Method and apparatus for processing natural language using auto-intersection

[0001] Priority is claimed from U.S. Provisional Application No. 60/481,639, filed Nov. 14, 2003, entitled “Method And Apparatus For Processing Natural Language Using Multi-Tape Automata”, by the same inventors and assignee, which is hereby incorporated herein by reference. In addition, cross-reference is made to the following U.S. patent application that is concurrently filed with this patent application, assigned to the same assignee as this patent application, incorporated in this patent application by reference, and claims priority to U.S. Provisional Patent Application Ser. No. 60/481,639, filed Nov. 14, 2003: U.S. patent application Ser. No. 10/849,919, entitled “Method And Apparatus For Processing Natural Language Using Tape-Intersection”.

[0002] The present invention relates to a method an apparatus for processing natural language using operations performed on weighted and non-weighted multi-tape automata.

[0003] Finite state automata (FSAs) are mathematically well defined and offer many practical advantages. They allow for fast processing of input data and are easily modifiable and combinable by well defined operations. Consequently, FSAs are widely used in Natural Language Processing (NLP) as well as many other fields. A general discussion of FSAs is described in Patent Application Publication U.S. 2003/0004705 A1 and in “Finite State Morphology” by Beesley and Karttunen (CSLI Publications, 2003), which are incorporated herein by reference.

[0004] Weighted finite state automata (WFSAs) combine the advantages of ordinary FSAs with the advantages of statistical models, such as Hidden Markov Models (HMMs), and hence have a potentially wider scope of application than FSAs. Weighted multi-tape automata (WMTAs) have yet more advantages. For example, WMTAs permit the separation of different types of information used in NLP (e.g., surface word form, lemma, POS-tag, domain-specific information) over different tapes, and preserve intermediate results of different steps of NLP on different tapes. Operations on WMTAs may be specified to operate on one, several, or all tapes.

[0005] While some basic WMTAs operations, such as union, concatenation, projection, and complementary projection, have been defined for a sub-class of non-weighted multi-tape automata (see for example the publication by Kaplan and Kay, “Regular models of phonological rule systems”, in Computational Linguistics, 20(3):331-378, 1994) and implemented (see for example the publication by Kiraz and Grimley-Evans, “Multi-tape automata for speech and language systems: A prolog implementation”, in D. Woods and S. Yu, editors, Automata Implementation, number 1436 in Lecture Notes in Computer Science, Springer Verlag, Berlin, Germany, 1998), there continues to be a need for improved, simplified, and more efficient operations for processing WMTAs to make use of these advantages in natural language processing.

[0006] In accordance with the invention, there is provided a method and apparatus for using weighted multi-tape automata (WMTAs) in natural language processing (NLP) that includes morphological analysis, part-of-speech (POS) tagging, disambiguation, and entity extraction. In performing NLP, operations are employed that perform cross-product, auto-intersection, and tape-intersection (i.e., single-tape intersection and multi-tape intersection) of automata. Such operations may be performed using transition-wise processing on weighted or non-weighted multi-tape automata.

[0007] In accordance with one aspect of the invention there is provided in a system for processing natural language, a method for intersecting a first tape and a second tape of a multi-tape automaton (MTA) that has a plurality of n tapes and a plurality of paths. The method includes generating a string tuple <s₁, . . . , s_n> having a string s for each of the n tapes of each path of the MTA and comparing the string s_j of the first tape with the string s_k of the second tape in the string tuple. If the strings s_j and s_k equal, the string tuple is retained in the MTA. However, if the strings s_j and s_k do not equal, the MTA is restructured to remove the string tuple.

[0008] In accordance with another aspect of the invention, there is provided in a system for processing natural language, a method for intersecting a first tape and a second tape of an input multi-tape automaton (MTA) that has a plurality of tapes and a plurality of paths. The method includes computing a first limit and a second limit of the input MTA. An output MTA is constructed that intersects the first tape and the second tape using the second limit to delimit its construction. Transitions are removed along paths in the output MTA during construction except for those transitions of paths having similar labels on the first selected tape and the second selected tape. The constructed output MTA is determined to be regular using the first limit. If the output MTA is determined to be regular, then the output MTA is provided as a complete solution to the intersection of the first tape and the second tape of the input MTA. If the output MTA is determined not to be regular, then the output MTA is provided as a partial solution to the intersection of the first tape and the second tape of the input MTA.

[0009] It will be appreciated that the present invention has the following advantages over weighted 1-tape or 2-tape processing of automata because it allows for: (a) the separation of different types of information used in NLP (e.g., surface form, lemma, POS-tag, domain-specific information, etc.) over different tapes; (b) the preservation of some or all intermediate results of various NLP steps on different tapes; and (c) the possibility of defining and implementing contextual replace rules referring to different types of information on different tapes.

[0032] Outline of Detailed Description

• • [0033] A. Definitions
    • [0034] A.1 Semirings
    • [0035] A.2 Weighted Automata
    • [0036] A.3 Weighted Multi-Tape Automata
  • [0037] B. Operations On Multi-Tape Automata
    • [0038] B.1 Pairing and Concatenation
    • [0039] B.2 Projection and Complementary Projection
    • [0040] B.3 Cross-Product
    • [0041] B.4 Auto-Intersection
    • [0042] B.5 Single-Tape Intersection
    • [0043] B.6 Multi-Tape Intersection
    • [0044] B.7 Transition Automata and Transition-Wise Processing
  • [0045] C. Methods For Performing MTA Operations
    • [0046] C.1 Cross-Product
      • [0047] C.1.1 Conditions
      • [0048] C.1.2 Path Concatenation Method
      • [0049] C.1.3 Path Alignment Method
      • [0050] C.1.4 Complexity
    • [0051] C.2 Auto-Intersection
      • [0052] C.2.1 Conditions Of First Method
      • [0053] C.2.2 First Method
      • [0054] C.2.3 Example Of First Method
      • [0055] C.2.4 Conditions Of Second Method
      • [0056] C.2.5 Second Method
        • [0057] C.2.5.A Compile Limits
        • [0058] C.2.5.B Construct Auto-Intersection
        • [0059] C.2.5.C Test Regularity
      • [0060] C.2.6 Examples Of Second Method
    • [0061] C.3 Single-Tape Intersection
      • [0062] C.3.1 Conditions
      • [0063] C.3.2 First Embodiment
      • [0064] C.3.3 Mohri's Epsilon-Filter
      • [0065] C.3.4 Second Embodiment
      • [0066] C.3.5 Complexity
    • [0067] C.4 Multi-Tape Intersection
      • [0068] C.4.1 Conditions
      • [0069] C.4.2 Embodiments
      • [0070] C.4.3 Example
  • [0071] D. Applications
    • [0072] D.1 General Use
    • [0073] D.2 Building A Lexicon From A Corpus
    • [0074] D.3 Enhancing A Lexicon With Lemmas
    • [0075] D.4 Normalizing A Lexicon
    • [0076] D.5 Using A Lexicon
    • [0077] D.6 Searching For Similarities
    • [0078] D.7 Preserving Intermediate Transduction Results
    • [0079] D.8 Example System
  • [0080] E. Miscellaneous

[0081] A. Definitions

[0082] This section recites basic definitions of algebraic structures that are used in describing the present invention, such as for “monoid” and “semiring” and “weighted automaton” (which are described in more detail in the following publications, which are incorporated herein by reference, by: Eilenberg, “Automata, Languages, and Machines”, volume A, Academic Press, San Diego, Calif., USA, 1974; and Kuich and Salomaa, “Semirings, Automata, Languages”, Number 5 in EATCS Monographs on Theoretical Computer Science, Springer Verlag, Berlin, Germany, 1986), and for weighted multitape automaton, based on the definitions of multi-tape automaton (which are described in more detail in the following publication, which is incorporated herein by reference, by: Elgot and Mezei, “On relations defined by generalized finite automata”, IBM Journal of Research and Development, 9:47-68, 1965).

[0083] A.1 Semirings

[0084] A monoid consists of a set M, an associative binary operation “o” on M, and a neutral element 1 such that 1 o a=a o 1=a for all a ε M. A monoid is called commutative iff a o b=b o a for all a, b ε M.

[0085] The set K with two binary operations ⊕ (collection) and {circle around (×)} (extension) and two elements 0 and 1 is called a semiring, if it satisfies the following properties:

• • [0086] (a) <K, ⊕, 0> is a commutative monoid;
  • [0087] (b) <K, {circle around (×)}, 1> is a monoid;
  • [0088] (c) {circle around (×)} is left-distributive and right-distributive over ⊕:
    a {circle around (×)}(b⊕c)=(a{circle around (×)}b)⊕(a{circle around (×)}c), (a⊕b){circle around (×)}c=(a{circle around (×)}c)⊕(b{circle around (×)}c), ∀a, b, c εK;
  • [0089] (d) 0 is an annihilator for {circle around (×)}: 0{circle around (×)}a=a{circle around (×)} 0= 0, ∀aεK.

[0090] A generic semiring K is denoted as <K, ⊕,{circle around (×)}, 0, 1>.

[0091] Some methods for processing automata require semirings to have specific properties. Composition, for example, requires a semiring to be commutative (which is described in more detail in the following publications, incorporated herein by reference, by: Pereira and Riley, “Speech recognition by composition of weighted finite automata”, in Emmanuel Roche and Yves Schabes, editors, Finite-State Language Processing, MIT Press, Cambridge, Mass., USA, pages 431-453, 1997; and Mohri, Pereira, and Riley, “A rational design for a weighted finite-state transducer library”, Lecture Notes in Computer Science, 1436:144-158, 1998), and ε (i.e., epsilon) removal requires it to be k-closed (which is described in more detail in the following publication incorporated herein by reference by: Mohri, “Generic epsilon-removal and input epsilon-normalization algorithms for weighted transducers”, International Journal of Foundations of Computer Science, 13(1):129-143, 2002). These properties are defined as follows:

• • [0092] (a) commutativity: a{circle around (×)}b=b{circle around (×)}a, ∀a, b εK; and

[0093] $\left(b\right)k\text{-}\mathrm{closedness}\text{:}\underset{n=0}{\overset{k+1}{\oplus }}{a}^n=\underset{n=0}{\overset{k}{\oplus }}{a}^n,\forall a\in K.$

[0094] The following well-known examples are all commutative semirings:

• • [0095] (a) <IB,+,×, 0, 1>: boolean semiring, with IB={0, 1} and 1+1=1;
  • [0096] (b) <IN,+,×, 0, 1>: integer semiring with the usual addition and multiplication;
  • [0097] (c) <IR⁺,+,×, 0, 1>: real positive sum times semiring;
  • [0098] (d) < IR⁺, min,+,∞, 0>: a real tropical semiring where IR⁺ denotes IR⁺ ∪{∞}.

[0099] A number of methods for processing automata require semirings to be equipped with an order or partial order denoted by <K. Each idempotent semiring K (i.e., ∀aεK: a⊕a=a) has a natural partial order defined by a <K ba⊕b=a. In the above examples, the boolean and the real tropical semiring are idempotent, and hence have a natural partial order.

[0100] A.2 Weighted Automata

[0101] A weighted automaton A over a semiring K is defined as a six-tuple:
A=_def<Σ, Q, I, F, E, K>
with:

[0102]

	Σ	being a finite alphabet
	Q	the finite set of states
	I  Q	the set of initial states
	F  Q	the set of final states
	E  Q × (Σ∪{ε}) × Q	the finite set of transitions and
	κ = <K, 0, 1, ⊕, >	the semiring

[0103] For any state qεQ, there is defined:

[0104]

λ(q)	λ : I → κ	the initial weight function (with λ(q) = 0,
		∀q∉I)
(q)	: F → κ	the final weight function (with (q) = 0,
		∀q∉F)
E(q)	= {e | p(e) = q}	the finite set of out-going transitions

and for any transition eεE, e=<p, l, w,n>, there is defined:

[0105]

p(e)	p : E → Q	the source state
l(e)	l : E → Σ∪{ε}	the label (with ε being the empty string)
w(e)	w : E → κ	the weight (with w(e) ≠ 0, ∀e∈E)
n(e)	n : E → Q	the target state

[0106] A path π of length r=|π| is a sequence of transitions e₁e₂ . . . e_r such that n(e_i)=p(e_i+1) for all iε[[1,r−1]]. A path is said to be successful iff p(e₁)εI and n(e_r)εF. In the following description only successful paths are considered. The label, {circumflex over (l)}(π), of any successful path π equals the concatenation of the labels of its transitions:
{circumflex over (l)}(π)=l(e₁)l(e₂) . . . l(e_r)
and its “weight” or “accepting weight” w(π) is:

[0107] $w\left(\pi \right)=\lambda \left(p\left(e_1\right)\right)\otimes \left(\underset{j=〚1,l〛}{\otimes }w\left(e_j\right)\right)\otimes \varrho \left(n\left(e_r\right)\right)$

[0108] Π(s) denotes the (possibly infinite) set of successful paths of A and Π(s) denotes the (possibly infinite) set of successful paths for the string s:
Π(s)={π|∀π ε Π(A), s={circumflex over (l)}(π)}

[0109] (A) is defined as the language of A. It is the (possibly infinite) set of strings s having successful paths in A:
(A)={{circumflex over (l)}(π)|π ε Π(A)}

[0110] The accepting weight for any string sε(A) is defined by:

[0111] $w\left(s\right)=\underset{\pi \in \prod \left(s\right)}{\oplus }w\left(\pi \right)$

[0112] A.3 Weighted Multi-Tape Automata

[0113] In analogy to a weighted automaton, a weighted multi-tape automaton (WMTA), also called weighted n-tape automaton, over a semiring K is defined as a six-tuple:
A⁽ⁿ⁾=_def<Σ, Q, I, F, E⁽ⁿ⁾, K>
with Σ, Q, I, F, and K being defined as above in section A.2, and with:

[0114]

E(n)  Q × (Σ∪{ε})n × Q	being the finite set of n-tape transitions
n	and the arity, i.e., the number of tapes
	in A

[0115] Most of the definitions given for weighted automata are also valid for WMTAs, except that each transition e⁽ⁿ⁾ ε E⁽ⁿ⁾ is labeled with:

[0116]

	l(e(n))	l : E(n) → (Σ∪{ε})n	an n-tuple of symbols

[0117] If all symbols σ ε(Σ∪{ε}) of a tuple are equal, the short-hand notation σ⁽ⁿ⁾ may be used on the terminal symbol. For example:
a⁽³⁾=<a, a, a>
ε⁽²⁾=<ε, ε>

[0118] The label of a successful path π⁽ⁿ⁾ of length r=|π⁽ⁿ⁾| equals the concatenation of the labels of its transitions, all of which must have the same arity n:
{circumflex over (l)}(π⁽ⁿ⁾)=l(e₁⁽ⁿ⁾)l(e₂⁽ⁿ⁾) . . . l(e_r⁽ⁿ⁾)
which is an n-tuple of strings:
{circumflex over (l)}(π⁽ⁿ⁾)=s⁽ⁿ⁾=<s₁, s₂, . . . , s_n>
where each string s_j is a concatenation of the j-th element of all l(e_i⁽ⁿ⁾) of π⁽ⁿ⁾ (with iε[[1,r]]). In anticipation of the projection operation, P_j(), (defined in section B.2) this can be expressed as:
s_j=P_j({circumflex over (l)}(π⁽ⁿ⁾))=P_j(l(e₁⁽ⁿ⁾)) P_j(l(e₂⁽ⁿ⁾)) . . . P_j(l(e_r⁽ⁿ⁾))

[0119] The symbols on e⁽ⁿ⁾ are not “bound” to each other. For example, the string triple s⁽³⁾=<aaa, bb, ccc> can be encoded, among others, by any of the following sequences of transitions: (a:b:c)(a:b:c)(a:ε:c) or (a:b:c)(a:ε:c)(a:b:c) or (ε:ε:c)(a:b:c)(a:b:c)(a:ε:ε), etc.

[0120] Π(s) denotes the (possibly infinite) set of successful paths for the n-tuple of strings s⁽ⁿ⁾:
Π(s⁽ⁿ⁾)={π⁽ⁿ⁾|∀π⁽ⁿ⁾ εΠ(A⁽ⁿ⁾), s⁽ⁿ⁾=l(π⁽ⁿ⁾)}

[0121] (A⁽ⁿ⁾) is called the n-tape language of A⁽ⁿ⁾ (which may also be referred to as a relation of arity(n)). It is the (possibly infinite) set of n-tuples of strings s⁽ⁿ⁾ having successful paths in A⁽ⁿ⁾:
(A⁽ⁿ⁾)={l⁽ⁿ⁾|l⁽ⁿ⁾=l(π⁽ⁿ⁾), ∀π⁽ⁿ⁾ εΠ(A⁽ⁿ⁾)}

[0122] The accepting weight for any n-tuple of strings s⁽ⁿ⁾ε(A⁽ⁿ⁾) is defined by:

[0123] $w\left(s^{\left(n\right)}\right)=\underset{{\pi }^{\left(n\right)}\in \prod \left(s^{\left(n\right)}\right)}{\oplus }w\left({\pi }^{\left(n\right)}\right)$

[0124] It is possible to define an arbitrary weighted relation between the different tapes of (A⁽ⁿ⁾). For example, (A⁽²⁾) of the weighted transducer (i.e., two-tape automata) A⁽²⁾ is usually considered as a weighted relation between its two tapes, P₁((A⁽²⁾)) and P₂((A⁽²⁾)). One of the two tapes is considered to be the input tape, the other as the output tape.

[0125] B. Operations on Multi-Tape Automata

[0126] All operations defined in this section are defined on symbol tuples, string tuples, or n-tape languages, taking their accepting weights into account. Whenever these operations are used on transitions, paths, or automata, they are actually applied to their labels or languages respectively. For example, the binary operation ö on two automata, A₁⁽ⁿ⁾ öA₂⁽ⁿ⁾, actually means (A₁⁽ⁿ⁾ öA₂⁽ⁿ⁾)=(A₁⁽ⁿ⁾)ö(A₂⁽ⁿ⁾), and the unary operation {dot over (o)} on one automaton, {dot over (o)} A⁽ⁿ⁾, actually means ({dot over (o)} A⁽ⁿ⁾)={dot over (o)} (A⁽ⁿ⁾).

[0127] B.1 Pairing and Concatenation

[0128] The pairing of two string tuples, s⁽ⁿ⁾: v^(m)=u^(n+m), and its accepting weight is defined as:
<s₁, . . . , s_n>:<v₁, . . . , v_m>=_def<s₁, . . . , s_n, v₁, . . . , v_m>
w(<s₁, . . . , s_n>:<v₁, . . . , v_m>)=_def w(<s₁, . . . , s_n>){circle around (×)}w(<v₁, . . . , v_m>)

[0129] 1-tuples of strings are not distinguished herein from strings, and hence, instead of writing s⁽¹⁾:v⁽¹⁾ or <s>:<v>, s:v is simply written. If strings contain only one symbol σε(Σ∪ε), they are not distinguished from the strings and their only symbol, and instead the pairing σ₁:σ₂ is written.

[0130] Pairing is associative:
s₁⁽ⁿ¹⁾:s₂⁽ⁿ²⁾:s₃⁽ⁿ³⁾=(s₁⁽ⁿ¹⁾:s₂⁽ⁿ²⁾ ):s₃⁽ⁿ³⁾=s₁⁽ⁿ¹⁾:(s₂⁽ⁿ²⁾:s₃⁽ⁿ³⁾)=s^(n1+n2+n3)

[0131] The concatenation of two sting tuples of equal arity, s⁽ⁿ⁾ v⁽ⁿ⁾=u⁽ⁿ⁾, and its accepting weight are defined as:
<s₁, . . . , s_n><v₁, . . . , v_n>=_def<s₁v₁, . . . , s_nv_n>
w(<s₁, . . . , s_n><v₁, . . . , v_n>)=_def w(<s₁, . . . , s_n>){circle around (×)}w(<v₁, . . . , v_n>)

[0132] Again, 1-tuples of strings are not distinguished herein from strings, and hence, instead of writing s⁽¹⁾v⁽¹⁾ or <s><v>, sv is simply written. If strings contain only one symbol σε(Σ∪ε), they are not distinguished from the strings and their only symbol, and instead the concatenation σ₁σ₂ is written.

[0133] Concatenation is associative:
s₁⁽ⁿ⁾s₂⁽ⁿ⁾s₃⁽ⁿ⁾=(s₁⁽ⁿ⁾s₂⁽ⁿ⁾)s₃⁽ⁿ⁾=s₁⁽ⁿ⁾(s₂⁽ⁿ⁾s₃⁽ⁿ⁾)=s⁽ⁿ⁾

[0134] The relation between pairing and concatenation can be expressed through a matrix of string tuples s_jk^(nj) given by:

[0135] $\left[\begin{array}{ccc}{s}_{11}^{\left(n_1\right)}& \dots & s_{1r}^{\left(n_1\right)}\\ ⋮& & ⋮\\ s_{\mathrm{m1}}^{\left(n_m\right)}& \dots & s_{\mathrm{mr}}^{\left(n_m\right)}\end{array}\right]\hspace{1em}$
that are horizontally concatenated and vertically paired:

[0136] $\begin{array}{c}{s}^{\left(n_1+\dots +n_m\right)}=(\begin{array}{ccc}{s}_{11}^{\left(n_1\right)}& \dots & s_{1r}^{\left(n_1\right)})\begin{array}{ccc}⋮& \dots & ⋮\end{array}\end{array}(\begin{array}{ccc}{s}_{\mathrm{m1}}^{\left(n_m\right)}& \dots & s_{\mathrm{mr}}^{\left(n_m\right)})\end{array}\\ =(\begin{array}{ccccc}{s}_{11}^{\left(n_1\right)}& ⋮& \dots & ⋮& s_{\mathrm{m1}}^{\left(n_m\right)})\begin{array}{c}\dots (\begin{array}{ccccc}{s}_{1r}^{\left(n_1\right)}& ⋮& \dots & ⋮& s_{\mathrm{mr}}^{\left(n_m\right)})\end{array}\end{array}\end{array}\end{array}$
where the equation above does not hold for the accepting weights unless they are defined over a commutative semiring K.

[0137] B.2 Projection and Complementary Projection

[0138] A projection P_j,k, . . . (s⁽ⁿ⁾) retains only those strings (i.e., tapes) of the tuple s⁽ⁿ⁾ that are specified by the indices j, k, . . . ε[[(1,n]], and places them in the specified order. The projection and its accepting weight are defined as:
P_j,k, . . . (<s₁, . . . , s_n>)=_def<s_j, s_k, . . . >
w(P_j,k, . . . (<s₁, . . . , s_n>))=_defw(<s₁, . . . , s_n>)
where the weights are not modified by the projection. Projection indices can occur in any order and more than once. Thus, the tapes of s⁽ⁿ⁾ can, for example, be reversed or duplicated:
P_{n, . . . , 1}(<s₁, . . . , s_n>)=<s_n, . . . , s₁>
P_j,j,j(<s₁, . . . , s_n>)=<s_j, s_j, s_j>

[0139] The relation between projection and pairing, and between their respective accepting weights is:

[0140] $s^{\left(n\right)}=\begin{array}{ccccc}{𝒫}_1\left(s^{\left(n\right)}\right)& ⋮& \dots & ⋮& {𝒫}_n\left(s^{\left(n\right)}\right)\end{array}$ $w\left(s^{\left(n\right)}\right)\ne \begin{array}{ccccc}w({𝒫}_1\left(s^{\left(n\right)}\right)& ⋮& \dots & ⋮& {𝒫}_n\left(s^{\left(n\right)}\right))=\underset{\underset{n\mathrm{times}}{︸}}{w\left(s^{\left(n\right)}\right)\otimes \dots \otimes w\left(s^{\left(n\right)}\right)}\end{array}$

[0141] A complementary projection P_j,k, . . . (s⁽ⁿ⁾) removes those strings (i.e., tapes) of the tuple s⁽ⁿ⁾ that are specified by the indices j,k, . . . ε[[1,n]], and preserves all other strings in their original order. Complementary projection and its accepting weight are defined as:
P_j,k, . . . (<s₁, . . . , s_n>)=_def< . . . , s_j−1, s_{j +1}, . . . , s_k−1, s_k+1, . . . >
w( P_j,k, . . . (<s₁, . . . , s_n>))=_defw(<s₁, . . . , s_n>)

[0142] Complimentary projection indices can occur only once, but in any order.

[0143] The projection of an n-tape language is the projection of all its string tuples and complimentary projection of an n-tape language is defined in the same way, respectively as:
P_j,k, . . . (⁽ⁿ⁾)={P_j,k, . . . (s⁽ⁿ⁾)|∀s⁽ⁿ⁾ ε⁽ⁿ⁾}
P_j,k, . . . (⁽ⁿ⁾)={ P_,j,k, . . . (s⁽ⁿ⁾)|∀s⁽ⁿ⁾ ε⁽ⁿ⁾}

[0144] B.3 Cross-Product

[0145] The cross-product of two n-tape languages is defined as:
₁⁽ⁿ⁾×₂^(m)=_def{s⁽ⁿ⁾: v^(m)|∀s⁽ⁿ⁾ε₁⁽ⁿ⁾, ∀v^(m)ε₂^(m)}

[0146] The accepting weight of each string tuple in ₁⁽ⁿ⁾×₂^(m) follows from the definition of pairing. The cross product operation is associative.

[0147] A well known example (and special case) is the cross-product of two acceptors (i.e., a 1-tape automata) leading to a transducer (i.e., a 2-tape automaton):
A⁽²⁾=A₁⁽¹⁾×A₂⁽¹⁾
(A₁⁽¹⁾×A₂⁽¹⁾)={s:v|∀s ε (A₁⁽¹⁾), ∀v ε (A₂⁽¹⁾)}
w(s:v)=w_A1(s){circle around (×)}w_A2(v)

[0148] B.4 Auto-Intersection

[0149] Operations for auto-intersection are described in this section. More specifically, this section describes operations for performing auto-intersection on string tuples and languages.

[0150] The auto-intersection I_j,k(s⁽ⁿ⁾) on string tuples succeeds, if the two strings s_j and s_k of the tuple s⁽ⁿ⁾=<s₁, . . . , s_n> are equal (with j,k ε [[1,n]]), and fails otherwise (⊥). Auto-intersection and its accepting weight are defined as:

[0151] $\begin{array}{ccc}{ℐ}_{j,k}\left(s^{\left(n\right)}\right)& {=}_{\mathrm{def}}& \{\begin{array}{cc}{s}^{\left(n\right)}& \mathrm{for}{s}_j=s_k\\ \perp & \mathrm{for}{s}_j\ne s_k\end{array}\\ w\left({ℐ}_{j,k}\left(s^{\left(n\right)}\right)\right)& {=}_{\mathrm{def}}& \{\begin{array}{cc}w\left(s^{\left(n\right)}\right)& \mathrm{for}{s}_j=s_k\\ \stackrel{\_}{0}& \mathrm{for}{s}_j\ne s_k\end{array}\end{array}$

[0152] This means the weight of a successfully auto-intersected string tuple is not modified, whereas the weight of a string tuple where the auto-intersection failed is 0, which corresponds to the invalidation or elimination of that tuple.

[0153] FIG. 1 is a flow diagram that sets forth steps for performing the auto-intersection operation on a first tape and a second tape of a path of a weighted multi-tape automaton. At 102, a string tuple <s₁, . . . , s_n> is generated that has a string s for each of the n tapes of a selected path of the WMTA. At 104, the string s_j of the first tape is compared with the string s_k of the second tape in the string tuple. If at 106, the strings s_j and s_k are equal, then the string tuple is retained in the WMTA at 108; otherwise, the WMTA is restructured to remove the string tuple at 110. At 112, if the last of the paths of the WMTA has been processed, then auto-intersection completes; otherwise, it continues with the next selected path of the WMTA at 102.

[0154] For example, the result of performing the auto-intersection operation I_j,k(s⁽ⁿ⁾) on tapes 1 and 3 of the example three-tape WMTA (i.e., s⁽³⁾) shown FIG. 2 fails for the first of its two paths, producing a first string tuple <ax, by, az> (with weight w₁w₂), but succeeds for the second of its two paths, producing a second string tuple <ax, by, ax> (with weight w₁w₃). The final state of the WMTA has weight w₄. Because the strings of tape 1 (i.e., “ax”) and tape 3 (i.e., “az”) of the first string tuple <ax, by, az> are not equal unlike the strings of tape 1 (i.e., “ax”) and tape 3 (i.e., “ax”) of the second string tuple <ax, by, ax>, the WMTA shown in FIG. 2 is restructured to remove the arc x:y:x/w₂ of the first path because there are no other successful paths in the WMTA that depend on that arc (i.e., I_(1,3)(s⁽³⁾)). The result of the auto-intersection of the two selected tapes 1 and 3 of the WMTA is to filter out all string tuples of the WMTA except for those that have strings that are equal on two selected tapes.

[0155] More generally, auto-intersection of a language, I_j,k(⁽ⁿ⁾), equals the auto-intersection of all of its string tuples such that only the successfully auto-intersected string tuples are retained, which may be defined as:

[0156] $\begin{array}{ccc}{ℐ}_{j,k}\left({ℒ}^{\left(n\right)}\right)& {=}_{\mathrm{def}}& \left\{s^{\left(n\right)}|s_j=s_k,s^{\left(n\right)}\in {ℒ}^{\left(n\right)}\right\}\\ {ℐ}_{j,k}\left({ℒ}^{\left(n\right)}\right)& =& \left\{{ℐ}_{j,k}\left(s^{\left(n\right)}\right)|s^{\left(n\right)}\in {ℒ}^{\left(n\right)}\right\}\\ {{ℐ}_{j,k}\left({ℒ}^{\left(n\right)}\right)}_1& \subseteq & {ℒ}^{\left(n\right)}\end{array}$

[0157] For example, given a language ₁⁽³⁾ (where “*” denotes kleen star):
₁⁽³⁾=<a, x, ε><b, y, a>*<ε, z, b>=ab*, xy*z, a*b>,
the results of its auto-intersection I_1,3(₁⁽³⁾) of tapes 1 and 3 is:
I_1,3(₁⁽³⁾)={<ab, xyz, ab>}
which means the auto-intersection admits a single iteration through the cycle <b,y,a>* (i.e., “*” takes only the value 1).

[0158] B.5 Single-Tape Intersection

[0159] Single-tape intersection of two multi-tape languages, ₁⁽ⁿ⁾ and ₂^(m), is based on one single tape j and k from each of their sets of n and m tapes, respectively, and may be defined as:

[0160] ${ℒ}_1^{\left(n\right)}\bigcap _{j,k}{ℒ}_2^{\left(m\right)}{=}_{\mathrm{def}}{\stackrel{\_}{𝒫}}_{n+k}\left({ℐ}_{j,n+k}\left({ℒ}_1^{\left(n\right)}\times {ℒ}_2^{\left(m\right)}\right)\right).$

[0161] The single-ape intersection operation pairs each string tuple s⁽ⁿ⁾ ε₁⁽ⁿ⁾ with each string tuple v^(m) ε ₂^(m) iff s_j=v_k. The resulting language ^(n+m−1) is defined as:
^(n+m−1)={u^(n+m−1)|u^(n+m−1)= P_n+k(s⁽ⁿ⁾:v^(m)), s⁽ⁿ⁾ ε ₁⁽ⁿ⁾, v^(m) ε ₂^(m), s_j=v_k}
with weight w:
w(u^(n+m−1))=w(s⁽ⁿ⁾){circle around (×)}w(v^(m)).

[0162] Single-tape intersection can intuitively be understood as a “composition” of the two languages such that tape j of ₁⁽ⁿ⁾ is intersected with tape k of ₂^(m). Tape k, which due to the intersection becomes equal to tape j, is then removed, and all other tapes of both languages are preserved without modification. For example, if one language contains the ordered pair <x,y> and another language contains the ordered pair <y,z>, then composing <x,y> and <y,z>, in that order, results in a language that contains the ordered pair <x,z>, whereas intersecting <x,y> and <y,z>, at tapes 2 and 1 respectively, results in the language that contains the ordered pair <x,y,z>.

[0163] Single-tape intersection is neither associative nor commutative, except for special cases. A first special case of single-tape intersection is the intersection of two acceptors (i.e., 1-tape automata) leading to an acceptor, which may be defined as:

[0164] $A_1^{\left(1\right)}\bigcap A_2^{\left(1\right)}=A_1^{\left(1\right)}\bigcap _{1,1}{A}_2^{\left(1\right)}={\stackrel{\_}{𝒫}}_2\left({ℐ}_{1,2}\left(A_1^{\left(1\right)}\times A_2^{\left(1\right)}\right)\right)$
where the first special case of single-tape intersection has the language and weight w:
(A₁⁽¹⁾ ∩ A₂⁽¹⁾)={s|s ε (A₁), s ε (A₂)}
w(s)=w_A1(s){circle around (×)}w_A2(s)
and where single-tape intersection has the same language:
(A₁⁽¹⁾×A₂⁽¹⁾)={<s₁, s₂>|s₁ ε (A₁), s₂ ε (A₂)}
(I_1,2(A₁⁽¹⁾×A₂⁽¹⁾))={<s, s>|s ε (A₁), s ε (A₂)}
( P₂(I_1,2(A₁⁽¹⁾×A₂⁽¹⁾)))={s|s ε (A₁), s ε (A₂)}
w(s)=w(<s, s>)=w(<s₁, s₂>)=w_A1(s){circle around (×)}w_A2(s)

[0165] A second special case of single-tape intersection is the composition of two transducers (i.e., 2-tape automata) leading to a transducer. The second special case of single-tape intersection requires an additional complementary projection and may be defined as:

[0166] $A_1^{\left(2\right)}\diamond A_2^{\left(2\right)}={\stackrel{\_}{𝒫}}_2\left(A_1^{\left(2\right)}\bigcap _{2,1}{A}_2^{\left(2\right)}\right)={\stackrel{\_}{𝒫}}_{2,3}\left({ℐ}_{2,3}\left(A_1^{\left(2\right)}\times A_2^{\left(2\right)}\right)\right).$

[0167] FIG. 3 is a flow diagram that sets forth steps for performing a single-tape intersection operation of a first WMTA and a second WMTA. At 302, a cross-product WMTA is computed using the first WMTA and the second WMTA. At 304, a string tuple for each path of the cross-product WMTA is generated. At 306 and 316, for each string tuple generated at 304, the string of a first selected tape is compared with the string of a second selected tape at 308. If at 310, the strings compared at 308 are equal, then the corresponding string tuple is retained in the cross-product WMTA at 312; otherwise, the corresponding string tuple is restructured at 314. When strings of the last tuple have been compared at 316, redundant strings retained in the string tuples at 312 are removed in the cross-product WMTA at 318.

[0168] For example, FIG. 4 presents two WMTAs A₁⁽³⁾ (with string tuple <ax, by, cz> and weight w₁{circle around (×)}w₂ and string tuple <ae, bf, cg> and weight w₁{circle around (×)}w₃) and A₂⁽²⁾ (with string tuple <aa, cg> and weight w₄{circle around (×)}w₅) for illustrating a simple example of the single-tape intersection operation

[0169] $A_1^{\left(3\right)}\bigcap _{3,2}{A}_2^{\left(2\right)}$
(where the final state of each automaton also has a weight). The resulting cross-product WMTA A₃⁽⁵⁾ of the two WMTAs A₁⁽³⁾ and A₂⁽²⁾ results in the following two string tuples (see 302 FIG. 3): <ax, by, cz, aa, cg> having weight w₁{circle around (×)}w₂{circle around (×)}w₄{circle around (×)}w₅ and <ea, bf, cg, aa, cg> having weight w₁{circle around (×)}w₃{circle around (×)}w₄{circle around (×)}w₅. In comparing tapes 3 and 5 of the cross-product WMTA A₃⁽⁵⁾ (see 308 in FIG. 3), the following strings are compared for each string tuple of the cross-product WMTA A₃⁽⁵⁾, respectively: “cz” and “cg”; and “cg” and “cg”.

[0170] In the example shown in FIG. 4, one set of strings at the selected tapes 3 and 2 are equal, which results in the retention of the string tuple <ae, bf, cg, aa, cg> in the cross-product WMTA A₃⁽⁵⁾ (see 312 in FIG. 3). Also in the example, one set of strings at the selected tapes 3 and 5 in the cross-product WMTA A₃⁽⁵⁾ are not equal, resulting in the restructuring of the cross-product WMTA to remove the string tuple <ax, by, cz, aa, cg> through auto-intersection as WMTA A₄⁽⁵⁾=I_(3,5)(A₃⁽⁵⁾) (see 314 in FIG. 3). Finally, redundant strings in the string tuple retained in the cross-product WMTA A₄⁽⁵⁾ are removed through complementary projection A₅⁽⁴⁾= P(A₄⁽⁵⁾) to result in the simplified string tuple <ae, bf, cg, aa> for the WMTA A₅⁽⁴⁾ (see 318 in FIG. 3).

[0171] B.6 Multi-Tape Intersection

[0172] Multi-tape intersection of two multi-tape languages, ₁⁽ⁿ⁾ and ₂^(m), uses r tapes in each language, and intersects them pair-wise. In other words, multi-tape intersection is an operation that involves intersecting several tapes of a first MTA with the same number of tapes of a second MTA. Multi-tape intersection is a generalization of single-tape intersection, and is defined as:

[0173] ${𝔏}_1^{\left(n\right)}\bigcap _{\underset{\underset{j_r,k_r}{\cdots }}{j_1,k_1}}{𝔏}_2^{\left(m\right)}{=}_{\mathrm{def}}{\stackrel{\_}{𝒫}}_{n+k_1,\dots ,n+k_r}\left({ℐ}_{j_r,n+k_r}\left(\dots {ℐ}_{j_1,n+k_1}\left({𝔏}_1^{\left(n\right)}\times {𝔏}_2^{\left(m\right)}\right)\dots \right)\right)$

[0174] The multi-tape intersection operation pairs each string tuple s⁽ⁿ⁾ ε ₁⁽ⁿ⁾ with each string tuple v^(m) ε ₂^(m) iff s_j1=v_k1 until s_jr=v_kr. The resulting language ^(n+m−r) is defined as:
^(n+m−r)={u^(n+m−r)|u^(n+m−r)= P_{n+k1, . . . , n+kr}(s⁽ⁿ⁾:v^(m)), s⁽ⁿ⁾ ε₁⁽ⁿ⁾, v^(m)ε₂^(m), s_j1=v_k1, . . . , s_jr=v_kr}
weight w:
w(u^(n+m−r))=w(s⁽ⁿ⁾){circle around (×)}w(v^(m)).

[0175] All tapes k_i of language ₂⁽ⁿ²⁾ that have directly participated in the intersection are afterwards equal to the tapes j_i of ₁⁽ⁿ¹⁾, and are removed. Multi-tape intersection is neither associative nor commutative (except for special cases).

[0176] B.7. Transition Automata and Transition-Wise Processing

[0177] A transition automaton A(e) is defined herein as an automaton containing only one single transition e that actually belongs to another automaton A. Any automaton operation is allowed to be performed on A(e), which means that the operation is performed on one single transition of A rather than on A. This can be thought of either as the operation being performed in place (i.e., inside A on one single transition) or as the transition being extracted from A and placed into a new automaton A(e) that participates in an operation whose result is then placed back into A, at the original location of e.

[0178] The concept of transition automata allows a method to be defined where an automaton A is transition-wise processed (i.e., each of its transitions is processed independently through a sequence of automaton operations). In the following example that illustrates the transition automata A(e) being transition-wise processed, e is one transition in a set of transitions E of automata A:

[0179]

	1	for ∀e ∈ E do
	2	A(e) ← A1(2) ⋄ A(e) ⋄ A2(2)
	3	A(e) ← . . . A(e) . . .

[0180] C. Methods for Performing MTA Operations

[0181] This section sets forth methods for performing multi-tape operations for automata defined in section B, while referring to the variables and definitions in Table 1. Note that in Table 1 the following variables serve for assigning temporarily additional data to a state q: μ[q], v[q], ξ[q], v[q], and χ[q].

[0182]

TABLE 1
Aj = <Σj, Qj, ij, Fj, Ej, κj>	Specific (original) weighted automaton
	from which a new weighted automaton A
	is constructed
A = <Σ, Q, i, F, E, κ>	New weighted automaton resulting from
	the construction
ν[q] = q1	State q1 of an original automaton A1
	assigned to a state q of a new auto-
	maton A
μ[q] = (q1, q2)	Pair of states (q1, q2) of two ordinal
	automata, A1 and A2, assigned to a
	state q of a new automaton A
ψ[q] = {umlaut over (q)}	Previous state {umlaut over (q)} in the new automaton
	A on the same path as q (back pointer)
υ[q] = (q1, q2, qε)	Triple of states q1, q2, qε belonging
	to the original automata, A1 and A2,
	and to a simulated filter automaton,
	Aε, respectively; assigned to a state
	q of a new automaton A
ξ[q] = (s, u)	Pair of “leftover” substrings (s, u)
	assigned to a state q of a new auto-
	maton A
lcp(s, s′)	Longest common prefix of the strings s
	and s′
lj,k, . . . (x) =	Short-hand notation for the projection
Pj,k, . . . (l(x))	of the label x
δ(s, u) = |s| − |u|	Delay between two string (or leftover
	substrings) s and u, where |s| is the
	length of string s and |u| is the
	length of string u. Example: δ(ξ[q])
χ[q] = (χ1, χ2)	Pair of integers assigned to a state
	q, expressing the lengths of two
	strings s and u on different tape of
	the same path ending at q

[0183] C.1 Cross-Product

[0184] Generally, this section sets forth two alternate embodiments for performing the cross-product operation defined in section B-3, and more specifically for compiling the cross product of two WMTAs, A₁⁽ⁿ⁾ and A₂^(m).

[0185] The first embodiment pairs the label of each transition e₁ ε E₁ with ε^(m) (producing l(e₁):ε^(m), and the label of each transition e₂ε E₂ with ε⁽ⁿ⁾ (producing ε⁽ⁿ⁾: l(e₂)), and finally concatenates A₁^(n+m) with A₂^(n+m). This operation is referred to herein as “CrossPC (A₁, A₂)” where the suffix “PC” stands for “path concatenation” and can be expressed as:

[0186] $l\left({\pi }_1^{\left(n\right)}:{\pi }_2^{\left(m\right)}\right)=\left(l\left(e_{1,1}^{\left(n\right)}\right):{\varepsilon }^{\left(m\right)}\right)\dots \left(l\left(e_{1,\alpha }^{\left(n\right)}\right):{\varepsilon }^{\left(m\right)}\right)·\left({\varepsilon }^{\left(m\right)}\right)·\left({\varepsilon }^{\left(n\right)}:l\left(e_{2,1}^{\left(m\right)}\right)\right)\dots \left({\varepsilon }^{\left(n\right)}:l\left(e_{2,\beta }^{\left(m\right)}\right)\right).$

[0187] The second embodiment pairs each string tuple of A₁⁽ⁿ⁾ with each string tuple of A₂^(m), following the definition in section B-3. This embodiment in actuality pairs each path π₁ of A₁⁽ⁿ⁾ with each path π₂ of A₂^(m) transition-wise, and appends epsilon transitions (i.e., ε-transitions) to the shorter of two paired paths, so that both have equal length. This operation is referred to herein as “CrossPA (A₁, A₂)” where the suffix “PA” stands for “path alignment” and can be expressed as:

[0188] $l\left({\pi }_1^{\left(n\right)}:{\pi }_2^{\left(m\right)}\right)=\left(l\left(e_{1,1}^{\left(\stackrel{.}{n}\right)}\right):l\left(e_{2,1}^{\left(m\right)}\right)\right)\dots \left(l\left(e_{1,\alpha }^{\left(n\right)}\right):l\left(e_{2,\alpha }^{\left(m\right)}\right)\right)·\left({\varepsilon }^{\left(n\right)}:l\left(e_{2,\alpha +1}^{\left(m\right)}\right)\right)·\left({\varepsilon }^{\left(n\right)}:l\left(e_{2,\beta }^{\left(m\right)}\right)\right)$
for α<β, and similarly otherwise.

[0189] C.1.1 Conditions

[0190] Both embodiments for performing the cross-product operation operate the conditions that:

• • [0191] (A) the semirings of the two automata A₁⁽ⁿ⁾ and A₂^(m) are equal to: K₁=K₂; and
  • [0192] (B) the common semiring K=K₁=K₂ is commutative (which holds in the case of CrossPA embodiment only):
    ∀w₁, w₂ ε K: w₁{circle around (×)}w₂=w₂{circle around (×)}w₁

[0193] C.1.2 Path Concatenation Method

[0194] A (brute force) method for performing the cross-product operation in accordance with an embodiment, defined as “CrossPC()”, with path concatenation “PC” may be described as follows in pseudocode:

[0195]

	CROSSPC(A1(n), A2(m)) → A :
	1	A ← <Σ1 ∪ Σ2, Q1 ∪ Q2, i1, F2, E1 ∪ E2, κ1>
	2	for ∀e1 ∈ E1 do
	3	l(e1) ← l(e1):ε(m)
	4	for ∀e2 ∈ E2 do
	5	l(e2) ← ε(n):l(e2)
	6	for ∀q ∈ F1 do
	7	E ← E ∪ { <q, ε(n+m), (q), i2> }
	8	(q) ← 0
	9	return A

[0196] The pseudocode set forth above for cross-product path concatenation (i.e., CrossPC()) starts with a WMTA A that is equipped with the union of the alphabets (i.e., the union of the state sets of transducers A₁ and A₂). The initial state of A equals that of A₁, its set of final states equals that of A₂, and its semiring equal those of A₁ and A₂ (see line 1). First, the labels of all transitions originally coming from A₁ are (post-) paired with ε^(m)-transitions, and the labels of all transitions originally coming from A₂ are (pre-) paired with ε⁽ⁿ⁾-transitions. Subsequently, all final states of A₁ are connected with the initial state of A₂ through ε^(m+m)-transitions. As a result, each string n-tuple of A₁ will be physically followed by each string m-tuple of A₂. However, logically those string tuples will be paired since they are on different tapes.

[0197] It will be appreciated by those skilled in the art that the paths of the WMTA A become longer in this embodiment than the other embodiment with path alignment. In addition, it will be appreciated that each transition of A in this embodiment is partially labeled with an epsilon, which may increase the runtime of subsequent operations performed on A. Further it will be appreciated that this embodiment may be readily adapted to operate with non-weighted multi-tape automata (MTAs) by removing the weight Q(q) from line 7 and the semiring K₁ from line 1 and by replacing line 8 with “Final(q)←false”, in the pseudocode above for path concatenation (i.e., CrossPC()).

[0198] C.1.3 Path Alignment Method

[0199] FIG. 5 sets forth a second embodiment in pseudocode for performing the cross-product operation with path alignment “PA” (i.e., CrossPA()). In FIG. 5, the final weight of an undefined state q=⊥ is assumed to be 1: Q(⊥)= 1. In the embodiment shown in FIG. 5, the pseudocode starts with a WMTA A whose alphabet is the union of the alphabets of A₁ and A₂, whose semiring equals those of A₁ and A₂, and that is otherwise empty (see line 1). First, the initial state i of A is created from the initial states A₁ and A₂ (at line 3), and i is pushed onto the stack (at line 4) which was previously initialized (at line 2). While the stack is not empty, the states q are popped from it to access the states q₁ and q₂ that are assigned to q through u[q] (see lines 5 and 6).

[0200] Further in the embodiment shown in FIG. 5, if both q₁ and q₂ are defined (i.e., ≠⊥), each outgoing transition e₁ of q₁ is paired with each outgoing transition of e₂ of q₂ (see lines 7 to 9). Also (at line 13), a transition in A is created whose label is the pair l(e₁):l(e₂) and whose target q′ corresponds to the tuple of targets (n(e₁), n(e₂)). If q′ does not exist yet, it is created and pushed onto the stack (see lines 10 to 12).

[0201] In addition in the embodiment shown in FIG. 5 (at lines 14 and 15), if a final state q₁ (with Q(q₁)≠ 0) in A₁ is encountered, the path is followed beyond q₁ on an epsilon-transition that exists only locally (i.e., virtually) but not physically in A₁. The target of the resulting transition in A corresponds to the tuple of targets (n(e₁), n(e₂)) with n(e₁) being undefined (=⊥) because e₁ does not exist physically (see line 17). If a final state q₂ (with Q(q₂)≠ 0) in A₂ is encountered, it is processed similarly (see lines 20 to 25). It will be; appreciated by those skilled in the art that this embodiment may be readily adapted to operate with non-weighted multi-tape automata (MTAs) by removing the weights from lines 13, 19, and 25, and the semiring K₁ from line 1, and the and by replacing line 28 with “Final(q)←Final(q₁) Final(q₂)”, in the pseudocode shown in FIG. 5.

[0202] C.1.4 Complexity

[0203] The space complexity of the (brute-force) cross-product path, concatenation embodiment described in section C.1.2 is |Q₁|+|Q₂| (i.e., on the order of O(n)) and its running complexity is |F₁|. In contrast, the space complexity of the cross-product path alignment embodiment described in section C.1.3 is (|Q₁|+1)·(|Q₂|+1) (i.e., on the order of O(n²) and its running complexity is (|E₁|+1)·(|E₂|+1).

[0204] C.2 Auto-Intersection

[0205] This section describes two methods for performing the auto-intersection operation defined in section B.4.

[0206] C.2.1 Conditions of First Method

[0207] The method described for performing auto-intersection operation in this section operates under the condition that the original automaton A₁⁽ⁿ⁾ does not contain cycles labeled only with an epsilon on one and not only with an epsilon on the other of the two tapes involved in the operation. If condition occurs the method will stop without producing a result, rather than attempting to create an infinite number of states. It is assumed this undesirable condition occurs rarely (if at all) in natural language processing applications (see for example FIG. 8).

[0208] C.2.2 First Method

[0209] FIG. 6 sets forth a first method in pseudocode for performing the auto-intersection operation (i.e., AutoIntersect()). Line 1 of the pseudocode begins with a WMTA A whose alphabet and semiring equal those of A₁ and that is otherwise empty. To each state q that will be created in A (see line 3), three variables are assigned: (i) v[q]=q₁ that indicates the corresponding state q₁ in A₁ (see line 24), (ii) Ψ[q]={umlaut over (q)} that indicates the previous state {umlaut over (q)} in A on the current path (back pointer) (see line 25), and (iii) ξ[q]=(s,u) that states the leftover string s of tape j (yet unmatched in tape k) and leftover string u of tape k (yet unmatched in tape j) (see line 26).

[0210] At lines 3-4 and 20-27, an initial state i in A is created and pushed onto the stack defined at line 2. As long as the stack is not empty, the states q are popped from the stack and each of the outgoing transitions e₁εE(q) in A with the same label and weight are followed (see line 5). To compile the leftover strings ξ[q′]=(s′, u′) of its target q′=n(e) in A, the leftover strings ξ[q]=(s,u) of its source q=p(e) are concatenated with the j-th and k-th component of its label, l_j(e₁) and l_k(e₁), and the longest common prefix of the resulting string s·l_j(e₁) and u·l_k(e₁) is removed (see lines 7 and 16-19).

[0211] If both leftover strings s′ and u′ of q′ are non-empty (i.e., ≠ε) then they are incompatible and the path that is being followed is invalid. In this case, the transition eεE(q) and its target q′ in A are not constructed. If either s′ or u′ is empty (i.e., =ε), then the current path is valid (at least up to this point) (see line 8).

[0212] At line 9, a test is made to determine whether the process will not terminate, which is the case if a cycle in A₁ was traversed and the ξ[{circumflex over (q)}] at the beginning of the cycle differs from the ξ[n(e)]=(s′,u′) at its end. In this case the states {circumflex over (q)} and n(e) are not equivalent, and cannot be represented through one state in A, although they correspond to the same state q₁ in A₁. In order that the process does not traverse the cycle an infinite number of times and create a new state on each transversal, the process aborts at line 10.

[0213] At line 14 (if the process did not abort at line 10), a transition e in A is constructed. If its target q′=n(e) does not exist yet, it is created and pushed onto the stack at lines 11-13. It will be appreciated by those skilled in the art that this embodiment may be readily adapted to operate with non-weighted multi-tape automata (MTAs) by removing the weight w(e₁) from line 14 and the semiring K₁ from line 1, and by replacing line 22 with “Final(q)←Final(q₁)” and line 23 with “Final(q)←false”, in the pseudocode shown in FIG. 6.

[0214] C.2.3 Example of First Method

[0215] FIG. 7 illustrates an example of the first method for performing the auto-intersection operation shown in FIG. 6. In the example shown in FIG. 7, the language of A₁⁽²⁾ is the infinite set of string tuples <ab*¹, a*¹b>. Only one of those tuples, namely <ab, ab>, is in the language of the auto-intersection with A⁽²⁾=I_1,2(A₁⁽²⁾) because all other tuples contain different strings on tapes 1 and 2. In FIG. 7, weights of each WMTA are omitted and dashed states and transitions are not constructed.

[0216] The method described above in section C.2.2 builds first the initial state 0 of A⁽²⁾ with v[0]=0, Ψ[0]=⊥, and ξ[0]=(ε,ε). Then the only outgoing transition of the state referenced by v[0] is selected, which is labeled a:ε, and the leftover strings of its target, state 1, is compiled by concatenating ξ[0]=(ε,ε) with the label a:ε. This gives first (a,ε) and then, after removal of the longest comment prefix (ε in this case), ξ[1]=(a,ε). State 1 is created because ξ[1] meets the constrains defined in the method. It is assigned v[1]=1, because it corresponds to state 1 in A₁, Ψ[1]=0, because it is (at present) reached from state 0 in A, and ξ[1]=(a,ε). State 1 in A is not final (unlike state 1 in A₁) because ξ[1]≠(ε,ε).

[0217] The remaining strings of state 2 in A result from ξ[1]=(a,ε) and the transition label ε:b, and are ξ[2]=(a,b). State 2 and its incoming transition are not created because ξ[2] does not meet the defined constraints. All:other states and transitions are similarly constructed.

[0218] FIG. 8 illustrates an example where the first method for performing the auto-intersection operation shown in FIG. 6 fails to construct the auto-intersection whose language, <a*¹a, aa*¹, x*¹yz*¹>, and is actually not finite-state (see conditions in section C.2.1). The failure condition is met at states 2 and 3 of the new automaton A. In FIG. 8, weights of each WMTA are omitted and dashed states and transitions are not constructed.

[0219] C.2.4 Conditions of Second Method

[0220] The second method unlike the first method has no conditions. In accordance with the second method for performing auto-intersection, the second method detects whether the auto-intersection of a given WMTA A₁⁽ⁿ⁾ is regular (i.e., whether it can be represented as an automaton). If it is found to be regular, the second method creates a WMTA A=I_j,k(A₁⁽ⁿ⁾); otherwise if it is found not to be regular and the complete result cannot be represented by an automaton, the second method creates the automaton A ⊂ I_j,k(A₁⁽ⁿ⁾), which is a partial result.

[0221] Briefly, the second method performs auto-intersection by assigning leftover-strings to states using variable ξ[q]=(s,u) and makes use of the concept of delay using variable δ(s,u), which are both defined in Table 1. The variable χ[q] defines a pair of integers assigned to a state q, which expresses the lengths of two strings s and u on different tape of the same path in a WMTA ending at q. The concept of delay provides that given a path in a WMTA, the delay of its states q is the difference of lengths of the strings on the tapes j and k up to q. It is expressed as the function δ(s,u)=|s|−|u|, where |s| is the length of string s and |u| is the length of string u.

[0222] C.2.5 Second Method

[0223] FIG. 9 sets forth the second method in pseudocode for performing the auto-intersection operation (i.e., AutoIntersect()). The method is based on the observation that if the auto-intersection A⁽ⁿ⁾=I_j,k(A₁⁽ⁿ⁾) of a WMTA A₁⁽ⁿ⁾ is regular, the delay will not exceed a limit δ_max at any state q of A⁽ⁿ⁾. If it is not regular, the delay will exceed any limit, but it is possible to construct a regular part, A_p⁽ⁿ⁾, of the auto-intersection within the limit of δ_max and a larger regular part, A_p2⁽ⁿ⁾, within a larger limit δ_max2 (i.e., A_p⁽ⁿ⁾ ⊂ A_p2⁽ⁿ⁾ ⊂ I_j,k(A₁⁽ⁿ⁾)).

[0224] By way of overview, the second method set forth in FIG. 9 and described in detail below involves three operations. First, two limits are computed corresponding to delays δ_max and δ_max2 of the auto-intersection (i.e., line 1). The first delay δ_max is computed by traversing the input automaton and measuring the delays along all its paths. The second delay is computed similar to the first delay while traversing an additional cycle of the input automaton. Next, the auto-intersection of the automaton is constructed using the delay δ_max2 (i.e., lines 2-10, where the second limit serves to delimit construction of the automaton). Finally, the constructed automaton is tested for regularity using the delay δ_max (i.e., line 11, where the first limit serves to determine whether the auto-intersection is regular).

[0225] C.2.5.A Compile Limits

[0226] In FIG. 9, a maximal delay, δ_max, is compiled. The maximal delay, δ_max, can occur at any point between the two strings l_j(π) and l_k(π) on any path π of I_j,k(A₁⁽ⁿ⁾) if it is regular. Let R(A₁⁽²⁾)=({<aa, ε>} ∪ {<ε, aaa>})*, encoded by two cycles (as provided at lines 1 and 27-42), where R(A⁽ⁿ⁾) is the n-tape relation of A⁽ⁿ⁾. To obtain a match between l₁(π) and l₂(π) in A⁽²⁾=I_1,2(A₁⁽²⁾), the first cycle must be traversed three times and the second cycle two times, allowing for any permutation: A⁽²⁾=(<aa, ε>³<ε, aaa>²∪<aa, ε>²>ε, aaa>²<aa, ε>¹∪ . . . )*. This illustrates that in a match between any two cycles of A₁⁽ⁿ⁾, the absolute value of the delay does not exceed δ_cyc={circumflex over (δ)}_cyc·max(1, δ_cyc−1), with {circumflex over (δ)}_cyc being the maximal absolute value of the delay of any cycle (as provided at line 30). For any other kind of match, the difference between the maximal and the minimal delay, {circumflex over (δ)}_max and {circumflex over (δ)}_min, encountered at any (cyclic or acyclic) path π of A₁⁽ⁿ⁾ is taken into account. Therefore, the absolute value of the delay in A⁽ⁿ⁾ does not exceed δ_max=max ({circumflex over (δ)}_max−{circumflex over (δ)}_min, δ_cyc) if I_j,k(A₁⁽ⁿ⁾) is regular (as provided at line 31). If it is non-regular, then δ_max will limit A to a regular subset of the auto-intersection, A⁽ⁿ⁾ ⊂ I_j,k(A₁⁽ⁿ⁾).

[0227] Next, a second limit, δ_max2, is compiled that permits, in case of non-regularity, to construct a larger regular subset A⁽ⁿ⁾ ⊂ I_j,k(A₁⁽ⁿ⁾) than δ_max does. Non-regularity can only result from matching cycles in A₁⁽ⁿ⁾. To obtain a larger subset of I_j,k(A₁⁽ⁿ⁾), the cycles of A₁⁽ⁿ⁾ must be unrolled further until one more match between two cycles is reached. Therefore, δ_max2=δ_max+δ_cyc (as provided at line 32).

[0228] C.2.5.B Construct Auto-Intersection

[0229] Construction starts with a WMTA A whose alphabet and semiring equal those of A₁ and that is otherwise empty (as provided at line 2). To each state q that will be created in WMTA A, two variables are assigned: (i) v[q]=q₁ indicating the corresponding state q₁ in A₁; and (ii) ξ[q]=(s,u), which sets forth the leftover string s of tape j (yet unmatched in tape k) and the leftover string u of tape k (yet unmatched in tape j).

[0230] Subsequently, an initial state i in A is created and pushed onto a stack (as provided at lines 4 and 17-26). As long as the stack is not empty, states q are taken from it and each of the outgoing transitions e₁εE(q₁) of the corresponding state q₁=v[q] in A₁ are followed (as provided in lines 5 and 6). A transition e₁ in A₁ is represented as eεE(q) in A, with the same label and weight. To compile the leftover strings ξ[q′]=(s′, u′) of its target q′=n(e) in A, the leftover strings ξ[q]=(s, u) of its source q=p(e) are concatenated with the j-th and k-th component of its label, l_j(e₁) and l_k(e₁), and the longest common prefix of the resulting strings s·l_j(e₁) and u·l_k(e₁) is removed (as provided in lines 7 and 13-16).

[0231] If both leftover strings s′ and u′ of q′ are non-empty (i.e., ≠ε) then they are incompatible and the path that is being followed is invalid. If either s′ or u′ is empty ε (i.e., =ε) then the current path is valid (at least up to this point) (as provided in line 8). Only in this case and only if the delay between s′ and u′ does not exceed δ_max2, a transition e in A is constructed corresponding to e₁ in A₁ (as provided in line 10). If its target q′=n(e) does not exist yet, it is created and pushed onto the stack (as provided in lines 9 and 17-26). The infinite unrolling of cycles is prevented by δ_max2.

[0232] C.2.5.C Test Regularity of Auto-Intersection

[0233] Finally, the constructed auto-intersection of the WMTA A is tested for regularity. From the above discussion of δ_max and δ_max2 it follows that if I_j,k(A₁⁽ⁿ⁾) is regular then none of the states that are both reachable and coreachable have |δ(ξ[q])|>δ_max. Further, if I_j,k(A₁⁽ⁿ⁾) is non-regular then a WMTA A⁽ⁿ⁾ built having a δ_max2, is bigger then a WMTA A⁽ⁿ⁾ built having δ_max, and hence has states with |δ(ξ[q])|>δ_max that are both reachable and coreachable. Since all states of A⁽ⁿ⁾ are reachable, due to the manner in which the WMTA A is constructed, it is sufficient to test for their coreachability (as provided at line 11) to know whether the WMTA A is regular (i.e., the delay of WMTA A⁽ⁿ⁾ will not exceed a limit δ max at any state q of A⁽ⁿ⁾).

[0234] C.2.6 Examples of Second Method

[0235] FIGS. 10 and 11 each present two different automata for illustrating the method for performing the auto-intersection operation set forth in FIG. 9 which results in a WMTA A⁽ⁿ⁾ that is regular. In contrast, FIG. 12 presents two automata for illustrating an example of the method for performing the auto-intersection operation set forth in FIG. 9 which results in a WMTA A⁽ⁿ⁾ that is not regular. In the FIGS. 10-12, the dashed portions are not construstructed by the method set forth in FIG. 9, and the states q in the Figures identified with the arrow “” have delay such that |δ(ξ[q])|>δ_max.

[0236] FIG. 10 illustrates the WMTA A₁⁽²⁾ and its auto-intersection A⁽²⁾. In FIG. 10, the WMTA A₁⁽²⁾ is the infinite set of string tuples {<ab^k, a^kb>|kεN} (where N is the set of natural numbers). Only one of those tuples, namely <ab, ab>, is in the relation of the auto-intersection A⁽²⁾=I_1,2(A₁⁽²⁾) because all other tuples contain different strings on tape 1 and tape 2. In accordance with the method for performing the auto-intersection operation set forth in FIG. 9, the following four steps are performed (namely, at (1) the limits are computed, at (2)-(3) the auto-intersection is computed, and at (3) the auto-intersection is tested for regularity):

[0237] $\begin{array}{cc}{\delta }_{\max }={\delta }_{\max 2}=1& \left(1\right)\\ ℛ\left(A_1^{\left(2\right)}\right)=\left\{\langle {\mathrm{ab}}^k,a^{k}b\rangle ❘k\in N\right\}& \left(2\right)\\ {ℐ}_{1,2}\left(ℛ\left(A_1^{\left(2\right)}\right)\right)=ℛ\left(A^{\left(2\right)}\right)=\left\{\langle {\mathrm{ab}}^1,a^{1}b\rangle \right\}& \left(3\right)\\ & \left(4\right)\end{array}$

[0238] FIG. 11 illustrates the WMTA A₁⁽³⁾ and its auto-intersection A⁽³⁾=I_1,2(A₁⁽³⁾), where the state 3 of A⁽²⁾ in FIG. 11 identified with the arrow “” has a delay such that |δ(ξ[q])|>δ_max. In accordance with the method for performing the auto-intersection operation set forth in FIG. 9, the following four steps are performed (namely, at (1)-(2) the limits are computed, at (3)-(4) the auto-intersection is computed, and at (5) the auto-intersection is tested for regularity):

[0239] $\begin{array}{cc}{\delta }_{\max }=2& \left(1\right)\\ {\delta }_{\max 2}=3& \left(2\right)\\ ℛ\left(A_1^{\left(3\right)}\right)=\left\{\langle a^k,a,x^{k}y\rangle ❘k\in N\right\}& \left(3\right)\\ {ℐ}_{1,2}\left(ℛ\left(A_1^{\left(3\right)}\right)\right)=ℛ\left(A^{\left(3\right)}\right)=\left\{\langle a^1,a,x^{1}y\rangle \right\}& \left(4\right)\\ q\in Q:\delta \left(\xi \left[q\right]\right)>{\delta }_{\max }\bigwedge \mathrm{coreachable}\left(q\right)\to \mathrm{regular}& \left(5\right)\end{array}$

[0240] FIG. 12 illustrates the WMTA A₁⁽³⁾ and its auto-intersection A⁽³⁾=I_1,2(A₁⁽³⁾), where the states 3, 8, and 11 of A⁽²⁾ in FIG. 12 identified with the arrow “” have a delay such that |δ(ξ[q])|>δ_max. In accordance with the method for performing the auto-intersection operation set forth in FIG. 9, the following four steps are performed (namely, at (1)-(2) the limits are computed, at (3)-(5) the auto-intersection is computed, and at (6) the auto-intersection is tested for regularity):

[0241] $\begin{array}{cc}{\delta }_{\max }=2& \left(1\right)\\ {\delta }_{\max 2}=3& \left(2\right)\\ ℛ\left(A_1^{\left(3\right)}\right)=\left\{\langle a^{k}a,{\mathrm{aa}}^h,x^k{\mathrm{yz}}^h\rangle ❘k,h\in N\right\}& \left(3\right)\\ {ℐ}_{1,2}\left(ℛ\left(A_1^{\left(3\right)}\right)\right)=\left\{\langle a^{k}a,{\mathrm{aa}}^h,x^k{\mathrm{yz}}^k\rangle ❘k\in N\right\}& \left(4\right)\\ {ℐ}_{1,2}\left(ℛ\left(A_1^{\left(3\right)}\right)\right)\supset ℛ\left(A^{\left(3\right)}\right)=\left\{\langle a^{k}a,{\mathrm{aa}}^k,x^k{\mathrm{yz}}^k\rangle ❘k\in \left[0,3\right]\right\}& \left(5\right)\\ \exists q\in Q:\delta \left(\xi \left[q\right]\right)>{\delta }_{\max }\bigwedge \mathrm{coreachable}\left(q\right)\to \mathrm{non}\text{-}\mathrm{regular}& \left(6\right)\end{array}$

[0242] C.3 Single-Tape Intersection

[0243] This section sets forth two alternate embodiments for performing, in one step, the single-tape intersection operation of two WMTAs A₁⁽ⁿ⁾ and A₂⁽ⁿ⁾ defined in section B.5 above. Instead of first building the cross-product, A₁⁽ⁿ⁾×A₂^(m), and then deleting most of its paths by the auto-intersection, I_j,n+k() operation defined in section B.4, both embodiments construct only the useful part of the cross-product. However, it will be understood by those skilled in the art that an alternate embodiment follows the definition and performs single-tape intersection in multiple steps.

[0244] The first embodiment is a method for performing single-tape intersection similar to known methods for performing composition, and is not adapted to handle WMTAs that contain epsilons on the intersected tapes, j and k. The first embodiment is referred to herein as IntersectCross (A₁, A₂,j, k), which is defined as:

[0245] $\begin{array}{c}\mathrm{IntersectCross}\left(A_1,A_2,j,k\right)={ℐ}_{j,n+k}\left(A_1^{\left(n\right)}\times A_2^{\left(m\right)}\right)\\ A_1^{\left(n\right)}\bigcap _{j,k}{A}_2^{\left(m\right)}={\stackrel{\_}{𝒫}}_{n+k}\left(\mathrm{IntersectCross}\left(A_1,A_2,j,k\right)\right)\end{array}$

[0246] It will be appreciated by those skilled in the art that the complementary projection, P_n+k(), operation defined in section B.2 may be integrated into the first embodiment in order to avoid an additional pass. However, it is kept separate from IntersectCross() because IntersectCross() may serve also as a building block of another operation where the complementary projection operation must be postponed (e.g., multi-tape intersection).

[0247] The second embodiment simulates the behavior of an epsilon-filter transducer for composing transducers with epsilon-transitions that is disclosed by Mohri, Pereira, and Riley in “A rational design for a weighted finite-state transducers”, published in Lecture Notes in Computer Science, 1436:144-158, 1998, which is incorporated herein by reference and hereinafter referred to as “Mohri's Epsilon-Filter Publication”. The second embodiment is referred to herein as IntersectCrossEps(A₁, A₂, j, k), where the suffix “Eps” expresses its suitability for WMTAs with epsilons on the intersected tapes, j and k.

[0248] Similar to the first embodiment, it will be appreciated by those skilled in the art that the complementary projection, P_n+k(), operation defined in section B.2 may be integrated into the second embodiment in order to avoid an additional pass. However, it is kept separate from lntersectCrossEps() because IntersectCrossEps() may serve also as a building block of other operations where the complementary projection operation must be postponed or extended (e.g., classical composition and multi-tape intersection).

[0249] C.3.1 Conditions

[0250] Both embodiments for performing the single-tape intersection operation operate under the conditions that:

• • [0251] (A) the semirings of the two automata A₁⁽ⁿ¹⁾ and A₂⁽ⁿ²⁾ equal to: K₁=K₂;
  • [0252] (B) the common semiring K=K₁=K₂ is commutative:
    ∀w₁, w₂ ε K: w₁{circle around (×)}w₂=w₂{circle around (×)}w₁;
  • [0253] (C) For lntersectCross(): Neither of the two intersected tapes contains ε:
    (∃e₁ ε E₁: l_j(e₁)=ε)Λ(∃e₂ ε E₂: l_k(e₂)=ε).

[0254] C.3.2 First Embodiment

[0255] FIG. 13 sets forth a method in pseudocode of the first embodiment for performing the single-tape intersection operation (i.e., A=IntersectCross(A₁, A₂, j, k)). Line 1 begins with a WMTA A whose semiring equals those of A₁ and A₂ and that is otherwise empty. At lines 3 and 12-18, the initial state i of the WMTA A is created from the initial states A₁ and A₂ and pushed onto the stack initialized at line 2. While the stack is not empty, the states q are popped from it and the states q₁ and q₂ are accessed that are assigned to q through μ[q] at lines 4 and 5.

[0256] At lines 6 and 7, each outgoing transition e₁ of q₁ is intersected with outgoing transitions e₂ of q₂. This succeeds at line 8 only if the j-th labeled component of e₁ equals the k-th labeled component of e₂, where j and k are the two intersected tapes of A₁ and A₂, respectively. Only if it succeeds at line 8 will a transition be created at line 10 for A whose label results from pairing l(e₁) with l(e₂) and whose target q′ corresponds with the pair of targets (n(e₁), n(e₂)). If q′ does not exist yet, it is created and pushed onto the stack at lines 9 and 12-18.

[0257] It will be appreciated by those skilled in the art that this embodiment may be readily adapted to operate with non-weighted multi-tape automata (MTAs) by removing the weights from line 10 and the semiring K₁ from line 1, and by replacing line 15 with “Final(q)←Final(q₁)ΛFinal(q₂)” in the pseudocode shown in FIG. 13.

[0258] C.3.3 Mohri's Epsilon-Filter

[0259] The second embodiment for performing the single-tape intersection operation makes use of an epsilon-filter transducer similar to its use in the composition of two transducers with an epsilon-transition that is described in by Mohri et al. in Mohri's Epsilon-Filter Publication. This section describes the use of Mohri's epsilon-filter by simulating its use.

[0260] FIG. 14 illustrates Mohri's epsilon-filter A_ε and two transducers A₁ and A₂. The transducers A₁ and A₂ are pre-processed for filtered composition (where x={φ₁, φ₂, ε₁, ε₂}) and each ε in tape 2 of A₁⁽²⁾ is replaced by an ε₁ and each ε in tape 1 of A₂⁽²⁾ by an ε₂. In addition, a looping transition labeled with ε:φ₁ is added to each state of A₁⁽²⁾, and a loop labeled with φ₂:ε to each state of A₂⁽²⁾. The pre-processed transducers are then composed with the filter A_ε⁽²⁾ in between: A₁ ⋄ A_ε ⋄ A₂. The filter controls how epsilon-transitions are composed along each pair of paths in A₁ and A₂, respectively. As long as there are equal symbols (ε or not) on the two paths, they are composed with each other, and the state does not change in A_ε from state zero. If a sequence of ε in A₁ but not in A₂ is encountered, the state advances in A₁, and the state does not change in A₂ from the state shown and in A_ε from state 1. If a sequence of ε in A₂ but not in A₁ is encountered, the state advances in A₂, and the state does not change in A₁ from the state shown and in A_ε from state 2.

[0261] C.3.4 Second Embodiment

[0262] FIG. 15 sets forth a method in pseudocode of the second embodiment for performing the single-tape intersection operation (i.e., A=IntersectCrossEps(A₁, A₂, j, k)). Unlike the embodiment set forth in section C.3.2 which builds the cross-product, A₁×A₂, and then deletes some of its paths by auto intersection, I_j,n+k(), this embodiment simulates the behavior of Mohri's epsilon-filter transducer described in section C.3.3 without Mohri's epsilon-filter transducer being present by adding an attribute at each state of the resulting WMTA A, thereby allowing only the useful parts of the cross-product to be constructed.

[0263] More specifically, in FIG. 15, line 1 begins with a WMTA A whose alphabet is the union of the alphabets of transducers A₁ and A₂, whose semiring equals those of A₁ and A₂, and that is otherwise empty. At lines 3 and 20-26, the initial state i of A is created from the states of A₁, A₂, and A_ε, and the initial state i is pushed onto the stack initialized at line 2. While the stack is not empty, states q are popped from it and the states q₁, q₂, and q_ε that are assigned to states q through V[q] at lines 4-5.

[0264] At lines 6-7, each outgoing transition e₁ of state q₁ is intersected with each outgoing transition e₂ of state q₂. This succeeds at line 8 only if the j-th labeled component of e₁ equals the k-th labeled component of e₂, where j and k are the two intersected tapes of A₁ and A₂, respectively, and if the corresponding transition in A_ε has target zero. Only if it succeeds at line 8 then at line 10, a transition in A is created (from the current source state) whose label results from pairing l(e₁) with l(e₂) and whose target state q′ corresponds with the triple of targets (n(e₁), n(e₂), 0). If state q′ does not exist yet, it is created and pushed onto the stack at lines 20-26.

[0265] Subsequently, all epsilon-transitions are handled in A₁ at lines 11-14 and in A₂ at lines 15-18. If an epsilon is encountered in A₁ and the automaton A_ε is in state 0 or in state 1, the current state advances in A₁, does not change in A₂, and advances to state 1 in A_ε. At lines 11-14, a transition in A is therefore created whose target corresponds to the triple (n(e₁), q₂, 1). Corresponding actions take place at lines 15-18 if an epsilon is encountered in A₂.

[0266] It will be appreciated by those skilled in the art that this embodiment may be readily adapted to operate with non-weighted multi-tape automata (MTAs) by removing the weights from lines 10, 14, and 18 and the semiring K₁ from line 1, and by replacing line 23 with “Final(q)←Final(q₁)ΛFinal(q₂)” in the pseudocode shown in FIG. 15.

[0267] It will further be appreciated by those skilled in the art that the embodiment shown in FIG. 15 for performing single-tape intersection of transducers A₁ and A₂ may be readily adapted to permit the composition of a first tape of the automata A₁ and a second tape of the automata A₂ while retaining one of the first and the second tapes and all other tapes of both transducers at the transitions where the tapes intersect by revising the labels at each transition E created at lines 10, 14, and 18 to remove one component (i.e., removing either the first tape or the second tape to eliminate redundant paths through complementary projection).

[0268] In addition it will be appreciated by those skilled in the art that the embodiment set forth in FIG. 15 may be readily adapted to perform a classical-composition operation where both the first tape of the automata A₁ and the second tape of the automata A₂ are removed while retaining all other tapes of both transducers at the transitions where the tapes intersect by revising the labels at each transition E created at lines 10, 14, and 18 to remove both components (i.e., removing both the first tape and the second tape).

[0269] C.3.5 Complexity

[0270] The worst-case complexity of both the first embodiment and the second embodiment for carrying out the a single-tape intersection is |E₁|·|E₂| in space and runtime (i.e., on the order of O(n²)). The complexity of the second embodiment is only linearly greater than that of the first, which may be significant for large WMTAs.

[0271] C.4 Multi-Tape Intersection

[0272] This section sets forth two embodiments for performing the multi-tape intersection operation of two WMTAs A₁⁽ⁿ⁾ and A₂^(m) defined in section B.6. The first embodiment is referred to herein as Intersect1(A₁⁽ⁿ⁾ and A₂^(m), j₁ . . . j_r, k₁ . . . k_r), follows the definition of multi-tape intersection while performing operations for cross-product, auto-intersection, and complementary projection. The second embodiment, which is more efficient, is referred to herein as Intersect2(A₁⁽ⁿ⁾ and A₂^(m), j₁ . . . j_r, k₁ . . . k_r), makes use of methods that perform cross-product and auto-intersection in one step (for intersection tape j₁ and k₁), and then the auto-intersection (for any intersecting tapes j_i with k_i, for i>1).

[0273] C.4.1 Conditions

[0274] Both embodiments for performing the multi-tape intersection operation operate under the conditions that:

• • [0275] (a) the semirings of the two automata A₁⁽ⁿ⁾ and A₂^(m) are equal to: K₁=K₂; and
  • [0276] (b) the common semiring K=K₁=K₂ is commutative:
    ∀w₁, w₂ ε K: w₁{circle around (×)}w₂=w₂{circle around (×)}w₁
  • [0277] (c) A₁⁽ⁿ⁾ and A₂^(m) have no cycles exclusively labeled with epsilon on any of the intersected tapes:
    j_iε[[1,n]] and k_iε[[1,m]], for i ε[[1,r]].

[0278] C.4.2 Embodiments

[0279] The first embodiment, Intersect1(), for carrying out the multi-tape intersection operation defined in section B.6 may be expressed in pseudocode as follows:

• [0280] (a) in one embodiment, while referring at line 1 to CrossPA() (to perform a cross product operation on the WMTAs A₁⁽ⁿ⁾ and A₂^(m)) set forth in FIG. 5, at line 3 to the first AutoIntersect() method (to perform the auto-intersection operation for each intersecting tapes j_i and k_i) set forth in FIG. 6, and at line 4 to complementary projection P (to eliminate redundant tapes) described in section B.2:

[0281]

	INTERSECT1(A1(n), A2(m), j1 . . . jr, k1 . . . kr) → A :
	1	A ← CROSSPA(A1(n), A2(m))
	2	for ∀i ∈ 1, r do
	3	A ← AUTOINTERSECT(A, ji, n + ki)
	4	A ← Pn+k1, . . . , n+kr(A)
	5	return A

or (b) in another embodiment, while referring at line 1 to CrossPA() (to perform a cross product operation on the WMTAs A₁⁽ⁿ⁾ and A₂^(m)) set forth in FIG. 5, at line 4 to the second AutoIntersect() method (to perform the auto-intersection operation for each intersecting tapes j_i and k_i) set forth in FIG. 9, and at line 6 to complementary projection P (to eliminate redundant tapes) described in section B.2:

[0282]

	INTERSECT1(A1(n), A2(m), j1 . . . jr, k1 . . . kr) → (A, boolean) :
	1	A ← CROSSPA(A1(n), A2(m))
	2	regular ← true
	3	for ∀i ∈ 1, r  do
	4	(A, reg) ← AUTOINTERSECT(A, ji, n + ki)
	5	regular ← regular  reg
	6	A ← Pn+k1, ... , n+kr(A)
	7	return (A, regular)

[0283] The second embodiment, Intersect2(), for carrying out the multi-tape intersection operation defined in section B.6 may be expressed in pseudocode as follows:

• [0284] (a) in one embodiment, while referring at line 1 to IntersectCrossEps() set forth in FIG. 15 (to perform the single-tape intersection operation on one pair of tapes, for tapes j₁ with k₁ on the WMTAs A₁⁽ⁿ⁾ and A₂^(m)), at line 3 to AutoIntersect() set forth in FIG. 6 (to perform the auto-intersection operation for any intersecting tapes j_i with k_i, for i>1), and at line 4 to complementary projection P (to eliminate redundant tapes) described in section B.2:

[0285]

	INTERSECT2(A1(n), A2(m), j1 . . . jr, k1 . . . kr) → A :
	1	A ← INTERSECTCROSSEPS(A1(n), A2(m), j1, k1)
	2	for ∀i ∈ 2, r do
	3	A ← AUTOINTERSECT(A, ji, n + ki)
	4	A ← Pn+k1, . . . , n+kr(A)
	5	return A

or (b) in another embodiment, while referring at line 1 to lntersectCrossEps() set forth in FIG. 15 (to perform the single-tape intersection operation on one pair of tapes j_i=1 with k_i=1 on the WMTAs A₁⁽ⁿ⁾ and A₂^(m)), at line 4 to AutoIntersect() set forth in FIG. 9 (to perform the auto-intersection operation for any intersecting tapes j_i with k_i, for i>1), and at line 6 to complementary projection P (to eliminate redundant tapes) described in section B.2:

[0286]

	INTERSECT2(A1(n), A2(m), j1 . . . jr, k1 . . . kr) → (A, boolean) :
	1	A ← INTERSECTCROSSEPS(A1(n), A2(m), j1, k1)
	2	regular ← true
	3	for ∀i ∈ 2, r do
	4	(A, reg) ← AUTOINTERSECT(A, ji, n + ki)
	5	regular ← regular  reg
	6	A ← Pn+k1, ... , n+kr(A)
	7	return (A, regular)

[0287] C.4.3 Example

[0288] By way of example, a solution is presented for compiling a multi-tape intersection of the MTA A₁⁽²⁾ and the MTA A₂⁽²⁾ in accordance with the last method presented immediately above to produce the regular MTA A⁽²⁾ as follows:

[0289] $A^{\left(2\right)}=A_1^{\left(2\right)}\bigcap _{\begin{array}{c}1,1\\ 2,2\end{array}}{A}_2^{\left(2\right)}={\stackrel{\_}{𝒫}}_{3,4}({ℐ}_{2,4}({ℐ}_{1,3}\left(A_1^{\left(2\right)}\times A_2^{\left(2\right)}\right)\mathrm{where}{A}_1^{\left(2\right)}\bigcap _{\begin{array}{c}1,1\\ 2,2\end{array}}{A}_2^{\left(2\right)}$
is given by:

[0290] $\begin{array}{cc}a& b\\ \varepsilon & A\end{array}{\left(\begin{array}{ccc}c& a& b\\ B& \varepsilon & C\end{array}\right)}^{*}\begin{array}{ccccc}\varepsilon & \varepsilon & \varepsilon & c& \varepsilon \\ A& B& C& \varepsilon & A\end{array}\underset{2,2}{\bigcap _{1,1}}\begin{array}{c}\varepsilon \\ A\end{array}{\left(\begin{array}{cccc}a& b& \varepsilon & c\\ B& \varepsilon & C& A\end{array}\right)}^{*}$
where the preceding each row of the matrix-like representation sets forth a tape of each WMTA (e.g., A₁⁽²⁾=<a, ε><b, A>(<c, B><a, ε><b, C>)*<ε, A><ε, B><ε, C><c, ε><ε, A>).

[0291] In accordance with the method set forth above, the multi-tape intersection operation is performed in three steps. First, the following automaton B₁⁽⁴⁾=I_1,3(A₁⁽²⁾×A₂⁽²⁾) is computed using single-tape intersection at line 1 (i.e., IntersectCrossEps() shown in FIG. 15) to obtain:

[0292] $\begin{array}{ccc}\varepsilon & a& b\\ \varepsilon & \varepsilon & A\\ \varepsilon & a& b\\ A& B& \varepsilon \end{array}{\left(\begin{array}{cccc}\varepsilon & c& a& b\\ \varepsilon & B& \varepsilon & C\\ \varepsilon & c& a& b\\ C& A& B& \varepsilon \end{array}\right)}^{*}\begin{array}{ccccc}\varepsilon & \varepsilon & \varepsilon & c& \varepsilon \\ A& B& C& \varepsilon & A\\ \varepsilon & \varepsilon & \varepsilon & c& \varepsilon \\ C& \varepsilon & \varepsilon & A& \varepsilon \end{array}$

[0293] Next, the following automaton B₂⁽⁴⁾=I_2,4(B₁⁽⁴⁾) is computed using auto-intersection at line 4 (i.e., AutoIntersect() shown in FIG. 9) to obtain:

[0294] $\begin{array}{ccc}\varepsilon & a& b\\ \varepsilon & \varepsilon & A\\ \varepsilon & a& b\\ A& B& \varepsilon \end{array}{\left(\begin{array}{cccc}\varepsilon & c& a& b\\ \varepsilon & B& \varepsilon & C\\ \varepsilon & c& a& b\\ C& A& B& \varepsilon \end{array}\right)}^1\begin{array}{ccccc}\varepsilon & \varepsilon & \varepsilon & c& \varepsilon \\ A& B& C& \varepsilon & A\\ \varepsilon & \varepsilon & \varepsilon & c& \varepsilon \\ C& \varepsilon & \varepsilon & A& \varepsilon \end{array}$

[0295] Finally, the following automaton A²= P_3,4(B₂⁽⁴⁾) is computed using complementary projection P at line 6 (described in section B.2) to obtain:

[0296] $\begin{array}{ccc}\varepsilon & a& b\\ \varepsilon & \varepsilon & A\end{array}{\left(\begin{array}{cccc}\varepsilon & c& a& b\\ \varepsilon & B& \varepsilon & C\end{array}\right)}^1\begin{array}{ccccc}\varepsilon & \varepsilon & \varepsilon & c& \varepsilon \\ A& B& C& \varepsilon & A\end{array}$

[0297] D. Applications

[0298] This section describes different applications for using the operations on WMTAs that are described herein.

[0299] D.1 General Use

[0300] FIG. 16 sets forth an automaton W⁽¹⁾ for illustrating a proposed operation for part-of-speech (POS) disambiguation and how it may be used in natural language processing (NLP). The automaton W⁽¹⁾ shown in FIG. 16 is an automaton that represents a natural-language sentence, with one word w on each transition. In general, such an automaton can contain one or several paths, according to whether the sentence was unambiguously tokenized or not. In the example shown in FIG. 16, the automaton W⁽¹⁾ contains two paths, corresponding to an ambiguous tokenization into either three or four words. FIG. 17 illustrates the process for one path π_j(W) (which occurs similarly for all paths) of the automaton W⁽¹⁾.

[0301] Table 2 sets forth resources that may be used in this example, which are encoded as WMTAs.

[0302]

	TABLE 2
		Type Of
	Automaton	Resource	Tapes
	N(2)	normalizer	original word form,
			normalized form
	L(3)	morphological	surface form,
		lexicon	lemma,
			POS-tag
	H(1)	Hidden	POS-tag sequence
		Markov Model

[0303] This example can be expressed in pseudocode as follows:

[0304]

1	for ∀e ∈ E(W) do
2	⁢ ( e ) ( 1 ) ← 2 ⁢ ( ⁢ ( e ) ( 1 ) ⁢ ⋂ 1 , 1 ⁢ N ( 2 ) )
3	⁢ ( e ) ( 3 ) ← ⁢ ( e ) ( 1 ) ⁢ ⋂ 1 , 1 ⁢ L ( 3 )
4	if (e)(3) = ⊥
5	then (e) ← . . .
6	W ( 4 ) ← W ( 4 ) ⁢ ⋂ 3 , 1 ⁢ H ( 1 )
7	W(4) ← bestPath(W(4))

[0305] At line 1 of the pseudocode above, each transition of 1-tape automaton W⁽¹⁾ i.e., each word of the sentence) is processed separately: A(e), representing the transition e ε E(W), is intersected with a normalizer N⁽²⁾ and only the second tape of the result is retained at line 2. Subsequently, A(e) is intersected with a 3-tape morphological lexicon ⁽³⁾ and becomes itself 3-tape as set forth in line 3 of the pseudocode and in FIG. 18. Specifically, FIG. 18 illustrates the intersection of an arc e of W⁽¹⁾ with a path π⁽³⁾ of the lexicon automaton ⁽³⁾. If at line 4 this operation fails, then something else is done with A(e) at line 5 such as intersecting A(e) with another n-tape automaton.

[0306] When all transitions have been separately processed, as described above, processing begins again on the sentence automaton W⁽⁴⁾, which at this point has four tapes due to previous operations. At line 6, the sentence automaton W⁽⁴⁾ is intersected with a 1-tape automaton H⁽¹⁾ that represents an HMM as shown in FIG. 19. More specifically, FIG. 19 illustrates the intersection of a path π_j⁽⁴⁾ of the sentence automaton W⁽⁴⁾ with a path π⁽¹⁾ of the HMM automaton H⁽¹⁾. Finally at line 7, only the best path of the sentence automaton W⁽⁴⁾ is retained.

[0307] D.2 Building a Lexicon from a Corpus

[0308] Using basic software programming utilities such as those forming part of UNIX, a list of inflected words with their POS-tags and their frequencies from an annotated corpus may be generated and stored in a file. Such a file may for example contain entries as shown in Table 3.

[0309]

	TABLE 3
	Inflected	POS-
	Word	tag	Frequency
	leave	NN	7
	leave	VB	154
	leaves	NNS	18
	leaves	VBZ	25
	leaving	VBG	67
	left	JJ	47
	left	VBD	118
	left	VBN	147

[0310] In one implementation, a WMTA lexicon may be created over the semiring < IR⁺, +, ×, 0, 1>, where each line of a file (as illustrated in Table 3) becomes the n-tape label of a path, such that each token of the line (or each column of Table 3) is placed onto a different tape, except for the last token (i.e., the frequency) which becomes the weight of the path. Using such an implementation and the example data in Table 3, a 2-tape corpus lexicon C⁽²⁾ may be constructed with paths having the labels l⁽²⁾ and the weights w as shown in Table 4.

[0311]

	TABLE 4
	Label l(2)	Weight w
	<leave, NN>	7
	<leave, VB>	154
	<leaves, NNS>	18
	<leaves, VBZ>	25
	<leaving, VBG>	67
	<left, JJ>	47
	<left, VBD>	118
	<left, VBN>	147

[0312] D.3 Enhancing a Lexicon with Lemmas

[0313] The corpus of lexicons and lemmas defined in section D.2 may be enhanced using another lexicon encoded as a non-weighted MTA, T⁽³⁾ having entries of the form T⁽³⁾<InflectedWord, Lemma, PosTag>, and containing at least those entries in Table 5.

[0314]

	TABLE 5
	Inflected
	Word	Lemma	PosTag
	leave	leave	NN
	leave	leave	VB
	leaves	leaf	NNS
	leaves	leave	NNS
	leaves	leave	VBZ
	leaving	leave	VBG
	left	left	JJ
	left	leave	VBD
	left	leave	VBN

[0315] Each of the entries in Table 5 is uniformly weighted with w=1.0 by assigning 1.0 to each transition and final state. To distribute the probability among different lemmas for equal inflected forms and POS tags, the lexicon T⁽³⁾ is normalized with respect to tape 1 and tape 3 using multi-tape intersection resulting in the entries shown in Table 6.

[0316]

	TABLE 6
	Lexicon T(3)	Weight w
	<leave, leave, NN>	1.0
	<leave, leave, VB>	1.0
	<leaves, leaf, NNS>	0.5
	<leaves, leave, NNS>	0.5
	<leaves, leave, VBZ>	1.0
	<leaving, leave, VBG>	1.0
	<left, left, JJ>	1.0
	<left, leave, VBD>	1.0
	<left, leave, VBN>	1.0

[0317] The intersection of the lexicon T⁽³⁾ (having values shown in column 1 of Table 6) with the corpus lexicon C⁽²⁾ (having values shown in Table 4) on the tapes of inflected forms and POS-tags, respectively, is defined as:

[0318] $L^{\left(3\right)}=T^{\left(3\right)}\underset{3,2}{\bigcap _{1,1}}{C}^{\left(2\right)}$
and results in the lexicon L⁽³⁾ the entries of which are shown in Table 7.

[0319]

	TABLE 7
	Lexicon L(3)	Weight w
	<leave, leave, NN>	7
	<leave, leave, VB>	154
	<leaves, leaf, NNS>	9
	<leaves, leave, NNS>	9
	<leaves, leave, VBZ>	25
	<leaving, leave, VBG>	67
	<left, left, JJ>	47
	<left, leave, VBD>	118
	<left, leave, VBN>	147

[0320] Finally, the lexicon L⁽³⁾ (having values shown in Table 7) is normalized (i.e., all entries with the same inflected form have a weight w that sums to one) with respect to inflected forms to provide the entries shown in Table 8.

[0321]

	TABLE 8
	Lexicon L(3)	Weight w
	<leave, leave, NN>	0.043
	<leave, leave, VB>	0.957
	<leaves, leaf, NNS>	0.209
	<leaves, leave, NNS>	0.209
	<leaves, leave, VBZ>	0.581
	<leaving, leave, VBG>	1.000
	<left, left, JJ>	0.151
	<left, leave, VBD>	0.378
	<left, leave, VBN>	0.471

[0322] D.4 Normalizing a Lexicon

[0323] In this example, the conditional normalization of a WMTA, A₁^(n+m), is compiled over the semiring < IR⁺, +, ×, 0, 1>, with respect to some of its tapes, using a generalization of the method proposed by Eisner in “Parameter Estimation For Probabilistic Finite-state Transducers”, in Proceedings of the 40th Annual Meeting, pages 1-8, Philadelphia, Pa., USA, Association For Computer Linguistics, 2002, which is incorporated herein by reference. Consider, for instance, each string tuple u^(n+m) in the language ₁^(n+m) of A₁^(n+m) to be a pair of string tuples (i.e., u^(n+m)=s⁽ⁿ⁾: v^(m)). Normalizing A₁^(n+m) conditionally with respect to n+1 to n+m means compiling for each u^(n+m) the probability of s⁽ⁿ⁾ in the context of v^(m).

[0324] Suppose, originally the weight of each u^(n+m) is its frequency (i.e., the frequency of concurrence of tuples s⁽ⁿ⁾ and v^(m)) is given by:
w_A1(u^(n+m))=f(s⁽ⁿ⁾:v^(m))

[0325] The context tapes of z,900 ₁^(n+m) are projected and the following context language is obtained:
_c^(m)=P_{n+1, . . . , n+m}(₁^(n+m))

[0326] Each string tuple v^(m) ε _c^(m) of the determinized context automaton A_c^(m) has the weight given by:

[0327] $w\left(v^{\left(m\right)}\right)=\underset{v_i^{\left(m\right)}=v^{\left(m\right)}}{\oplus }{w}_{A_1}\left(s_i^{\left(n\right)}:v_i^{\left(m\right)}\right)=f\left(v^{\left(m\right)}\right)$

[0328] The weight of all string tuples v^(m) ε _c^(m) is inversed by inversing the weight of each transition and each final state of A_c^(m). The resulting weight of each v^(m) is given by:
w_Ac(v^(m))=f(v^(m))⁻¹

[0329] Finally, the original A₁^(n+m) is intersected with A_c^(m) on all context tapes, and an automaton A₂^(n+m) obtained with the language given by:

[0330] ${ℒ}_2^{\left(n+m\right)}={ℒ}_1^{\left(n+m\right)}\underset{n+m,m}{\underset{\dots }{\bigcap _{n+1,1}}}{ℒ}_c^{\left(m\right)}$

[0331] All string tuples u^(n+m)=s⁽ⁿ⁾: v^(m) ε ₂^(n+m) have the weight given by:

[0332] $\begin{array}{c}{w}_{A_2}\left(u^{\left(n+m\right)}\right)=w_{A_1}\left(s^{\left(n\right)}:v^{\left(m\right)}\right)\otimes w_{A_c}\left(v^{\left(m\right)}\right)\\ =f\left(s^{\left(n\right)}:v^{\left(m\right)}\right)·{f\left(v^{\left(m\right)}\right)}^{-1}\\ =p\left(s^{\left(n\right)}|v^{\left(m\right)}\right)\end{array}$

[0333] Those skilled in the art will appreciate that this approach does not require the context tapes to be consecutive.

[0334] To compile the joint normalization of a WMTA, A₁⁽ⁿ⁾, over the semiring < IR⁺, +, ×, 0, 1>, the frequency of each string tuple s(n)ε (A₁⁽ⁿ⁾) is multiplied with the inverse of the total frequency f_T of all string tuples. The total frequency f_T can be obtained by replacing the label of each transition of A₁⁽ⁿ⁾ with ε and applying an epsilon removal (as described for example by Mohri in “Generic epsilon-removal and input epsilon-normalization algorithms for weighted transducers”, in International Journal of Foundations of Computer Science, 13(1):129-143, 2002, which is incorporated herein by reference). The resulting automaton, A_T, has one single state, without any outgoing transitions, and a final weight that equals the total frequency, Q=f_T. The final weight Q is inversed and A₁⁽ⁿ⁾ is concatenated with A_T. All string tuples s⁽ⁿ⁾ ε (A₂⁽ⁿ⁾) of the resulting automaton A₂⁽ⁿ⁾=A₁⁽ⁿ⁾A have the weight given by:

[0335] $\begin{array}{c}{w}_{A_2}\left(s^{\left(n\right)}\right)=w_{A_1}\left(s^{\left(n\right)}\right)\otimes w_{A_T}\left({\varepsilon }^{\left(n\right)}\right)\\ =f\left(s^{\left(n\right)}\right)·f_T^{-1}\\ =p\left(s^{\left(n\right)}\right)\end{array}$

[0336] D.5 Using a Lexicon

[0337] When using a lexicon WMTA, A(n), the following may be specified: r input tapes, j₁ to j_r, and x output tapes, k₁ to k_x, which do not have to be consecutive. A weighted r-tuple of input strings, s^(r), may first be converted into an input WMTA, I^(r), having one single path with the specified label, s^(r), and weight, w(s^(r)). Subsequently to obtaining the output WMTA, O^(x), whose language contains all weighted x-tuples of output strings, v^(x), the following multi-tape intersection and projection may be used:

[0338] $O^{\left(x\right)}={𝒫}_{k_1,\dots ,k_x}\left(A^{\left(n\right)}\bigcap _{\underset{\underset{j_r,r}{\cdots }}{j_1,1}}{I}^{\left(r\right)}\right)$

[0339] D.6 Searching for Similarities

[0340] In the applications described in this section, string tuples s(n) may searched for in the language ₁⁽ⁿ⁾ of a WMTA A₁⁽ⁿ⁾ whose strings s_j1 to s_jr are similar to its strings s_k1 to s_kr, respectively. The comparison of each pair of tapes, j_i and k_i, may be done independently form all other pairs of tapes. Hence the task may be reduced to comparing two tapes, j and k.

[0341] First, a 2-tape automaton, R⁽²⁾, may be created, whose language, _R⁽²⁾, describes the requested relation between tape j and k of ₁⁽ⁿ⁾. To obtain the language given by the relation:
₂⁽ⁿ⁾ ⊂ ₁⁽ⁿ⁾
and where when this relation holds between the languages, the following language may be compiled:

[0342] ${ℒ}_2^{\left(n\right)}={ℒ}_1^{\left(n\right)}\bigcap _{\underset{k,2}{j,1}}{R}^{\left(2\right)}$

[0343] In one specific example, suppose there exists an English-German dictionary that is encoded as a 3-tape WMTA, with each entry being of the form <English, German, PosTag>. To find all words that are similar in the two languages, while having the same POS tag, a 2-tape automaton, R⁽²⁾, may be created that describes this similarity by using, for example, either of the approaches referred to herein as the “Levenshtein Distance” or the “List of Grapheme Correspondences” described below.

[0344] The Levenshtein Distance between two strings is the minimal number of intersections, deletions or substitutions of symbols that are needed to transform one string into the other, as disclosed by Schulz and Mihov in “Fast string correction with Levenshtein automata”, International Journal on Document Analysis and Recognition, 5(1):67-85, 2002, which is incorporated herein by reference. A WMTA, Lv_d⁽²⁾, having a Levenshtein Distance d between the members s₁ and s₂ of all string tuples s⁽²⁾ ε (Lv_d⁽²⁾) can be compiled from the regular expression:
Lv_d⁽²⁾=((?:i?)*(?:? ∪ ?:ε ∪ ε:?)(?:i?)*)^d
where ? means any symbol (i.e., ? ε {a, b, c, . . . }), and :i is an identity paring such that (?:i ?) ε {a:a, b:b, c:c, . . . }, whereas (?:?) ε {a:a, a:b, b:a, . . . }.

[0345] The List of Grapheme Correspondences by manually writing a list of synchronic grapheme correspondences resulting from historical phonological alterations in the English and German languages may, for example, contain the entries in Table 9 (for English in column 1 and German in column 2).

[0346]

	TABLE 9
	ENGLISH	GERMAN
	th	d
	th	ss
	d	t

[0347] Using the lexicon construction method described in sections D.2-D.5, the list of entries in Table 9 may be used to construct a lexicon-like automaton, Ch⁽²⁾, that encodes these changes in the form set forth in Table 10 (where column 1 identifies the lexicon, and column 2 identifies the weight).

[0348]

	TABLE 10
	Lexicon Ch(2)	Weight w
	<th, d>	1
	<th, ss>	1
	<d, t>	1

[0349] The lexicon-like automaton Ch⁽²⁾ may then be used to obtain a WMTA, R⁽²⁾, whose language describes the relation between any English word and its potential German form that is given by:
R⁽²⁾=(Ch⁽²⁾ ∪ ?:i?)⁺

[0350] D.7 Preserving Intermediate Transduction Results

[0351] In this section, the advantage of WMTAs through transduction cascades, which are frequently used in language and speech processing, is illustrated. In a (classical) weighted transduction cascade, T₁⁽²⁾ . . . T_r⁽²⁾, a set of weighted strings, encoded as a weighted acceptor, L₀⁽¹⁾, is composed with the transducer, T₁⁽²⁾, on its input tape as shown in FIG. 20. The output projection of this composition is the first intermediate result, L₁⁽¹⁾, of the cascade. It is further composed with the second transducer, T₂⁽²⁾, which leads to the second intermediate result, L₂⁽¹⁾, etc. The projection of the last transducer is the final result, L_r⁽¹⁾, which is defined as follows:
L_i⁽¹⁾=P₂(L_i−1⁽¹⁾⋄T_i⁽²⁾) for i ε[[1,r]]
At any point in this cascade, previous results cannot be accessed.

[0352] In a weighted transduction cascade, A₁⁽ⁿ¹⁾ . . . A_r^(nr), that uses WMTAs and multi-tape intersection, intermediate results can be preserved and used by all subsequent transductions. For example, assuming the two previous results at each point in the cascade (except in the first transduction) are to be used in computing the results. This operation requires all intermediate results, L₁⁽²⁾, to have two tapes as shown in FIG. 2 and defined as follows:

[0353] $\begin{array}{c}{L}_1^{\left(2\right)}=L_0^{\left(1\right)}\bigcap _{1,1}{A}_1^{\left(2\right)}\\ L_i^{\left(2\right)}={𝒫}_{2,3}\left(L_{i-1}^{\left(2\right)}\bigcap _{\underset{2,2}{1,1}}{A}_i^{\left(3\right)}\right)\mathrm{for}i\in 〚2,r-1〛\\ L_r^{\left(2\right)}={𝒫}_3\left(L_{i-1}^{\left(2\right)}\bigcap _{\underset{2,2}{1,1}}{A}_r^{\left(3\right)}\right)\end{array}$