Admin GRAMMARS. Formally. Context free grammar 2/20/19. Assignment 3 out: due next Monday. Quiz #1 G = (NT, T, P, S)
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1 Admin Assignment 3 out: due next Monday Quiz #1 GRAMMAR David Kauchak C159 pring 2019 some slides adapted from Ray Mooney Context free grammar Formally G = (NT, T, P, ) left hand side (single symbol) right hand side (one or more symbols) NT: finite set of nonterminal symbols T: finite set of terminal symbols, NT and T are disjoint P: finite set of productions of the form A a, A Î NT and a Î (T È NT)* Î NT: start symbol 1
2 CFG: Example Many possible CFGs for English, here is an example (fragment): What can we do? What can we do? 2
3 DetP N DetP N the boy 3
4 the boy likes the boy likes a girl ; Order of Derivation Irrelevant Derivations of CFGs tring rewriting system: we derive a string DetP N V Derivation history shows constituent tree: the boy likes a girl DetP N V the boy likes DetP N the boy likes a girl a girl 4
5 Parsing Parsing is the field of NLP interested in automatically determining the syntactic structure of a sentence Parsing can be thought of as determining what sentences are valid English sentences As a byproduct, we often can get the structure Parsing Given a CFG and a sentence, determine the possible parse tree(s) -> -> N -> PRP -> N PP -> V -> V PP PP -> IN N PRP -> I V -> eat N -> sushi N -> tuna IN -> with What parse trees are possible for this sentence? How did you do it? What if the grammar is much larger? Parsing Parsing ambiguity PP PP -> -> PRP -> N PP -> N -> V -> V PP PP -> IN N PRP -> I V -> eat N -> sushi N -> tuna IN -> with PP PP -> -> PRP -> N PP -> N -> V -> V PP PP -> IN N PRP -> I V -> eat N -> sushi N -> tuna IN -> with PRP V N IN N PRP V N IN N PRP V N IN N PRP V N IN N What is the difference between these parses? How can we decide between these? 5
6 A imple PCFG Probabilities! PP 0.3 PP P 1.0 P with 1.0 V saw 1.0 PP 0.4 astronomers 0.1 ears 0.18 saw 0.04 stars 0.18 telescope 0.1 Just like n-gram language modeling, PCFGs break the sentence generation process into smaller steps/probabilities The probability of a parse is the product of the PCFG rules What are the different interpretations here? Which do you think is more likely? = 1.0 * 0.1 * 0.7 * 1.0 * 0.4 * 0.18 * 1.0 * 1.0 * 0.18 = = 1.0 * 0.1 * 0.3 * 0.7 * 1.0 * 0.18 * 1.0 * 1.0 * 0.18 =
7 Parsing problems Pick a model e.g. CFG, PCFG, PCFG: Training If we have example parsed sentences, how can we learn a set of PCFGs? Train (or learn) a model What CFG/PCFG rules should I use? Parameters (e.g. PCFG probabilities)? What kind of data do we have? Parsing Determine the parse tree(s) given a sentence Tree Bank John V PP put the dog in the pen John V PP put the dog in the pen.. upervised PCFG Training Det A N 0.5 PP 0.3 PropN 0.2 A ε 0.6 A Adj A 0.4 PP Prep 1.0 V 0.7 PP 0.3 English Extracting the rules Estimating PCFG Probabilities PRP V N IN PP N PRP PRP I V eat N PP N sushi PP IN N IN with N tuna What CFG rules occur in this tree? We can extract the rules from the trees PRP PRP I V eat N PP N sushi PP 0.3 PP P 1.0 P with 1.0 V saw 1.0 How do we go from the extracted CFG rules to PCFG rules? 7
8 Estimating PCFG Probabilities Estimating PCFG Probabilities Extract the rules from the trees Calculate the probabilities using MLE α β P(α β α) = count(α β) count(α β) = count(α γ) count(α) γ p(α β α) Occurrences 10 V 3 PP 2 N 7 N PP 3 DT N 6 P( V ) = P( V ) = P( V ) =? count( V ) count() = 3 15 Grammar Equivalence Grammar Equivalence What does it mean for two grammars to be equal? Weak equivalence: grammars generate the same set of strings Grammar 1: DetP N and Grammar 2: a N the N trong equivalence: grammars have the same set of derivation trees With CFGs, possible only with useless rules Grammar 2: a N the N Grammar 3: a N the N, DetP many 8
9 Normal Forms CNF Grammar There are weakly equivalent normal forms (Chomsky Normal Form, Greibach Normal Form) A CFG is in Chomsky Normal Form (CNF) if all productions are of one of two forms: A B C with A, B, C nonterminals A a, with A a nonterminal and a a terminal Every CFG has a weakly equivalent CFG in CNF -> -> VB -> VB PP -> DT NN -> NN -> PP PP -> IN DT -> the IN -> with VB -> film VB -> trust NN -> man NN -> film NN -> trust -> -> VB -> 2 PP 2 -> VB -> DT NN -> NN -> PP PP -> IN DT -> the IN -> with VB -> film VB -> trust NN -> man NN -> film NN -> trust Probabilistic Grammar Conversion Original Grammar Chomsky Normal Form tates Aux Pronoun Proper-Noun Det Nominal Nominal Noun Nominal Nominal Noun Nominal Nominal PP Verb Verb PP PP Prep X1 X1 Aux book include prefer Verb PP I he she me Houston NWA Det Nominal Nominal book flight meal money Nominal Nominal Noun Nominal Nominal PP book include prefer Verb PP PP Prep What is the capitol of this state? Helena (Montana) 9
10 Grammar questions Can we determine if a sentence is grammatical? Given a sentence, can we determine the syntactic structure? Next time: parsing Can we determine how likely a sentence is to be grammatical? to be an English sentence? Can we generate candidate, grammatical sentences? 10
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