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

Size-Change Abstraction and Max-Plus Automata

Max-plus automata (over ℕ ∪ − ∞) are finite devices that map input words to non-negative integers or − ∞. In this paper we present (a) an algorithm allowing to compute the asymptotic behaviour of max-plus automata, and (b) an application of this technique to the evaluation of the computational time complexity of programs

The Reversed q-Exponential Functional Relation

After obtaining some useful identities, we prove an additional functional relation for $q$ exponentials with reversed order of multiplication, as well as the well known direct one in a completely rigorous manner.Comment: 6 pages, LaTeX, no figure

Linguistics

Contains reports on three research projects.National Science Foundatio

Mathematics

Contains reports on three research projects

Revisiting the Equivalence Problem for Finite Multitape Automata

The decidability of determining equivalence of deterministic multitape automata (or transducers) was a longstanding open problem until it was resolved by Harju and Karhum\"{a}ki in the early 1990s. Their proof of decidability yields a co_NP upper bound, but apparently not much more is known about the complexity of the problem. In this paper we give an alternative proof of decidability, which follows the basic strategy of Harju and Karhumaki but replaces their use of group theory with results on matrix algebras. From our proof we obtain a simple randomised algorithm for deciding language equivalence of deterministic multitape automata and, more generally, multiplicity equivalence of nondeterministic multitape automata. The algorithm involves only matrix exponentiation and runs in polynomial time for each fixed number of tapes. If the two input automata are inequivalent then the algorithm outputs a word on which they differ

h analogue of Newton's binomial formula

In this letter, the $h$--analogue of Newton's binomial formula is obtained in the $h$--deformed quantum plane which does not have any $q$--analogue. For $h=0$, this is just the usual one as it should be. Furthermore, the binomial coefficients reduce to $\frac{n!}{(n-k)!}$ for $h=1$. \\ Some properties of the $h$--binomial coefficients are also given. \\ Finally, I hope that such results will contribute to an introduction of the $h$--analogue of the well--known functions, $h$--special functions and $h$--deformed analysis.Comment: 6 pages, latex Jounal-ref: J. Phys. A: Math. Gen. 31 (1998) L75

From algebra to logic: there and back again -- the story of a hierarchy

This is an extended survey of the results concerning a hierarchy of languages that is tightly connected with the quantifier alternation hierarchy within the two-variable fragment of first order logic of the linear order.Comment: Developments in Language Theory 2014, Ekaterinburg : Russian Federation (2014

Stability and Complexity of Minimising Probabilistic Automata

We consider the state-minimisation problem for weighted and probabilistic automata. We provide a numerically stable polynomial-time minimisation algorithm for weighted automata, with guaranteed bounds on the numerical error when run with floating-point arithmetic. Our algorithm can also be used for "lossy" minimisation with bounded error. We show an application in image compression. In the second part of the paper we study the complexity of the minimisation problem for probabilistic automata. We prove that the problem is NP-hard and in PSPACE, improving a recent EXPTIME-result.Comment: This is the full version of an ICALP'14 pape