171 research outputs found
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 Reversed q-Exponential Functional Relation
After obtaining some useful identities, we prove an additional functional
relation for 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
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
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
Noncommutative Schur polynomials and the crystal limit of the U_q sl(2)-vertex model
Starting from the Verma module of U_q sl(2) we consider the evaluation module
for affine U_q sl(2) and discuss its crystal limit (q=0). There exists an
associated integrable statistical mechanics model on a square lattice defined
in terms of vertex configurations. Its transfer matrix is the generating
function for noncommutative complete symmetric polynomials in the generators of
the affine plactic algebra, an extension of the finite plactic algebra first
discussed by Lascoux and Sch\"{u}tzenberger. The corresponding noncommutative
elementary symmetric polynomials were recently shown to be generated by the
transfer matrix of the so-called phase model discussed by Bogoliubov, Izergin
and Kitanine. Here we establish that both generating functions satisfy Baxter's
TQ-equation in the crystal limit by tying them to special U_q sl(2) solutions
of the Yang-Baxter equation. The TQ-equation amounts to the well-known
Jacobi-Trudy formula leading naturally to the definition of noncommutative
Schur polynomials. The latter can be employed to define a ring which has
applications in conformal field theory and enumerative geometry: it is
isomorphic to the fusion ring of the sl(n)_k -WZNW model whose structure
constants are the dimensions of spaces of generalized theta-functions over the
Riemann sphere with three punctures.Comment: 24 pages, 6 figures; v2: several typos fixe
The Word Problem for Omega-Terms over the Trotter-Weil Hierarchy
For two given -terms and , the word problem for
-terms over a variety asks whether
in all monoids in . We show that the
word problem for -terms over each level of the Trotter-Weil Hierarchy
is decidable. More precisely, for every fixed variety in the Trotter-Weil
Hierarchy, our approach yields an algorithm in nondeterministic logarithmic
space (NL). In addition, we provide deterministic polynomial time algorithms
which are more efficient than straightforward translations of the
NL-algorithms. As an application of our results, we show that separability by
the so-called corners of the Trotter-Weil Hierarchy is witnessed by
-terms (this property is also known as -reducibility). In
particular, the separation problem for the corners of the Trotter-Weil
Hierarchy is decidable
h analogue of Newton's binomial formula
In this letter, the --analogue of Newton's binomial formula is obtained in
the --deformed quantum plane which does not have any --analogue. For
, this is just the usual one as it should be. Furthermore, the binomial
coefficients reduce to for . \\ Some properties of the
--binomial coefficients are also given. \\ Finally, I hope that such results
will contribute to an introduction of the --analogue of the well--known
functions, --special functions and --deformed analysis.Comment: 6 pages, latex Jounal-ref: J. Phys. A: Math. Gen. 31 (1998) L75
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