54 research outputs found
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
Universality of Univariate Mixed Fractions in Divisive Meadows
Univariate fractions can be transformed to mixed fractions in the equational
theory of meadows of characteristic zero.Comment: 12 page
Discriminating Lambda-Terms Using Clocked Boehm Trees
As observed by Intrigila, there are hardly techniques available in the
lambda-calculus to prove that two lambda-terms are not beta-convertible.
Techniques employing the usual Boehm Trees are inadequate when we deal with
terms having the same Boehm Tree (BT). This is the case in particular for fixed
point combinators, as they all have the same BT. Another interesting equation,
whose consideration was suggested by Scott, is BY = BYS, an equation valid in
the classical model P-omega of lambda-calculus, and hence valid with respect to
BT-equality but nevertheless the terms are beta-inconvertible. To prove such
beta-inconvertibilities, we employ `clocked' BT's, with annotations that convey
information of the tempo in which the data in the BT are produced. Boehm Trees
are thus enriched with an intrinsic clock behaviour, leading to a refined
discrimination method for lambda-terms. The corresponding equality is strictly
intermediate between beta-convertibility and Boehm Tree equality, the equality
in the model P-omega. An analogous approach pertains to Levy-Longo and
Berarducci Trees. Our refined Boehm Trees find in particular an application in
beta-discriminating fixed point combinators (fpc's). It turns out that Scott's
equation BY = BYS is the key to unlocking a plethora of fpc's, generated by a
variety of production schemes of which the simplest was found by Boehm, stating
that new fpc's are obtained by postfixing the term SI, also known as Smullyan's
Owl. We prove that all these newly generated fpc's are indeed new, by
considering their clocked BT's. Even so, not all pairs of new fpc's can be
discriminated this way. For that purpose we increase the discrimination power
by a precision of the clock notion that we call `atomic clock'.Comment: arXiv admin note: substantial text overlap with arXiv:1002.257
On Periodically Iterated Morphisms
We investigate the computational power of periodically iterated morphisms,
also known as D0L systems with periodic control, PD0L systems for short. These
systems give rise to a class of one-sided infinite sequences, called PD0L
words.
We construct a PD0L word with exponential subword complexity, thereby
answering a question raised by Lepisto (1993) on the existence of such words.
We solve another open problem concerning the decidability of the first-order
theories of PD0L words; we show it is already undecidable whether a certain
letter occurs in a PD0L word. This stands in sharp contrast to the situation
for D0L words (purely morphic words), which are known to have at most quadratic
subword complexity, and for which the monadic theory is decidable.
The main result of our paper, leading to these answers, is that every
computable word w over an alphabet Sigma can be embedded in a PD0L word u over
an extended alphabet Gamma in the following two ways: (i) such that every
finite prefix of w is a subword of u, and (ii) such that w is obtained from u
by erasing all letters from Gamma not in Sigma. The PD0L system generating such
a word u is constructed by encoding a Fractran program that computes the word
w; Fractran is a programming language as powerful as Turing Machines.
As a consequence of (ii), if we allow the application of finite state
transducers to PD0L words, we obtain the set of all computable words. Thus the
set of PD0L words is not closed under finite state transduction, whereas the
set of D0L words is. It moreover follows that equality of PD0L words (given by
their PD0L system) is undecidable. Finally, we show that if erasing morphisms
are admitted, then the question of productivity becomes undecidable, that is,
the question whether a given PD0L system defines an infinite word
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