Step-Indexed Relational Reasoning for Countable Nondeterminism

Programming languages with countable nondeterministic choice are computationally interesting since countable nondeterminism arises when modeling fairness for concurrent systems. Because countable choice introduces non-continuous behaviour, it is well-known that developing semantic models for programming languages with countable nondeterminism is challenging. We present a step-indexed logical relations model of a higher-order functional programming language with countable nondeterminism and demonstrate how it can be used to reason about contextually defined may- and must-equivalence. In earlier step-indexed models, the indices have been drawn from {\omega}. Here the step-indexed relations for must-equivalence are indexed over an ordinal greater than {\omega}

Predicate-Transformer Semantics of General Recursion

We develop the semantics of a language with arbitrary atomic statements, unbounded nondeterminacy, and mutual recursion. The semantics is expressed in weakest preconditions and weakest liberal preconditions. Individual states are not mentioned. The predicates on the state space are treated as elements of a distributive lattice. The semantics of recursion is constructed by means of the theorem of Knaster-Tarski. It is proved that the law of the excluded miracle can be preserved, if that is wanted. The universal conjunctivity of the weakest liberal precondition, and the connection between the weakest precondition and the weakest liberal precondition are proved to remain valid. Finally we treat Hoare-triple methods for proving correctness and conditional correctness of programs

Constructive Design of a Hierarchy of Semantics of a Transition System by Abstract Interpretation

We construct a hierarchy of semantics by successive abstract interpretations. Starting from the maximal trace semantics of a transition system, we derive the big-step semantics, termination and nontermination semantics, Plotkin’s natural, Smyth’s demoniac and Hoare’s angelic relational semantics and equivalent nondeterministic denotational semantics (with alternative powerdomains to the Egli-Milner and Smyth constructions), D. Scott’s deterministic denotational semantics, the generalized and Dijkstra’s conservative/liberal predicate transformer semantics, the generalized/total and Hoare’s partial correctness axiomatic semantics and the corresponding proof methods. All the semantics are presented in a uniform fixpoint form and the correspondences between these semantics are established through composable Galois connections, each semantics being formally calculated by abstract interpretation of a more concrete one using Kleene and/or Tarsk