Bisimilarity as a Theory of Functional Programming

AbstractMorris-style contextual equivalence — invariance of termination under any context of ground type — is the usual notion of operational equivalence for deterministic functional languages such as FPC (PCF plus sums, products and recursive types). Contextual equivalence is hard to establish directly. Instead we define a labelled transition system for call-by-name FPC (and variants) and prove that CCS-style bisimilarity equals contextual equivalence — a form of operational extensionality. Using co-induction we establish equational laws for FPC. By considering variations of Milner's ‘bisimulations up to ∼’ we obtain a second co-inductive characterisation of contextual equivalence in terms of reduction behaviour and production of values. Hence we use co-inductive proofs to establish contextual equivalence in a series of stream-processing examples. Finally, we consider a form of Milner's original context lemma for FPC, but conclude that our form of bisimilarity supports simpler co-inductive proofs

Nested Term Graphs (Work In Progress)

We report on work in progress on 'nested term graphs' for formalizing higher-order terms (e.g. finite or infinite lambda-terms), including those expressing recursion (e.g. terms in the lambda-calculus with letrec). The idea is to represent the nested scope structure of a higher-order term by a nested structure of term graphs. Based on a signature that is partitioned into atomic and nested function symbols, we define nested term graphs both in a functional representation, as tree-like recursive graph specifications that associate nested symbols with usual term graphs, and in a structural representation, as enriched term graph structures. These definitions induce corresponding notions of bisimulation between nested term graphs. Our main result states that nested term graphs can be implemented faithfully by first-order term graphs. keywords: higher-order term graphs, context-free grammars, cyclic lambda-terms, higher-order rewrite systemsComment: In Proceedings TERMGRAPH 2014, arXiv:1505.0681

Coinductive Big-Step Semantics for Concurrency

In a paper presented at SOS 2010, we developed a framework for big-step semantics for interactive input-output in combination with divergence, based on coinductive and mixed inductive-coinductive notions of resumptions, evaluation and termination-sensitive weak bisimilarity. In contrast to standard inductively defined big-step semantics, this framework handles divergence properly; in particular, runs that produce some observable effects and then diverge, are not "lost". Here we scale this approach for shared-variable concurrency on a simple example language. We develop the metatheory of our semantics in a constructive logic.Comment: In Proceedings PLACES 2013, arXiv:1312.221

A uniform framework for modelling nondeterministic, probabilistic, stochastic, or mixed processes and their behavioral equivalences

Labeled transition systems are typically used as behavioral models of concurrent processes, and the labeled transitions define the a one-step state-to-state reachability relation. This model can be made generalized by modifying the transition relation to associate a state reachability distribution, rather than a single target state, with any pair of source state and transition label. The state reachability distribution becomes a function mapping each possible target state to a value that expresses the degree of one-step reachability of that state. Values are taken from a preordered set equipped with a minimum that denotes unreachability. By selecting suitable preordered sets, the resulting model, called ULTraS from Uniform Labeled Transition System, can be specialized to capture well-known models of fully nondeterministic processes (LTS), fully probabilistic processes (ADTMC), fully stochastic processes (ACTMC), and of nondeterministic and probabilistic (MDP) or nondeterministic and stochastic (CTMDP) processes. This uniform treatment of different behavioral models extends to behavioral equivalences. These can be defined on ULTraS by relying on appropriate measure functions that expresses the degree of reachability of a set of states when performing single-step or multi-step computations. It is shown that the specializations of bisimulation, trace, and testing equivalences for the different classes of ULTraS coincide with the behavioral equivalences defined in the literature over traditional models