An Operational Characterization of Mutual Information in Algorithmic Information Theory

We show that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings x and y is equal, up to logarithmic precision, to the length of the longest shared secret key that two parties, one having x and the complexity profile of the pair and the other one having y and the complexity profile of the pair, can establish via a probabilistic protocol with interaction on a public channel. For l > 2, the longest shared secret that can be established from a tuple of strings (x_1, . . .x_l) by l parties, each one having one component of the tuple and the complexity profile of the tuple, is equal, up to logarithmic precision, to the complexity of the tuple minus the minimum communication necessary for distributing the tuple to all parties. We establish the communication complexity of secret key agreement protocols that produce a secret key of maximal length, for protocols with public randomness. We also show that if the communication complexity drops below the established threshold then only very short secret keys can be obtained

Communication Complexity of the Secret Key Agreement in Algorithmic Information Theory

It is known that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings x and y is equal to the length of the longest shared secret key that two parties can establish via a probabilistic protocol with interaction on a public channel, assuming that the parties hold as their inputs x and y respectively. We determine the worst-case communication complexity of this problem for the setting where the parties can use private sources of random bits. We show that for some x, y the communication complexity of the secret key agreement does not decrease even if the parties have to agree on a secret key whose size is much smaller than the mutual information between x and y. On the other hand, we discuss examples of x, y such that the communication complexity of the protocol declines gradually with the size of the derived secret key. The proof of the main result uses spectral properties of appropriate graphs and the expander mixing lemma, as well as information theoretic techniques.Comment: 33 pages, 6 figures. v3: the full version of the MFCS 2020 pape

Model Checking Linear Logic Specifications

The overall goal of this paper is to investigate the theoretical foundations of algorithmic verification techniques for first order linear logic specifications. The fragment of linear logic we consider in this paper is based on the linear logic programming language called LO enriched with universally quantified goal formulas. Although LO was originally introduced as a theoretical foundation for extensions of logic programming languages, it can also be viewed as a very general language to specify a wide range of infinite-state concurrent systems. Our approach is based on the relation between backward reachability and provability highlighted in our previous work on propositional LO programs. Following this line of research, we define here a general framework for the bottom-up evaluation of first order linear logic specifications. The evaluation procedure is based on an effective fixpoint operator working on a symbolic representation of infinite collections of first order linear logic formulas. The theory of well quasi-orderings can be used to provide sufficient conditions for the termination of the evaluation of non trivial fragments of first order linear logic.Comment: 53 pages, 12 figures "Under consideration for publication in Theory and Practice of Logic Programming

Society-in-the-Loop: Programming the Algorithmic Social Contract

Recent rapid advances in Artificial Intelligence (AI) and Machine Learning have raised many questions about the regulatory and governance mechanisms for autonomous machines. Many commentators, scholars, and policy-makers now call for ensuring that algorithms governing our lives are transparent, fair, and accountable. Here, I propose a conceptual framework for the regulation of AI and algorithmic systems. I argue that we need tools to program, debug and maintain an algorithmic social contract, a pact between various human stakeholders, mediated by machines. To achieve this, we can adapt the concept of human-in-the-loop (HITL) from the fields of modeling and simulation, and interactive machine learning. In particular, I propose an agenda I call society-in-the-loop (SITL), which combines the HITL control paradigm with mechanisms for negotiating the values of various stakeholders affected by AI systems, and monitoring compliance with the agreement. In short, `SITL = HITL + Social Contract.'Comment: (in press), Ethics of Information Technology, 201