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**Non-deterministic Tree Automata Models for Statistical Machine Translation**

Chiara Moraglia

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**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.

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**Problem: translational ambiguity**

Words: anchor Sentences : Cleaning fluid can be dangerous Claire kicked the bucket.

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**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

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**Example of a syntax-based translation**

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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.

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Tree A hierarchical collection of labeled nodes connected by edges, starting at a root node https://upload.wikimedia.org/wikipedia/commons/f/f7/Binary_tree.svg

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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)

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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)

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**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)

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**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>

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Example input tree output tree f a a b b ε

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**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>

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**Non-deterministic Tree Transducer**

g(<f, >) <a, stay> qin a h(<f, >) Modified from (Fülöp, 1981)

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

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**Application to Statistical Machine Translation**

The possible output trees would be assigned probabilities Then the words would be translated into the target language

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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

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