Probabilistic Recursion Theory and Implicit Computational Complexity

In this thesis we provide a characterization of probabilistic computation in itself, from a recursion-theoretical perspective, without reducing it to deterministic computation. More specifically, we show that probabilistic computable functions, i.e., those functions which are computed by Probabilistic Turing Machines (PTM), can be characterized by a natural generalization of Kleene's partial recursive functions which includes, among initial functions, one that returns identity or successor with probability 1/2. We then prove the equi-expressivity of the obtained algebra and the class of functions computed by PTMs. In the the second part of the thesis we investigate the relations existing between our recursion-theoretical framework and sub-recursive classes, in the spirit of Implicit Computational Complexity. More precisely, endowing predicative recurrence with a random base function is proved to lead to a characterization of polynomial-time computable probabilistic functions.

How to explain: from theory to practice

Today we live in an age where the internet and artificial intelligence allow us to search for information through impressive amounts of data, opening up revolutionary new ways to make sense of reality and understand our world. However, it is still an area of improvement to exploit the full potential of large amounts of explainable information by distilling it automatically in an intuitive and user-centred explanation. For instance, different people (or artificial agents) may search for and request different types of information in a different order, so it is unlikely that a short explanation can suffice for all needs in the most generic case. Moreover, dumping a large portion of explainable information in a one-size-fits-all representation may also be sub-optimal, as the needed information may be scarce and dispersed across hundreds of pages. The aim of this work is to investigate how to automatically generate (user-centred) explanations from heterogeneous and large collections of data, with a focus on the concept of explanation in a broad sense, as a critical artefact for intelligence, regardless of whether it is human or robotic. Our approach builds on and extends Achinstein’s philosophical theory of explanations, where explaining is an illocutionary (i.e., broad but relevant) act of usefully answering questions. Specifically, we provide the theoretical foundations of Explanatory Artificial Intelligence (YAI), formally defining a user-centred explanatory tool and the space of all possible explanations, or explanatory space, generated by it. We present empirical results in support of our theory, showcasing the implementation of YAI tools and strategies for assessing explainability. To justify and evaluate the proposed theories and models, we considered case studies at the intersection of artificial intelligence and law, particularly European legislation. Our tools helped produce better explanations of software documentation and legal texts for humans and complex regulations for reinforcement learning agents.

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.