Transductions Computed by One-Dimensional Cellular Automata

Cellular automata are investigated towards their ability to compute transductions, that is, to transform inputs into outputs. The families of transductions computed are classified with regard to the time allowed to process the input and to compute the output. Since there is a particular interest in fast transductions, we mainly focus on the time complexities real time and linear time. We first investigate the computational capabilities of cellular automaton transducers by comparing them to iterative array transducers, that is, we compare parallel input/output mode to sequential input/output mode of massively parallel machines. By direct simulations, it turns out that the parallel mode is not weaker than the sequential one. Moreover, with regard to certain time complexities cellular automaton transducers are even more powerful than iterative arrays. In the second part of the paper, the model in question is compared with the sequential devices single-valued finite state transducers and deterministic pushdown transducers. It turns out that both models can be simulated by cellular automaton transducers faster than by iterative array transducers.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249

Transforming a Single-Valued Transducer Into a Mealy Machine

AbstractThis article deals with the transformation of a single-valued finite transducer into a Mealy machine. The following results are obtained: (1) LetMbe a single-valued real-time (or “letter-to-word”) transducer withnstates, input alphabetΣ, and output alphabetΔwhich is equivalent to some Mealy machineM′. Then,Mcan be effectively transformed into such anM′ having at most 2n+1·min{#Σ, #Δ}n−1states. A similar result holds ifMis not real time. As an important side effect three “Mealy” properties are obtained which characterize the fact that the given transducerMis equivalent to some Mealy machine. (2) The upper bound in result (1) improves to 2n−1 ifMis known to be a letter-to-letter transducer. (3) For every integert⩾2 and every odd integern⩾3 there is a single-valued real-time transducerMwithnstates and input and output alphabets of cardinalitytsuch thatMis equivalent to some Mealy machineM′ and every suchM′ has at leastt(n−1)/2states. (4) Ift=3, then result (3) holds true with letter-to-letter transducers rather than real-time transducers and with a lower bound of 2(n−1)/2. (5) It is a PSPACE-complete problem to decide whether or not a given single-valued transducerMis equivalent to some Mealy machine. The problem remains PSPACE-complete ifMis known to be a letter-to-letter transducer

Degree of Sequentiality of Weighted Automata

Weighted automata (WA) are an important formalism to describe quantitative properties. Obtaining equivalent deterministic machines is a longstanding research problem. In this paper we consider WA with a set semantics, meaning that the semantics is given by the set of weights of accepting runs. We focus on multi-sequential WA that are defined as finite unions of sequential WA. The problem we address is to minimize the size of this union. We call this minimum the degree of sequentiality of (the relation realized by) the WA. For a given positive integer k, we provide multiple characterizations of relations realized by a union of k sequential WA over an infinitary finitely generated group: a Lipschitz-like machine independent property, a pattern on the automaton (a new twinning property) and a subclass of cost register automata. When possible, we effectively translate a WA into an equivalent union of k sequential WA. We also provide a decision procedure for our twinning property for commutative computable groups thus allowing to compute the degree of sequentiality. Last, we show that these results also hold for word transducers and that the associated decision problem is PSPACE -complete

On Functionality of Visibly Pushdown Transducers

Visibly pushdown transducers form a subclass of pushdown transducers that (strictly) extends finite state transducers with a stack. Like visibly pushdown automata, the input symbols determine the stack operations. In this paper, we prove that functionality is decidable in PSpace for visibly pushdown transducers. The proof is done via a pumping argument: if a word with two outputs has a sufficiently large nesting depth, there exists a nested word with two outputs whose nesting depth is strictly smaller. The proof uses technics of word combinatorics. As a consequence of decidability of functionality, we also show that equivalence of functional visibly pushdown transducers is Exptime-Complete.Comment: 20 page

A Generalised Twinning Property for Minimisation of Cost Register Automata

Weighted automata (WA) extend finite-state automata by associating with transitions weights from a semiring S, defining functions from words to S. Recently, cost register automata (CRA) have been introduced as an alternative model to describe any function realised by a WA by means of a deterministic machine. Unambiguous WA over a monoid (M, ⊗) can equivalently be described by cost register automata whose registers take their values in M, and are updated by operations of the form x: = y ⊗ c, with c ∈ M. This class is denoted by CRA⊗c(M). We introduce a twinning property and a bounded variation property parametrised by an integer k, such that the corresponding notions introduced originally by Choffrut for finite-state transducers are obtained for k = 1. Given an unambiguous weighted automaton W over an infinitary group (G, ⊗) realizing some function f, we prove that the three following properties are equivalent: i) W satisfies the twinning property of order k, ii) f satisfies the k-bounded variation property, and iii) f can be described by a CRA⊗c(G) with at most k registers. In the spirit of tranducers, we actually prove this result in a more general setting by considering machines over the semiring of finite sets of elements from (G, ⊗): the three properties are still equivalent for such finite-valued weighted automata, that is the ones associating with words subsets of G of cardinality at most ℓ, for some natural ℓ. Moreover, we show that if the operation ⊗ of G is commutative and computable, then one can decide whether a WA satisfies the twinning property of order k. As a corollary, this allows to decide the register minimisation problem for the class CRA⊗c(G). Last, we prove that a similar result holds for finite-valued finite-state transducers, and that the register minimisation problem for the class CRA.c (B*) is Pspace-complete

The descriptive complexity approach to LOGCFL

Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers. Our work extends the elaborate theory relating monoidal quantifiers to NC1 and its subclasses. In the absence of the BIT predicate, we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's ``hardest context-free language'' is LOGCFL-complete under quantifier-free BIT-free projections. We then prove that FO with unary groupoidal quantifiers is strictly more expressive with the BIT predicate than without. Considering a particular groupoidal quantifier, we prove that first-order logic with majority of pairs is strictly more expressive than first-order with majority of individuals. As a technical tool of independent interest, we define the notion of an aperiodic nondeterministic finite automaton and prove that FO translations are precisely the mappings computed by single-valued aperiodic nondeterministic finite transducers.Comment: 10 pages, 1 figur

Granular State Effects on Wave Propagation

Sound and pressure wave propagation in a granular material is of interest not only for its intrinsic and practical value, but also because it provides a non-intrusive means of probing the state of a granular material. By examining wave speeds and attenuation, insight can be gained into the nature of the contacts between the particles. In the present paper, wave speeds and attenuation rates are first examined for a static granular bed for a variety of system parameters including particle size, composition and the overburden of the material above the measuring transducers. Agitation of the bed is then introduced by shaking the material vertically. This causes the bed to transition from a static granular state to a vibrofluidized state. The dilation of the bed allows for relative particle motion and this has a significant effect on the measured wave speeds and attenuation. Further, the fluid-like characteristics of the agitated bed distort the forcechain framework through which the waves are thought to travel. The consequences of bed consolidation, a natural result of shaking, are also examined