Interactive theorem provers: issues faced as a user and tackled as a developer

Interactive theorem provers (ITP for short) are tools whose final aim is to certify proofs written by human beings. To reach that objective they have to fill the gap between the high level language used by humans for communicating and reasoning about mathematics and the lower level language that a machine is able to “understand” and process. The user perceives this gap in terms of missing features or inefficiencies. The developer tries to accommodate the user requests without increasing the already high complexity of these applications. We believe that satisfactory solutions can only come from a strong synergy between users and developers. We devoted most part of our PHD designing and developing the Matita interactive theorem prover. The software was born in the computer science department of the University of Bologna as the result of composing together all the technologies developed by the HELM team (to which we belong) for the MoWGLI project. The MoWGLI project aimed at giving accessibility through the web to the libraries of formalised mathematics of various interactive theorem provers, taking Coq as the main test case. The motivations for giving life to a new ITP are: • study the architecture of these tools, with the aim of understanding the source of their complexity • exploit such a knowledge to experiment new solutions that, for backward compatibility reasons, would be hard (if not impossible) to test on a widely used system like Coq. Matita is based on the Curry-Howard isomorphism, adopting the Calculus of Inductive Constructions (CIC) as its logical foundation. Proof objects are thus, at some extent, compatible with the ones produced with the Coq ITP, that is itself able to import and process the ones generated using Matita. Although the systems have a lot in common, they share no code at all, and even most of the algorithmic solutions are different. The thesis is composed of two parts where we respectively describe our experience as a user and a developer of interactive provers. In particular, the first part is based on two different formalisation experiences: • our internship in the Mathematical Components team (INRIA), that is formalising the finite group theory required to attack the Feit Thompson Theorem. To tackle this result, giving an effective classification of finite groups of odd order, the team adopts the SSReflect Coq extension, developed by Georges Gonthier for the proof of the four colours theorem. • our collaboration at the D.A.M.A. Project, whose goal is the formalisation of abstract measure theory in Matita leading to a constructive proof of Lebesgue’s Dominated Convergence Theorem. The most notable issues we faced, analysed in this part of the thesis, are the following: the difficulties arising when using “black box” automation in large formalisations; the impossibility for a user (especially a newcomer) to master the context of a library of already formalised results; the uncomfortable big step execution of proof commands historically adopted in ITPs; the difficult encoding of mathematical structures with a notion of inheritance in a type theory without subtyping like CIC. In the second part of the manuscript many of these issues will be analysed with the looking glasses of an ITP developer, describing the solutions we adopted in the implementation of Matita to solve these problems: integrated searching facilities to assist the user in handling large libraries of formalised results; a small step execution semantic for proof commands; a flexible implementation of coercive subtyping allowing multiple inheritance with shared substructures; automatic tactics, integrated with the searching facilities, that generates proof commands (and not only proof objects, usually kept hidden to the user) one of which specifically designed to be user driven.

Extending Implicit Computational Complexity and Abstract Machines to Languages with Control

The Curry-Howard isomorphism is the idea that proofs in natural deduction can be put in correspondence with lambda terms in such a way that this correspondence is preserved by normalization. The concept can be extended from Intuitionistic Logic to other systems, such as Linear Logic. One of the nice conseguences of this isomorphism is that we can reason about functional programs with formal tools which are typical of proof systems: such analysis can also include quantitative qualities of programs, such as the number of steps it takes to terminate. Another is the possiblity to describe the execution of these programs in terms of abstract machines. In 1990 Griffin proved that the correspondence can be extended to Classical Logic and control operators. That is, Classical Logic adds the possiblity to manipulate continuations. In this thesis we see how the things we described above work in this larger context.

Towards a logical foundation of randomized computation

This dissertation investigates the relations between logic and TCS in the probabilistic setting. It is motivated by two main considerations. On the one hand, since their appearance in the 1960s-1970s, probabilistic models have become increasingly pervasive in several fast-growing areas of CS. On the other, the study and development of (deterministic) computational models has considerably benefitted from the mutual interchanges between logic and CS. Nevertheless, probabilistic computation was only marginally touched by such fruitful interactions. The goal of this thesis is precisely to (start) bring(ing) this gap, by developing logical systems corresponding to specific aspects of randomized computation and, therefore, by generalizing standard achievements to the probabilistic realm. To do so, our key ingredient is the introduction of new, measure-sensitive quantifiers associated with quantitative interpretations. The dissertation is tripartite. In the first part, we focus on the relation between logic and counting complexity classes. We show that, due to our classical counting propositional logic, it is possible to generalize to counting classes, the standard results by Cook and Meyer and Stockmeyer linking propositional logic and the polynomial hierarchy. Indeed, we show that the validity problem for counting-quantified formulae captures the corresponding level in Wagner's hierarchy. In the second part, we consider programming language theory. Type systems for randomized \lambda-calculi, also guaranteeing various forms of termination properties, were introduced in the last decades, but these are not "logically oriented" and no Curry-Howard correspondence is known for them. Following intuitions coming from counting logics, we define the first probabilistic version of the correspondence. Finally, we consider the relationship between arithmetic and computation. We present a quantitative extension of the language of arithmetic able to formalize basic results from probability theory. This language is also our starting point to define randomized bounded theories and, so, to generalize canonical results by Buss.