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In contexts including [[complex manifold]]s and [[algebraic geometry]], a '''logarithmic''' [[differential form]] is a meromorphic differential form with [[pole (complex analysis)|poles]] of a certain kind.
A '''finite state transducer''' ('''FST''') is a [[finite state machine]] with two tapes: an input tape and an output tape. This contrasts with an ordinary [[finite state automaton]] (or [[finite state acceptor]]), which has a single tape.


Let ''X'' be a complex manifold, and ''D'' ⊂ ''X'' a [[divisor]] and ω a holomorphic ''p''-form on ''X''−''D''. If ω and ''d''ω have a pole of order at most one along ''D'', then ω is said to have a logarithmic pole along ''D''. ω is also known as a logarithmic ''p''-form. The logarithmic ''p''-forms make up a [[Sheaf (mathematics)|subsheaf]] of the meromorphic ''p''-forms on ''X'' with a pole along ''D'', denoted
==Overview==


:<math>\Omega^p_X(\log D).</math>
An automaton can be said to ''recognize'' a string if we view the content of its tape as input.  In other words, the automaton computes a function that maps strings into the set {0,1}.  Alternatively, we can say that an automaton ''generates'' strings, which means viewing its tape as an output tape.  On this view, the automaton generates a [[formal language]], which is a set of strings.  The two views of automata are equivalent: the function that the automaton computes is precisely the [[indicator function]] of the set of strings it generates.  The class of languages generated by finite automata is known as the class of [[regular language]]s.


In the theory of [[Riemann surfaces]], one encounters logarithmic one-forms which have the local expression
The two tapes of a transducer are typically viewed as an input tape and an output tape.  On this view, a transducer is said to ''transduce'' (i.e., translate) the contents of its input tape to its output tape, by accepting a string on its input tape and generating another string on its output tape.  It may do so [[nondeterministic]]ally and it may produce more than one output for each input string.  A transducer may also produce no output for a given input string, in which case it is said to ''reject'' the input.  In general, a transducer computes a [[relation (mathematics)|relation]] between two formal languages.  The class of relations computed by finite state transducers is known as the class of [[rational relation]]s.


:<math>\omega = \frac{df}{f} =\left(\frac{m}{z} + \frac{g'(z)}{g(z)}\right)dz</math>
Finite-state transducers are often used for [[phonology|phonological]] and [[morphology (linguistics)|morphological analysis]] in [[natural language processing]] research and applications. Pioneers in this field include [[Ronald Kaplan]], [[Lauri Karttunen]], [[Martin Kay]] and [[Kimmo Koskenniemi]].<ref>{{Harvnb|Koskenniemi|1983}}</ref>
A common way of using transducers is in a so-called "cascade", where transducers for various operations are combined into a single transducer by repeated application of the composition operator (defined below).


for some [[meromorphic function]] (resp. [[rational function]]) <math> f(z) = z^mg(z) </math>, where ''g'' is holomorphic and non-vanishing at 0,  and ''m'' is the order of ''f'' at ''0''.. That is, for some [[open covering]], there are local representations of this differential form as a [[logarithmic derivative]] (modified slightly with the [[exterior derivative]] ''d'' in place of the usual [[differential operator]] ''d/dz''). Observe that ω has only simple poles with integer residues. On higher-dimensional complex manifolds, the [[Poincaré residue]] is used to describe the distinctive behavior of logarithmic forms along poles.
==Formal construction==


==Holomorphic log complex==
Formally, a finite transducer ''T'' is a 6-tuple (''Q'', Σ, Γ, ''I'', ''F'', δ) such that:
By definition of <math>\Omega^p_X(\log D)</math> and the fact that exterior differentiation ''d'' satisfies ''d''<sup>2</sup> = 0, one has
:<math> d\Omega^p_X(\log D)(U)\subset \Omega^{p+1}_X(\log D)(U) </math>.
This implies that there is a complex of sheaves <math>( \Omega^{\bullet}_X(\log D), d) </math>, known as the ''holomorphic log complex'' corresponding to the divisor ''D''. This is a subcomplex  of <math> j_*\Omega^{\bullet}_{X-D} </math>, where <math> j:X-D\rightarrow X </math> is the inclusion and <math> \Omega^{\bullet}_{X-D} </math> is the complex of sheaves of holomorphic forms on ''X''−''D''.


Of special interest is the case where ''D'' has simple [[normal crossings]]. Then if <math> \{D_{\nu}\} </math> are the smooth, irreducible components of ''D'', one has <math> D = \sum D_{\nu} </math> with the <math> D_{\nu} </math> meeting transversely. Locally ''D'' is the union of hyperplanes, with local defining equations of the form <math> z_1\cdots z_k = 0 </math> in some holomorphic coordinates. One can show that the stalk of <math> \Omega^1_X(\log D) </math> at ''p'' satisfies<ref name="foo">Chris A.M. Peters; Joseph H.M. Steenbrink (2007). Mixed Hodge Structures. Springer. ISBN 978-3-540-77015-6 {{Please check ISBN|reason=Check digit (6) does not correspond to calculated figure.}}</ref>
* ''Q'' is a [[finite set]], the set of ''states'';
:<math>\Omega_X^1(\log D)_p = \mathcal{O}_{X,p}\frac{dz_1}{z_1}\oplus\cdots\oplus\mathcal{O}_{X,p}\frac{dz_k}{z_k} \oplus \mathcal{O}_{X,p}dz_{k+1} \oplus \cdots \oplus \mathcal{O}_{X,p}dz_n</math>
* Σ is a finite set, called the ''input alphabet'';
and that
* Γ is a finite set, called the ''output alphabet'';
:<math> \Omega_X^k(\log D)_p = \bigwedge^k_{j=1} \Omega_X^1(\log D)_p </math>.
* ''I'' is a [[subset]] of ''Q'', the set of ''initial states'';
Some authors, e.g.,<ref name = "foo2">Phillip A. Griffiths; Joseph Harris (1979). Principles of Algebraic Geometry. Wiley-Interscience. ISBN 0-471-05059-8.</ref> use the term ''log complex'' to refer to the holomorphic log complex corresponding to a divisor with normal crossings.
* ''F'' is a subset of ''Q'', the set of ''final states''; and
* <math>\delta \subseteq Q \times (\Sigma\cup\{\epsilon\}) \times (\Gamma\cup\{\epsilon\}) \times Q</math> (where ε is the [[empty string]]) is the ''transition relation''.


===Higher-dimensional example===
We can view (''Q'', δ) as a labeled [[directed graph]], known as the ''transition graph'' of ''T'': the set of vertices is ''Q'', and  <math>(q,a,b,r)\in\delta</math> means that there is a labeled edge going from vertex ''q'' to vertex ''r''.  We also say that ''a'' is the ''input label'' and ''b'' the ''output label'' of that edge.
Consider a once-punctured elliptic curve, given as the locus ''D'' of complex points (''x'',''y'') satisfying <math> g(x,y) = y^2 - f(x) = 0 </math>, where <math>f(x) = x(x-1)(x-\lambda) </math> and <math> \lambda\neq 0,1 </math> is a complex number. Then ''D'' is a smooth irreducible [[hypersurface]] in '''C'''<sup>2</sup> and, in particular, a divisor with simple normal crossings. There is a meromorphic two-form on '''C'''<sup>2</sup>
:<math> \omega =\frac{dx\wedge dy}{g(x,y)} </math>
which has a simple pole along ''D''. The Poincaré residue <ref name = "foo2"/> of ω along ''D'' is given by the holomorphic one-form
:<math> \text{Res}_D(\omega) = \frac{dy}{\partial g/\partial x}|_D =-\frac{dx}{\partial g/\partial y}|_D = -\frac{1}{2}\frac{dx}{y}|_D </math>.
Vital to the residue theory of logarithmic forms is the [[Gysin sequence]], which is in some sense a generalization of the [[Residue Theorem]] for compact Riemann surfaces. This can be used to show, for example, that <math>dx/y|_D </math> extends to a holomorphic one-form on the [[Projective space#Projective space and affine space|projective closure]] of ''D'' in '''P'''<sup>2</sup>, a smooth elliptic curve.


=== Hodge theory ===
NOTE: This definition of finite transducer is also called ''letter transducer'' (Roche and Schabes 1997); alternative definitions are possible, but can all be converted into transducers following this one.
The holomorphic log complex can be brought to bear on the [[Hodge theory]] of complex algebraic varieties. Let ''X'' be a complex algebraic manifold and <math> j: X\hookrightarrow Y </math> a good compactification. This means that ''Y'' is a compact algebraic manifold and ''D'' = ''Y''−''X'' is a divisor on ''Y'' with simple normal crossings. The natural inclusion of complexes of sheaves
:<math> \Omega^{\bullet}_Y(\log D)\rightarrow j_*\Omega_{X}^{\bullet} </math>
turns out to be a quasi-isomorphism.  Thus
:<math> H^k(X;\mathbf{C}) = \mathbb{H}^k(Y, \Omega^{\bullet}_Y(\log D))</math>
where <math>\mathbb{H}^{\bullet}</math> denotes [[hypercohomology]] of a complex of abelian sheaves. There is<ref name="foo"/> a decreasing filtration <math>W_{\bullet} \Omega^p_Y(\log D) </math> given by
:<math>W_{m}\Omega^p_Y(\log D) =  \begin{cases}
0 & m < 0\\
\Omega^p_Y(\log D) & m\geq p \\
\Omega^{p-m}_Y\wedge \Omega^m_Y(\log D) & 0\leq m \leq p
\end{cases} </math>
which, along with the trivial  increasing filtration <math>F^{\bullet}\Omega^p_Y(\log D) </math> on logarithmic ''p''-forms, produces filtrations on cohomology
:<math> W_mH^k(X; \mathbf{C}) = \text{Im}(\mathbb{H}^k(Y, W_{m-k}\Omega^{\bullet}_Y(\log D))\rightarrow H^k(X; \mathbf{C})) </math>
:<math> F^pH^k(X; \mathbf{C}) = \text{Im}(\mathbb{H}^k(Y, F^p\Omega^{\bullet}_Y(\log D))\rightarrow H^k(X; \mathbf{C})) </math>.
One shows<ref name="foo"/> that <math> W_mH^k(X; \mathbf{C}) </math> can actually be defined over '''Q'''. Then the filtrations <math> W_{\bullet}, F^{\bullet}</math> on cohomology give rise to a mixed Hodge structure on <math> H^k(X; \mathbf{Z}) </math>.


Classically, for example in [[elliptic function]] theory, the logarithmic differential forms were recognised as complementary to the [[differentials of the first kind]]. They were sometimes called ''differentials of the second kind'' (and, with an unfortunate inconsistency, also sometimes ''of the third kind''). The classical theory has now been subsumed as an aspect of Hodge theory. For a Riemann surface ''S'', for example, the differentials of the first kind account for the term ''H''<sup>1,0</sup> in ''H''<sup>1</sup>(''S''), when by the [[Dolbeault isomorphism]] it is interpreted as the [[sheaf cohomology]] group ''H''<sup>0</sup>(''S'',Ω); this is tautologous considering their definition. The ''H''<sup>1,0</sup> direct summand in ''H''<sup>1</sup>(''S''), as well as being interpreted as ''H''<sup>1</sup>(''S'',O) where O is the sheaf of [[holomorphic function]]s on ''S'', can be identified more concretely with a vector space of logarithmic differentials.
Define the ''extended transition relation'' <math>\delta^*</math> as the smallest set such that:
 
* <math>\delta\subseteq\delta^*</math>;
* <math>(q,\epsilon,\epsilon,q)\in\delta^*</math> for all <math>q\in Q</math>; and
* whenever <math>(q,x,y,r) \in \delta^*</math> and <math>(r,a,b,s) \in \delta</math> then <math>(q,xa,yb,s) \in \delta^*</math>.
 
The extended transition relation is essentially the reflexive [[transitive closure]] of the transition graph that has been augmented to take edge labels into account.  The elements of <math>\delta^*</math> are known as ''paths''.  The edge labels of a path are obtained by concatenating the edge labels of its constituent transitions in order.
 
The ''behavior'' of the transducer ''T'' is the rational relation [''T''] defined as follows: <math>x[T]y</math> [[if and only if]] there exists <math>i \in I</math> and <math>f \in F</math> such that <math>(i,x,y,f) \in \delta^*</math>. This is to say that ''T'' transduces a string <math>x\in\Sigma^*</math> into a string <math>y\in\Gamma^*</math> if there exists a path from an initial state to a final state whose input label is ''x'' and whose output label is ''y''.
 
Finite State Transducers can be weighted, where each transition is labeled with a weight in addition to the input and output labels. This property makes FSTs very useful for machine learning applications. A Weighted Finite State Transducer (WFST) over a [[semiring]] ''K'' can be defined similarly to an unweighted one as an 8-tuple ''T''=(''Q'', Σ, Γ, ''I'', ''F'', E, λ, ρ), where:
* ''Q'', Σ, Γ, ''I'', ''F'' are defined as above;
* <math> E \subseteq Q \times (\Sigma\cup\{\epsilon\}) \times (\Gamma\cup\{\epsilon\}) \times Q \times K</math> (where ε is the [[empty string]]) is the finite set of transitions;
* <math>\lambda:  I \rightarrow K </math> maps initial states to weights;
* <math>\rho: F \rightarrow K </math> maps final states to weights.
In order to make certain operations on WFSTs well-defined, the weights have to form a [[Semiring]]. Two typical semirings used in practice are Log and Tropical Semirings.
 
==Operations on finite state transducers==
 
The following operations defined on finite automata also apply to finite transducers:
 
* [[Union (set theory)|Union]].  Given transducers ''T'' and ''S'', there exists a transducer <math>T\cup S</math> such that <math>x[T\cup S]y</math> if and only if <math>x[T]y</math> or <math>x[S]y</math>.
 
* Concatenation.  Given transducers ''T'' and ''S'', there exists a transducer <math>T\cdot S</math> such that <math>wx[T\cdot S]yz</math> if and only if <math>w[T]y</math> and <math>x[S]z</math>.
 
* [[Kleene closure]].  Given a transducer ''T'', there exists a transducer <math>T^*</math> with the following properties: (1) <math>\epsilon[T^*]\epsilon</math>; (2) if <math>w[T^*]y</math> and <math>x[T]z</math> then <math>wx[T^*]yz</math>; and <math>x[T^*]y</math> does not hold unless mandated by (1) or (2).
 
* [[Intersection (set theory)|Intersection]]. Given transducers ''T'', ''S'', the intersection of these transducers ''H'' has the property that (1) x[''H'']y if and only if  x[''T'']y and x[''S'']y.
 
* [[Composition of relations|Composition]]. Given a transducer ''T'' on alphabets Σ and Γ and a transducer ''S'' on alphabets Γ and Δ, there exists a transducer <math>T \circ S</math> on Σ and Δ such that <math>x[T \circ S]z</math> if and only if there exists a string <math>y\in\Gamma^*</math> such that <math>x[T]y</math> and <math>y[S]z</math>. This operation extends to the weighted case.<ref>{{harvnb|Mohri|2004|pp=3–5}}</ref>
 
:This definition uses the same notation which is used in mathematics for [[Composition of relations|relation composition]]. However, the conventional reading for relation composition is the other way around: given two relations <math>T</math> and <math>S</math>, <math>(x,z)\in T\circ S</math> when there exist some <math>y</math> such that <math>(x,y)\in S</math> and <math>(y,z)\in T</math>.
 
* [[Projection (mathematics)|Projection]] to an automaton.  There are two projection functions: <math>\pi_1</math> preserves the input tape, and <math>\pi_2</math> preserves the output tape.  The first projection, <math>\pi_1</math> is defined as follows:
 
:* Given a transducer ''T'', there exists a finite automaton <math>\pi_1 T</math> such that <math>\pi_1 T</math> accepts ''x'' if and only if there exists a string ''y'' for which <math>x[T]y</math>.
 
:The second projection, <math>\pi_2</math> is defined similarly.
 
* [[Determinization]]. Given a transducer ''T'', we want to build an equivalent transducer which has a unique initial state and such that no two transitions leaving any state share the same input label. The [[powerset construction]] can be extended to transducers, or even weighted transducers, but sometimes fails to halt; indeed, some non-deterministic transducers do not admit equivalent deterministic transducers.<ref>[http://www.let.rug.nl/~vannoord/papers/preds/node22.html]</ref> [[Characterization (mathematics)|Characterizations]] of determinizable transducers have been proposed<ref>{{harvnb|Mohri|2004|pp=5–6}}</ref> along with efficient algorithms to test them<ref>{{harvnb|Allauzen|2003}}</ref>: they rely on the [[semiring]] used in the weighted case as well as a general property on the structure of the transducer (the [[twins property]]).
 
* Weight pushing for the weighted case.<ref>{{harvnb|Mohri|2004|pp=7–9}}</ref>
 
* Minimization for the weighted case.<ref>{{harvnb|Mohri|2004|pp=9–11}}</ref>
 
* Removal of [[epsilon-transitions]].
 
==Additional properties of finite state transducers==
 
* It is [[decidable]] whether the relation [''T''] of a transducer ''T'' is empty.
 
* It is decidable whether there exists a string ''y'' such that ''x''[''T'']''y'' for a given string ''x''.
 
* It is [[undecidable]] whether two transducers are equivalent.
 
* If one defines the alphabet of labels <math>L=(\Sigma\cup\{\epsilon\}) \times (\Gamma\cup\{\epsilon\})</math>, finite state transducers are isomorphic to [[nondeterministic finite automata|NDFA]] over the alphabet <math>L</math>, and may therefore be determinized (turned into [[deterministic finite automaton|deterministic finite automata]] over the alphabet <math>L=[(\Sigma\cup\{\epsilon\}) \times \Gamma] \cup [\Sigma \times (\Gamma\cup\{\epsilon\})]</math> ) and subsequently minimized so that they have the minimum number of states.{{Citation needed|date=September 2008}}
 
==Applications==
 
* Context-sensitive rewriting rules of the form ''a → b / c _ d'', used in [[linguistics]] to model [[phonological rule]]s and [[sound change]], are computationally equivalent to finite-state transducers, provided that application is nonrecursive, i.e. the rule is not allowed to rewrite the same substring twice. <ref>{{cite web | url=http://acl.ldc.upenn.edu/J/J94/J94-3001.pdf | title=Regular Models of Phonological Rule Systems | accessdate=August 25, 2012}}</ref>


==See also==
==See also==
*[[Algebraic Geometry]]
*[[Adjunction formula]]
*[[Differential of the first kind]]
*[[Residue Theorem]]


==References==
=== Internal links ===
{{Reflist}}
* [[Mealy machine]]
* [[Moore machine]]
* [[Morphological dictionary]]
* [[foma (software)]]
 
=== External links ===
* [http://openfst.org/ OpenFst], an open-source library for FST operations.
* [http://www.ims.uni-stuttgart.de/tcl/SOFTWARE/SFST.html Stuttgart Finite State Transducer Tools], another open-source FST toolkit
* [http://jsalatas.ictpro.gr/java-fst-framework-api-review/ java FST Framework], an open-source java FST Framework capable of handling OpenFst text format.
 
==Notes==
{{reflist|3}}
 
== References ==
 
<references/>
<div class="references-small">
*{{cite journal
  | last1 = Allauzen
  | first1 = Cyril
  | last2 = Mohri
  | first2 = Mehryar
  | title = Efficient Algorithms for Testing the Twins Property
  | journal = Journal of Automata, Languages and Combinatorics
  | year = 2003
  | volume = 8
  | issue = 2
  | pages = 117–144
  | url = http://www.cs.nyu.edu/~mohri/pub/twins.pdf
}}
*{{citation
  | last = Koskenniemi
  | first = Kimmo
  | authorlink = Kimmo Koskenniemi
  | title = Two-level morphology: A general computational model of word-form recognition and production
  | publisher = Department of General Linguistics, [[University of Helsinki]]
  | year = 1983
  | url = http://www.ling.helsinki.fi/~koskenni/doc/Two-LevelMorphology.pdf
}}
*{{cite journal
  | last = Mohri
  | first = Mehryar
  | title = Weighted Finite-State Transducer Algorithms: An Overview
  | journal = Formal Languages and Applications
  | year = 2004
  | volume = 148
  | issue = 620
  | pages = 551–564
  | url = http://www.cs.nyu.edu/~mohri/pub/fla.pdf
}}
</div>
 
==Further reading==
 
* {{cite book |last= Jurafsky|first= Daniel |coauthors= James H. Martin|authorlink=Daniel Jurafsky|title= [[Speech and Language Processing]] |publisher= [[Prentice Hall]] |year= 2000 |isbn= 0-13-095069-6 |pages=71–83}}
* {{cite book |last= Kornai|first= Andras|authorlink=Andras Kornai|title=[[Extended Finite State Models of Language]] |publisher= [[Cambridge University Press]] |year=1999 |isbn= 0-521-63198-X}}
* {{cite book |last= Roche|first= Emmanuel |coauthors= Yves Schabes|authorlink=Emmanuel Roche|title= [[Finite-state language processing]] |publisher= [[MIT Press]] |year= 1997 |isbn= 0-262-18182-7|pages=1–65}}
* {{cite book |last= Beesley|first= Kenneth R. |coauthors= Lauri Karttunen|authorlink=Kenneth R. Beesley|title= [[Finite State Morphology]] |publisher= [[Center for the Study of Language and Information]] |year= 2003 |isbn= 1-57586-434-7}}
* {{cite book |last= Roark|first= Brian |coauthors= Richard Sproat|authorlink=Brian Roark|title= [[Computational Approaches to Morphology and Syntax]] |publisher= [[Oxford University Press]] |year= 2007 |isbn= 0-19-927478-9}}
* {{cite book |last= Berstel|first= Jean |title= [[Transductions and Context-free Languages]] |publisher= [[Teubner Verlag]] |year= 1979 }}. [http://www-igm.univ-mlv.fr/~berstel/LivreTransductions/LivreTransductions.html Free PDF version]
 
{{DEFAULTSORT:Finite State Transducer}}
[[Category:Models of computation]]
[[Category:Formal languages]]
[[Category:Automata theory]]


[[Category:Complex analysis]]
[[bs:Konačni transduktor]]
[[Category:Algebraic geometry]]
[[ca:Transductor d'estats finits]]
[[de:Transduktor (Informatik)]]
[[es:Transductor de estados finitos]]
[[fr:Transducteur fini]]
[[hr:Konačni pretvornik]]
[[sr:Коначни трансдуктор]]

Revision as of 05:34, 13 August 2014

A finite state transducer (FST) is a finite state machine with two tapes: an input tape and an output tape. This contrasts with an ordinary finite state automaton (or finite state acceptor), which has a single tape.

Overview

An automaton can be said to recognize a string if we view the content of its tape as input. In other words, the automaton computes a function that maps strings into the set {0,1}. Alternatively, we can say that an automaton generates strings, which means viewing its tape as an output tape. On this view, the automaton generates a formal language, which is a set of strings. The two views of automata are equivalent: the function that the automaton computes is precisely the indicator function of the set of strings it generates. The class of languages generated by finite automata is known as the class of regular languages.

The two tapes of a transducer are typically viewed as an input tape and an output tape. On this view, a transducer is said to transduce (i.e., translate) the contents of its input tape to its output tape, by accepting a string on its input tape and generating another string on its output tape. It may do so nondeterministically and it may produce more than one output for each input string. A transducer may also produce no output for a given input string, in which case it is said to reject the input. In general, a transducer computes a relation between two formal languages. The class of relations computed by finite state transducers is known as the class of rational relations.

Finite-state transducers are often used for phonological and morphological analysis in natural language processing research and applications. Pioneers in this field include Ronald Kaplan, Lauri Karttunen, Martin Kay and Kimmo Koskenniemi.[1] A common way of using transducers is in a so-called "cascade", where transducers for various operations are combined into a single transducer by repeated application of the composition operator (defined below).

Formal construction

Formally, a finite transducer T is a 6-tuple (Q, Σ, Γ, I, F, δ) such that:

We can view (Q, δ) as a labeled directed graph, known as the transition graph of T: the set of vertices is Q, and means that there is a labeled edge going from vertex q to vertex r. We also say that a is the input label and b the output label of that edge.

NOTE: This definition of finite transducer is also called letter transducer (Roche and Schabes 1997); alternative definitions are possible, but can all be converted into transducers following this one.

Define the extended transition relation as the smallest set such that:

The extended transition relation is essentially the reflexive transitive closure of the transition graph that has been augmented to take edge labels into account. The elements of are known as paths. The edge labels of a path are obtained by concatenating the edge labels of its constituent transitions in order.

The behavior of the transducer T is the rational relation [T] defined as follows: if and only if there exists and such that . This is to say that T transduces a string into a string if there exists a path from an initial state to a final state whose input label is x and whose output label is y.

Finite State Transducers can be weighted, where each transition is labeled with a weight in addition to the input and output labels. This property makes FSTs very useful for machine learning applications. A Weighted Finite State Transducer (WFST) over a semiring K can be defined similarly to an unweighted one as an 8-tuple T=(Q, Σ, Γ, I, F, E, λ, ρ), where:

In order to make certain operations on WFSTs well-defined, the weights have to form a Semiring. Two typical semirings used in practice are Log and Tropical Semirings.

Operations on finite state transducers

The following operations defined on finite automata also apply to finite transducers:

  • Intersection. Given transducers T, S, the intersection of these transducers H has the property that (1) x[H]y if and only if x[T]y and x[S]y.
This definition uses the same notation which is used in mathematics for relation composition. However, the conventional reading for relation composition is the other way around: given two relations and , when there exist some such that and .
The second projection, is defined similarly.
  • Determinization. Given a transducer T, we want to build an equivalent transducer which has a unique initial state and such that no two transitions leaving any state share the same input label. The powerset construction can be extended to transducers, or even weighted transducers, but sometimes fails to halt; indeed, some non-deterministic transducers do not admit equivalent deterministic transducers.[3] Characterizations of determinizable transducers have been proposed[4] along with efficient algorithms to test them[5]: they rely on the semiring used in the weighted case as well as a general property on the structure of the transducer (the twins property).
  • Weight pushing for the weighted case.[6]
  • Minimization for the weighted case.[7]

Additional properties of finite state transducers

  • It is decidable whether the relation [T] of a transducer T is empty.
  • It is decidable whether there exists a string y such that x[T]y for a given string x.
  • It is undecidable whether two transducers are equivalent.

Applications

  • Context-sensitive rewriting rules of the form a → b / c _ d, used in linguistics to model phonological rules and sound change, are computationally equivalent to finite-state transducers, provided that application is nonrecursive, i.e. the rule is not allowed to rewrite the same substring twice. [8]

See also

Internal links

External links

Notes

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References

  1. ↑ Template:Harvnb
  2. ↑ Template:Harvnb
  3. ↑ [1]
  4. ↑ Template:Harvnb
  5. ↑ Template:Harvnb
  6. ↑ Template:Harvnb
  7. ↑ Template:Harvnb
  8. ↑ Template:Cite web
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

Further reading

  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534. Free PDF version

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