Three hierarchies of transducers

Composition of top-down tree transducers yields a proper hierarchy of transductions and of output languages. The same is true for ETOL systems (viewed as transducers) and for two-way generalized sequential machines

A language theoretic analysis of combings

A group is combable if it can be represented by a language of words satisfying a fellow traveller property; an automatic group has a synchronous combing which is a regular language. This paper gives a systematic analysis of the properties of groups with combings in various formal language classes, and of the closure properties of the associated classes of groups. It generalises previous work, in particular of Epstein et al. and Bridson and Gilman.Comment: DVI and Post-Script files only, 21 pages. Submitted to International Journal of Algebra and Computatio

Hairdressing in groups: a survey of combings and formal languages

A group is combable if it can be represented by a language of words satisfying a fellow traveller property; an automatic group has a synchronous combing which is a regular language. This article surveys results for combable groups, in particular in the case where the combing is a formal language.Comment: 17 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon1/paper24.abs.htm

Step-Indexed Logical Relations for Probability (long version)

It is well-known that constructing models of higher-order probabilistic programming languages is challenging. We show how to construct step-indexed logical relations for a probabilistic extension of a higher-order programming language with impredicative polymorphism and recursive types. We show that the resulting logical relation is sound and complete with respect to the contextual preorder and, moreover, that it is convenient for reasoning about concrete program equivalences. Finally, we extend the language with dynamically allocated first-order references and show how to extend the logical relation to this language. We show that the resulting relation remains useful for reasoning about examples involving both state and probabilistic choice.Comment: Extended version with appendix of a FoSSaCS'15 pape

On Measuring Non-Recursive Trade-Offs

We investigate the phenomenon of non-recursive trade-offs between descriptional systems in an abstract fashion. We aim at categorizing non-recursive trade-offs by bounds on their growth rate, and show how to deduce such bounds in general. We also identify criteria which, in the spirit of abstract language theory, allow us to deduce non-recursive tradeoffs from effective closure properties of language families on the one hand, and differences in the decidability status of basic decision problems on the other. We develop a qualitative classification of non-recursive trade-offs in order to obtain a better understanding of this very fundamental behaviour of descriptional systems

Intensional and Extensional Semantics of Bounded and Unbounded Nondeterminism

We give extensional and intensional characterizations of nondeterministic functional programs: as structure preserving functions between biorders, and as nondeterministic sequential algorithms on ordered concrete data structures which compute them. A fundamental result establishes that the extensional and intensional representations of non-deterministic programs are equivalent, by showing how to construct a unique sequential algorithm which computes a given monotone and stable function, and describing the conditions on sequential algorithms which correspond to continuity with respect to each order. We illustrate by defining may and must-testing denotational semantics for a sequential functional language with bounded and unbounded choice operators. We prove that these are computationally adequate, despite the non-continuity of the must-testing semantics of unbounded nondeterminism. In the bounded case, we prove that our continuous models are fully abstract with respect to may and must-testing by identifying a simple universal type, which may also form the basis for models of the untyped lambda-calculus. In the unbounded case we observe that our model contains computable functions which are not denoted by terms, by identifying a further "weak continuity" property of the definable elements, and use this to establish that it is not fully abstract