Characterizing morphic sequences

Morphic sequences form a natural class of infinite sequences, extending the well-studied class of automatic sequences. Where automatic sequences are known to have several equivalent characterizations and the class of automatic sequences is known to have several closure properties, for the class of morphic sequences similar closure properties are known, but only limited equivalent characterizations. In this paper we extend the latter. We discuss a known characterization of morphic sequences based on automata and we give a characterization of morphic sequences by finiteness of a particular class of subsequences. Moreover, we relate morphic sequences to rationality of infinite terms and describe them by infinitary rewriting

Regularity Preserving but not Reflecting Encodings

Encodings, that is, injective functions from words to words, have been studied extensively in several settings. In computability theory the notion of encoding is crucial for defining computability on arbitrary domains, as well as for comparing the power of models of computation. In language theory much attention has been devoted to regularity preserving functions. A natural question arising in these contexts is: Is there a bijective encoding such that its image function preserves regularity of languages, but its pre-image function does not? Our main result answers this question in the affirmative: For every countable class C of languages there exists a bijective encoding f such that for every language L in C its image f[L] is regular. Our construction of such encodings has several noteworthy consequences. Firstly, anomalies arise when models of computation are compared with respect to a known concept of implementation that is based on encodings which are not required to be computable: Every countable decision model can be implemented, in this sense, by finite-state automata, even via bijective encodings. Hence deterministic finite-state automata would be equally powerful as Turing machine deciders. A second consequence concerns the recognizability of sets of natural numbers via number representations and finite automata. A set of numbers is said to be recognizable with respect to a representation if an automaton accepts the language of representations. Our result entails that there is one number representation with respect to which every recursive set is recognizable

The Guarded Lambda-Calculus: Programming and Reasoning with Guarded Recursion for Coinductive Types

We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. The use of guarded recursive types ensures the productivity of well-typed programs. Guarded recursive types may be transformed into coinductive types by a type-former inspired by modal logic and Atkey-McBride clock quantification, allowing the typing of acausal functions. We give a call-by-name operational semantics for the calculus, and define adequate denotational semantics in the topos of trees. The adequacy proof entails that the evaluation of a program always terminates. We introduce a program logic with L\"ob induction for reasoning about the contextual equivalence of programs. We demonstrate the expressiveness of the calculus by showing the definability of solutions to Rutten's behavioural differential equations.Comment: Accepted to Logical Methods in Computer Science special issue on the 18th International Conference on Foundations of Software Science and Computation Structures (FoSSaCS 2015

Automatic Sequences and Zip-Specifications

We consider infinite sequences of symbols, also known as streams, and the decidability question for equality of streams defined in a restricted format. This restricted format consists of prefixing a symbol at the head of a stream, of the stream function `zip', and recursion variables. Here `zip' interleaves the elements of two streams in alternating order, starting with the first stream. For example, the Thue-Morse sequence is obtained by the `zip-specification' {M = 0 : X, X = 1 : zip(X,Y), Y = 0 : zip(Y,X)}. Our analysis of such systems employs both term rewriting and coalgebraic techniques. We establish decidability for these zip-specifications, employing bisimilarity of observation graphs based on a suitably chosen cobasis. The importance of zip-specifications resides in their intimate connection with automatic sequences. We establish a new and simple characterization of automatic sequences. Thus we obtain for the binary zip that a stream is 2-automatic iff its observation graph using the cobasis (hd,even,odd) is finite. The generalization to zip-k specifications and their relation to k-automaticity is straightforward. In fact, zip-specifications can be perceived as a term rewriting syntax for automatic sequences. Our study of zip-specifications is placed in an even wider perspective by employing the observation graphs in a dynamic logic setting, leading to an alternative characterization of automatic sequences. We further obtain a natural extension of the class of automatic sequences, obtained by `zip-mix' specifications that use zips of different arities in one specification. We also show that equivalence is undecidable for a simple extension of the zip-mix format with projections like even and odd. However, it remains open whether zip-mix specifications have a decidable equivalence problem