofs of ∑ 1 0 sentences in Peano Arithmetic.

Some intuitionistic equivalents of classical principles for degree 2 formulas

Abstract We consider the restriction of classical principles like Excluded Middle, Markov’s Principle, Konig’s Lemma to arithmetical formulas of degree 2. For any such principle, we find simple mathematical statements which are intuitionistically equivalent to it, provided we restrict universal quantifications over maps to computable maps.

Using Subtyping in Program Optimization

Constructive logics can be used to write the specifications of programs as logic formulas to be proved. By using the Curry-Howard isomorphism, we can automatically extract executable code from constructive proofs. Normally, programs automatically generated are inefficent. They contains parts useless to compute the final result. In this paper, we show a technique to erase such useless parts from programs. Our technique is essentially an extension of another method, developed by S. Berardi, optimizing simply typed λ-terms. By using the notion of subtyping, we can overcome some intrinsecal limitations of the original method. We prove that optimized terms are equivalent to the original ones and we give an algorithm to find such optimizations

Classical logic as limit completion

We define a constructive model for

βη-complete models for System F

{\Delta^0_2}

A strong normalization result for classical logic

-maps, that is, maps recursively definable from a map deciding the halting problem. Our model refines an existing constructive interpretation for classical reasoning over one-quantifier formulas: it is compositional (Modus Ponens is interpreted as an application) and semantical (rather than translating classical proofs into intuitionistic ones, we define a mathematical structure intuitionistically validating excluded middle for one-quantifier formulas).

Interactive Realizers: A New Approach to Program Extraction from Nonconstructive Proofs

We show that Friedmans proof of the existence of non-trivial βη-complete models of λ→ can be extended to system F. We isolate a set of conditions that are sufficient to ensure βη-completeness for a model of F (and α-completeness at the level of types), and we discuss which class of models we get. In particular, the model introduced in Barbanera and Berardi (1997), having as polymorphic maps exactly all possible Scott continuous maps, is βη-complete, and is hence the first known complete non-syntactic model of F. In order to have a suitable framework in which to express the conditions and develop the proof, we also introduce the very natural notion of ‘polymax models’ of System F.

A New Use of Friedman’s Translation: Interactive Realizability

Abstract In this paper we give a strong normalization proof for a set of reduction rules for classical logic. These reductions, more general than the ones usually considered in literature, are inspired to the reductions of Felleisens lambda calculus with continuations.