1. Non-deterministic Tree Automata Models for Statistical Machine Translation
Chiara Moraglia

2. Mathematical Linguistics
Branch of computational linguistics
The study of mathematical structures and methods that pertain to linguistics.
Combines aspects of computer science, mathematics and linguistics.

3. Problem: translational ambiguity
Words: anchor
Sentences : Cleaning fluid can be dangerous Claire kicked the bucket.

4. Statistical Machine Translation
Machine translation that keeps in mind the problem of ambiguity.
A sequence of reordering decisions and word translation decisions, each with a probability assigned based upon linguistic data.
2 main reordering models:
1) phrase-based models: re-align phrases (strings of words)
2) syntax-based models: can use tree transducers to permute trees (syntactic structure) with words as leaves

5. Example of a syntax-based translation

6. My project
Generalize the work on tree automata and tree transductions to non-deterministic models and explore the equivalence properties that were proven to hold in the deterministic case.

7. Tree
A hierarchical collection of labeled nodes connected by edges, starting at a root node

8. Tree Transducers
A tree transducer is a 5-tuple <F,H,Q, qin,R> where
i) F is a functional signature of input symbols
ii) H is a functional signature of output symbols
iii) Q is a finite set of states
iv) qin∈Q is the initial state
v) R is a finite set of rules <q, φ> ζ where ζ is
<q’, ψ>
h(< q1, ψ1>,…,< qk, ψk>)
Φ gives the conditions the current node must satisfy,
Ψ says which node to go to from the current node
(Courcelle & Engelfriet, 2012)

9. Functional Signature
A functional signature is a set of function symbols, each with an associated arity ρ(f) (the number of arguments the function takes on)
E.g. f(x), ρ(f)=1
h(x,y,z), ρ(h)=3
(Courcelle & Engelfriet, 2012)

10. Example of a Tree Transduction
i) F={f,a,b} where ρ(f)=2, ρ(a)=ρ(b)=0
ii) H= {a,b,ε} where ρ(a)=ρ(b)= 1, ρ(ε)=0
iii) Q={qin,q1,q2}
iv) qin∈Q is the initial state
v) R=
<qin, labf(x1)> <qin, down1>
<qin, labx(x1)^bri(x1)> x(< qi, up>)
<q1, True> < qin, down2 >
<q2, bri(x1)> < qi, up >
<q2, rt(x1)> ε
<qin, labx(x1)^rt(x1)> x(< q2, stay>)
(Courcelle & Engelfriet, 2012)

11. Graphical Representation
a or b(<a or b &1st child, >) q1
<f, >
<True, >
<1st child, >
qin
<a or b & root, stay>
<2nd child, > q2 ε
a or b (<a or b & 2nd child, >)
<root, stay>

12. Example
input tree output tree f a a b b ε

13. Deterministic vs. Non-deterministic
A tree transducer is deterministic if the state and the position in the tree uniquely determine what rule should be applied
Otherwise, it is non-deterministic
E.g. <qin, labf(x1)> <qin, down>
<qin, labf(x1)> <q1, up>

14. Non-deterministic Tree Transducer
g(<f, >)
<a, stay>
qin a
h(<f, >)
Modified from (Fülöp, 1981)

15. Example
input
possible outputs
f g h g h
f g h h g
a a a a a

16. Application to Statistical Machine Translation
The possible output trees would be assigned probabilities
Then the words would be translated into the target language

17. References
Courcelle, B., & Engelfriet, J. (2012). Graph Structure and Monadic Second-Order Logic. Cambridge: University Press. Fülöp. (1981). On attributed tree transducers. Acta Cybernetica, 5, p Knight, K., & Koehn, P. What’s new in statistical machine translation [PDF document]. Retrieved from Tree (data structure). (n.d.). Retrieved from