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31
Duality for Labelled Markov Processes
"... Labelled Markov processes (LMPs) are automata whose transitions are given by probability distributions. In this paper we present a `universal' LMP as the spectrum of a commutative C # algebra consisting of formal linear combinations of labelled trees. We characterize the state space of the univ ..."
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Labelled Markov processes (LMPs) are automata whose transitions are given by probability distributions. In this paper we present a `universal' LMP as the spectrum of a commutative C # algebra consisting of formal linear combinations of labelled trees. We characterize the state space of the universal LMP as the set of homomorphims from an ordered commutative monoid of labelled trees into the multiplicative unit interval. This yields a simple semantics for LMPs which is fully abstract with respect to probabilistic bisimilarity. We also consider LMPs with entry points and exit points in the setting of iteration theories. We define an iteration theory of LMPs by specifying its categorical dual: a certain category of C*algebras. We find that the basic operations for composing LMPs have simple definitions in the dual category.
Compositions of Tree Series Transformations
, 2005
"... Tree series transformations computed by bottomup and topdown tree series transducers are called bottomup and topdown tree series transformations, respectively. (Functional) compositions of such transformations are investigated. It turns out that the class of bottomup tree series transformations ..."
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Cited by 8 (6 self)
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Tree series transformations computed by bottomup and topdown tree series transducers are called bottomup and topdown tree series transformations, respectively. (Functional) compositions of such transformations are investigated. It turns out that the class of bottomup tree series transformations over a commutative and complete semiring is closed under leftcomposition with linear bottomup tree series transformations and rightcomposition with boolean deterministic bottomup tree series transformations. Moreover, it is shown that the class of topdown tree series transformations over a commutative and complete semiring is closed under rightcomposition with linear, nondeleting topdown tree series transformations. Finally, the composition of a boolean, deterministic, total topdown tree series transformation with a linear topdown tree series transformation is shown to be a topdown tree series transformation.
Parsing algorithms based on tree automata
 IN PROC. IWPT
, 2009
"... We investigate several algorithms related to the parsing problem for weighted automata, under the assumption that the input is a string rather than a tree. This assumption is motivated by several natural language processing applications. We provide algorithms for the computation of parseforests, be ..."
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Cited by 8 (5 self)
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We investigate several algorithms related to the parsing problem for weighted automata, under the assumption that the input is a string rather than a tree. This assumption is motivated by several natural language processing applications. We provide algorithms for the computation of parseforests, best tree probability, inside probability (called partition function), and prefix probability. Our algorithms are obtained by extending to weighted tree automata the BarHillel technique, as defined for contextfree grammars.
Efficient Inference Through Cascades of Weighted Tree Transducers
, 2010
"... Weighted tree transducers have been proposed as useful formal models for representing syntactic natural language processing applications, but there has been little description of inference algorithms for these automata beyond formal foundations. We give a detailed description of algorithms for appli ..."
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Cited by 8 (1 self)
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Weighted tree transducers have been proposed as useful formal models for representing syntactic natural language processing applications, but there has been little description of inference algorithms for these automata beyond formal foundations. We give a detailed description of algorithms for application of cascades of weighted tree transducers to weighted tree acceptors, connecting formal theory with actual practice. Additionally, we present novel onthefly variants of these algorithms, and compare their performance on a syntax machine translation cascade based on (Yamada and Knight, 2001).
Behavioural Differential Equations and Coinduction for Binary Trees
"... Abstract. We study the set TA of infinite binary trees with nodes labelledinasemiringA from a coalgebraic perspective. We present coinductive definition and proof principles based on the fact that TA carries a final coalgebra structure. By viewing trees as formal power series, we develop a calculus ..."
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Cited by 5 (1 self)
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Abstract. We study the set TA of infinite binary trees with nodes labelledinasemiringA from a coalgebraic perspective. We present coinductive definition and proof principles based on the fact that TA carries a final coalgebra structure. By viewing trees as formal power series, we develop a calculus where definitions are presented as behavioural differential equations. We present a general format for these equations that guarantees the existence and uniqueness of solutions. Although technically not very difficult, the resulting framework has surprisingly nice applications, which is illustrated by various concrete examples. 1
Learning Rational Stochastic Tree Languages
"... Abstract. We consider the problem of learning stochastic tree languages, i.e. probability distributions over a set of trees T(F), from a sample of trees independently drawn according to an unknown target P. We consider the case where the target is a rational stochastic tree language, i.e. it can be ..."
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Cited by 5 (2 self)
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Abstract. We consider the problem of learning stochastic tree languages, i.e. probability distributions over a set of trees T(F), from a sample of trees independently drawn according to an unknown target P. We consider the case where the target is a rational stochastic tree language, i.e. it can be computed by a rational tree series or, equivalently, by a multiplicity tree automaton. In this paper, we provide two contributions. First, we show that rational tree series admit a canonical representation with parameters that can be efficiently estimated from samples. Then, we give an inference algorithm that identifies the class of rational stochastic tree languages in the limit with probability one. 1
Does osubstitution preserve recognizability?
 IN PROC. 11TH INT. CONF. IMPLEM. AND APPL. OF AUTOMATA
, 2006
"... Substitution operations on tree series are at the basis of systems of equations (over tree series) and tree series transducers. Tree series transducers seem to be an interesting transformation device in syntactic pattern matching. In this contribution, it is shown that osubstitution preserves reco ..."
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Cited by 4 (3 self)
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Substitution operations on tree series are at the basis of systems of equations (over tree series) and tree series transducers. Tree series transducers seem to be an interesting transformation device in syntactic pattern matching. In this contribution, it is shown that osubstitution preserves recognizable tree series provided that the target tree series is linear and the semiring is idempotent, commutative, and continuous. This result is applied to prove that the range of the otts transformation computed by a linear recognizable tree series transducer is pointwise recognizable.
Learning multiplicity tree automata
 In: Proceedings of the 8th International Colloquium on Grammatical Inference (ICGI’06). Volume 4201 of LNCS
, 2006
"... Abstract. In this paper, we present a theoretical approach for the problem of learning multiplicity tree automata. These automata allows one to define functions which compute a number for each tree. They can be seen as a strict generalization of stochastic tree automata since they allow to define fu ..."
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Cited by 4 (1 self)
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Abstract. In this paper, we present a theoretical approach for the problem of learning multiplicity tree automata. These automata allows one to define functions which compute a number for each tree. They can be seen as a strict generalization of stochastic tree automata since they allow to define functions over any field K. A multiplicity automaton admits a support which is a non deterministic automaton. From a grammatical inference point of view, this paper presents a contribution which is original due to the combination of two important aspects. This is the first time, as far as we now, that a learning method focuses on non deterministic tree automata which computes functions over a field. The algorithm proposed in this paper stands in Angluin’s exact model where a learner is allowed to use membership and equivalence queries. We show that this algorithm is polynomial in time in function of the size of the representation.
The power of tree series transducers of type I and II
 PROC. 9TH INT. CONF. DEVELOPMENTS IN LANGUAGE THEORY, VOLUME 3572 OF LNCS
, 2005
"... The power of tree series transducers of type I and II is studied for IO as well as OI tree series substitution. More precisely, it is shown that the IO tree series transformations of type I (respectively, type II) are characterized by the composition of homomorphism topdown IO tree series transfor ..."
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Cited by 4 (4 self)
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The power of tree series transducers of type I and II is studied for IO as well as OI tree series substitution. More precisely, it is shown that the IO tree series transformations of type I (respectively, type II) are characterized by the composition of homomorphism topdown IO tree series transformations with bottomup (respectively, linear bottomup) IO tree series transformations. On the other hand, polynomial OI tree series transducers of type I and II and topdown OI tree series transducers are equally powerful.
Bisimulation Minimisation of Weighted Automata on Unranked Trees
, 2008
"... Two examples of automatatheoretic models for the validation of xml documents against userde ned schema are the stepwise unranked tree automaton (suta) and the parallel unranked tree automaton (puta). By adding a weight, taken from some semiring, to every transition we generalise these two qualitat ..."
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Cited by 4 (1 self)
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Two examples of automatatheoretic models for the validation of xml documents against userde ned schema are the stepwise unranked tree automaton (suta) and the parallel unranked tree automaton (puta). By adding a weight, taken from some semiring, to every transition we generalise these two qualitative automata models to quantitative models, thereby obtaining weighted stepwise unranked tree automata (wsuta) and weighted parallel unranked tree automata (wputa); the qualitative automata models are reobtained by choosing the Boolean semiring. We deal with the minimisation problem of wsuta and wputa by using (forward and backward) bisimulations and we prove the following results: (1) for every wsuta an equivalent forward (resp. backward) bisimulation minimal wsuta can be computed in time O(mn) where n is the number of states and m is the number of transitions of the given wsuta; (2) the same result is proved for wputa instead of wsuta; (3) if the semiring is additive cancellative or the Boolean semiring, then the bound can be improved to O(m log n) for both wsuta and wputa; (4) for every deterministic puta we can compute a minimal equivalent deterministic puta in time O(m log n); (5) the automata models wsuta, wputa, and weighted unranked tree automaton have the same computational power.