The paper “Applications of the analogy between formulas and exponential polynomials to equivalence and normal forms” to be presented at PLS12 in Anogeia, Crete, Grece
We show some applications of the formulas-as-polynomials correspondence: 1) a method for (dis)proving formula isomorphism and equivalence based on showing (in)equality; 2) a constructive analogue of the arithmetical hierarchy, based on the exp-log normal form. The results are valid intuitionistically, as well as classically.
We revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials.
After observing that most of the invertible proof rules of intuitionistic (minimal) propositional sequent calculi are formula (i.e. sequent) isomorphisms corresponding to the high-school identities, we show that one can obtain a more compact variant of a proof system, consisting of non-invertible proof rules only, and where the invertible proof rules have been replaced by a formula normalisation procedure.
Moreover, for certain proof systems such as the G4ip sequent calculus of Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the non-invertible proof rules as strict inequalities between exponential polynomials; a careful combinatorial treatment is given in order to establish this fact.
Finally, we extend the exponential polynomial analogy to the first-order quantifiers, showing that it gives rise to an intuitionistic hierarchy of formulas, resembling the classical arithmetical hierarchy, and the first one that classifies formulas while preserving isomorphism.
Common Criteria [ISO/IEC 15408] is an international
standard for certifying the security of computer systems. When
high evaluation assurance levels (EAL6 and EAL7) are required of
a system, Common Criteria (CC) requires formal modelling and
mechanized proofs of security properties. In this talk, we will
first have an overall look at the schema for CC evaluations,
after which we will concentrate on the scientific aspects
relating to proofs and programming languages, and the special
role that proof assistants play during these
evaluations.
Abstract: In this chapter, we propose some future directions of work, potentially beneficial to Mathematics and its foundations, based on the recent import of methodology from the theory of programming languages into proof theory. This scientific essay, written for the audience of proof theorists as well as the working mathematician, is not a survey of the field, but rather a personal view of the author who hopes that it may inspire future and fellow researchers.
Abstract: Lambda calculi with algebraic data types lie at the core of functional programming languages and proof assistants, but conceal at least two fundamental theoretical problems already in the presence of the simplest non-trivial data type, the sum type. First, we do not know of an explicit and implemented algorithm for deciding the beta-eta-equality of terms—-and this in spite of the first decidability results proven two decades ago. Second, it is not clear how to decide when two types are essentially the same, i.e. isomorphic, in spite of the meta-theoretic results on decidability of the isomorphism.
In this paper, we present the exp-log normal form of types—-derived from the representation of exponential polynomials via the unary exponential and logarithmic functions—-that any type built from arrows, products, and sums, can be isomorphically mapped to. The type normal form can be used as a simple heuristic for deciding type isomorphism, thanks to the fact that it is a systematic application of the high-school identities.
We then show that the type normal form allows to reduce the standard beta-eta equational theory of the lambda calculus to a specialized version of itself, while preserving the completeness of equality on terms. We end by describing an alternative representation of normal terms of the lambda calculus with sums, together with a Coq-implemented converter into/from our new term calculus. The difference with the only other previously implemented heuristic for deciding interesting instances of eta-equality by Balat, Di Cosmo, and Fiore, is that we exploit the type information of terms substantially and this often allows us to obtain a canonical representation of terms without performing sophisticated term analyses.
Abstract: Logical incarnations of type isomorphism include the notions of constructive cardinality of sets and strong intuitionistic equivalence of formulas. These are challenging to study in simultaneous presence of functions (exponentials) and sums (disjoint unions, disjunction). In this talk, I will present a quasi-normal form of types that arises from the decomposition of binary exponentiation into the unary exponentiation and logarithm. This normal form can be applied for disentangling the equational theory (beta-eta) of the lambda calculus with sums. By an extension of the normal form from simple types (propositional logic) to quantifiers, one can also retrieve an “arithmetical” hierarchy for intuitionistic first order logic. Finally, this suggests a sequent calculus for intuitionistic logic that uses the notation of exponential polynomials and removes the need for most of the invertible proof rules of usual sequent calculi.