2008

Editor: Roberto Amadio

Local reasoning about programs exploits the natural local behaviour common in programs by focussing on the footprint - that part of the resource accessed by the program. We address the problem of formally characterising and analysing the footprint notion for abstract local functions introduced by Calcagno, O Hearn and Yang. With our definition, we prove that the footprints are the only essential elements required for a complete specification of a local function. We formalise the notion of small specifications in local reasoning and show that for well-founded resource models, a smallest specification always exists that only includes the footprints, and also present results for the non-well-founded case. Finally, we use this theory of footprints to investigate the conditions under which the footprints correspond to the smallest safe states. We present a new model of RAM in which, unlike the standard model, the footprints of every program correspond to the smallest safe states, and we also identify a general condition on the primitive commands of a programming language which guarantees this property for arbitrary models.

Simple stochastic games are two-player zero-sum stochastic games with turn-based moves, perfect information, and reachability winning conditions. We present two new algorithms computing the values of simple stochastic games. Both of them rely on the existence of optimal permutation strategies, a class of positional strategies derived from permutations of the random vertices. The "permutation-enumeration" algorithm performs an exhaustive search among these strategies, while the "permutation-improvement" algorithm is based on successive improvements, à la Hoffman-Karp. Our algorithms improve previously known algorithms in several aspects. First they run in polynomial time when the number of random vertices is fixed, so the problem of solving simple stochastic games is fixed-parameter tractable when the parameter is the number of random vertices. Furthermore, our algorithms do not require the input game to be transformed into a stopping game. Finally, the permutation-enumeration algorithm does not use linear programming, while the permutation-improvement algorithm may run in polynomial time.

First, we extend Leifer-Milner RPO theory, by giving general conditions to obtain IPO labelled transition systems (and bisimilarities) with a reduced set of transitions, and possibly finitely branching. Moreover, we study the weak variant of Leifer-Milner theory, by giving general conditions under which the weak bisimilarity is a congruence. Then, we apply such extended RPO technique to the lambda-calculus, endowed with lazy and call by value reduction strategies. We show that, contrary to process calculi, one can deal directly with the lambda-calculus syntax and apply Leifer-Milner technique to a category of contexts, provided that we work in the framework of weak bisimilarities. However, even in the case of the transition system with minimal contexts, the resulting bisimilarity is infinitely branching, due to the fact that, in standard context categories, parametric rules such as the beta-rule can be represented only by infinitely many ground rules. To overcome this problem, we introduce the general notion of second-order context category. We show that, by carrying out the RPO construction in this setting, the lazy observational equivalence can be captured as a weak bisimilarity equivalence on a finitely branching transition system. This result is achieved by considering an encoding of lambda-calculus in Combinatory Logic.