Structure and Ambiguity Removing Ambiguity

1. MA/CSSE 474 Theory of Computation Structure and Ambiguity Removing Ambiguity Chomsky Normal FormPushdown Automata Intro(who is he foolin', thinking that there will be time to get to this?)

2. Structure Context free languages: We care about structure. E E + E idE * E 3 idid 5 7

3. Derivations and parse trees Parse trees capture essential structure: 1 2 3 4 5 6 SSS (S)S ((S))S (())S (())(S)  (())() SSS (S)S ((S))S ((S))(S)  (())(S)  (())() 1 2 3 5 4 6 S S S ( S ) ( S ) ( S )  

4. Parse Trees A parse tree, derived from a grammar G = (V, , R, S), is a rooted, ordered tree in which: ● Every leaf node is labeled with an element of  {}, ● The root node is labeled S, ● Every other node is labeled with an element of N, and ● If m is a non-leaf node labeled X and the (ordered) children of m are labeled x1, x2, …, xn, then R contains the rule Xx1x2, … xn.

5. Structure in English S NP VP Nominal VNP AdjsNNominal AdjN the smart cat smells chocolate

6. Generative Capacity Because parse trees matter, it makes sense, given a grammar G, to distinguish between: ● G’s weak generative capacity, defined to be the set of strings, L(G), that G generates, and ● G’s strong generative capacity, defined to be the set of parse trees that G generates.

7. Algorithms Care How We Search S SS ( S ) ( S ) ( S )   Algorithms for generation and recognition must be systematic. They typically use either the leftmost derivation or the rightmost derivation.

8. Derivations of The Smart Cat • A left-most derivation is: SNP VP the NominalVP the AdjsNVP the AdjNVP the smart NVP the smart cat VP the smart cat V NP the smart cat smells NP the smart cat smells Nominal the smart cat smells N the smart cat smells chocolate • A right-most derivation is: SNP VPNPV NPNPV NominalNPV N NPV chocolate NP smellschocolate  the Nominal smellschocolate  the AdjsN smellschocolate  the Adjs cat smellschocolate  the Adj cat smellschocolate  the smart cat smellschocolate

9. Ambiguity A grammar is ambiguous iff there is at least one string in L(G) for which G produces more than one parse tree*. For many applications of context-free grammars, this is a problem. Example: A programming language. • If there can be two different structures for a string in the language, there can be two different meanings. • Not good! * Equivalently, more than one leftmost derivation, or more than one rightmost derivation.

10. An Arithmetic Expression Grammar EE + E EEE E (E) Eid

11. Inherent Ambiguity Some CF languages have the property that every grammar for them is ambiguous. We call such languages inherently ambiguous. Example: L = {anbncm: n, m 0}  {anbmcm: n, m 0}.

12. Inherent Ambiguity L = {anbncm: n, m 0}  {anbmcm: n, m 0}. One grammar for L has the rules: SS1 | S2 S1S1c | A /* Generate all strings in {anbncm}. AaAb |  S2aS2 | B /* Generate all strings in {anbmcm}. BbBc |  Consider any string of the form anbncn. It turns out that L is inherently ambiguous.

13. Ambiguity and undecidability Both of the following problems are undecidable*: • Given a context-free grammar G, is G ambiguous? • Given a context-free language L, is L inherently ambiguous? Informal definition of undecidable for the first problem:There is no algorithm (procedure that is guaranteed to always halt) that, given a grammar G, determines whether G is ambiguous.

14. But We Can Often Reduce Ambiguity We can get rid of: ●some rules like S, ● rules with symmetric right-hand sides, e.g., SSS EE + E ● rule sets that lead to ambiguous attachment of optional postfixes, such as if … else ….

15. A Highly Ambiguous Grammar S SSS S (S)

16. Resolving the Ambiguity with a Different Grammar The biggest problem is the  rule. A different grammar for the language of balanced parentheses: S*  S* S SSS S (S) S () We'd like to have an algorithm for removing all -productions… … except for the case where - is actually in the language; then we introduce a new start symbol and have one -production whose left side is that symbol.

17. Nullable Nonterminals Examples: SaTa T SaTa TA B A B A nonterminal X is nullable iff either: (1) there is a rule X, or (2) there is a rule XPQR… and P, Q, R, … are all nullable.

18. Nullable Nonterminals A nonterminal X is nullable iff either: (1) there is a rule X, or (2) there is a rule XPQR… and P, Q, R, … are all nullable. So compute N, the set of nullable nonterminals, as follows: 1. Set N to the set of nonterminals that satisfy (1). 2. Repeat until an entire pass is made without adding anything to N Evaluate all other nonterminals with respect to (2). If any nonterminal satisfies (2) and is not in N, insert it.

19. A General Technique for Getting Rid of -Rules Definition: a rule is modifiable iff it is of the form: PQ, for some nullable Q. removeEps(G: cfg) = 1. Let G = G. 2. Find the set N of nullable nonterminals in G. 3. Repeat until G contains no modifiable rules that haven’t been processed: Given the rule PQ, where QN, add the rule P if it is not already present and if  and if P. 4. Delete from G all rules of the form X. 5. Return G. L(G) = L(G) – {}

20. An Example G = {{S, T, A, B, C, a, b, c}, {a, b, c}, R, S), R = { SaTa TABC AaA | C BBb | C Cc |  } removeEps(G: cfg) = 1. Let G = G. 2. Find the set N of nullable nonterminals in G. 3. Repeat until G contains no modifiable rules that haven’t been processed: Given the rule PQ, where QN, add the rule P if it is not already present and if  and if P. 4. Delete from G all rules of the form X. 5. Return G.

21. What If L? atmostoneEps(G: cfg) = 1. G = removeEps(G). 2. If SG is nullable then /* i. e., L(G) 2.1 Create in G a new start symbol S*. 2.2 Add to RG the two rules: S*  S* SG. 3. Return G.

22. But There Can Still Be Ambiguity S*  What about ()()() ? S* S SSS S (S) S ()

23. Eliminating Symmetric Recursive Rules S*  S* S SSS S (S) S () Replace SSS with one of: SSS1 /* force branching to the left SS1S /* force branching to the right So we get: S*  SSS1 S* S SS1 S1 (S) S1 ()

24. Eliminating Symmetric Recursive Rules So we get: S*  S* S SSS1 SS1 S1 (S) S1 () S* S S S1 S S1 S1 ( ) ( ) ( )

25. Arithmetic Expressions EE + E EEE E (E) Eid } Problem 1: Associativity E E EE EE EE EE id  id  id id  id  id

26. Arithmetic Expressions EE + E EEE E (E) Eid } Problem 2: Precedence E E EE EE EE EE id  id + id id  id + id

27. Arithmetic Expressions - A Better Way EE + T ET TT * F TF F (E) F id

28. Ambiguous Attachment The dangling else problem: <stmt> ::= if <cond> then <stmt> <stmt> ::= if <cond> then <stmt> else <stmt> Consider: if cond1then if cond2then st1else st2

29. The Java Fix <Statement> ::= <IfThenStatement> | <IfThenElseStatement> | <IfThenElseStatementNoShortIf> <StatementNoShortIf> ::= <block> | <IfThenElseStatementNoShortIf> | … <IfThenStatement> ::= if ( <Expression> ) <Statement> <IfThenElseStatement> ::= if ( <Expression> ) <StatementNoShortIf> else <Statement> <IfThenElseStatementNoShortIf> ::= if ( <Expression> ) <StatementNoShortIf> else <StatementNoShortIf> <Statement> <IfThenElseStatement> if (cond) <StatementNoShortIf> else <Statement>

30. Hidden: Going Too Far SNPVP NP the Nominal | Nominal | ProperNoun | NPPP Nominal N | Adjs N Ncat | girl | dogs | ball | chocolate | bat ProperNoun Chris | Fluffy AdjsAdj Adjs | Adj Adjyoung | older | smart VPV | V NP | VPPP Vlike | likes | thinks | hits PPPrep NP Prepwith ● Chris likes the girl with the cat. ● Chris shot the bear with a rifle.

31. Normal Forms A normal form F for a set C of data objects is a form, i.e., a set of syntactically valid objects, with the following two properties: ● For every element c of C, except possibly a finite set of special cases, there exists some element f of F such that f is equivalent to c with respect to some set of tasks. ● F is simpler than the original form in which the elements of C are written. By “simpler” we mean that at least some tasks are easier to perform on elements of F than they would be on elements of C.

32. Normal Form Examples ● Disjunctive normal form for database queries so that they can be entered in a query-by-example grid. ● Jordan normal form for a square matrix, in which the matrix is almost diagonal in the sense that its only non-zero entries lie on the diagonal and the superdiagonal. ● Various normal forms for grammars to support specific parsing techniques.

33. Normal Forms for Grammars Chomsky Normal Form, in which all rules are of one of the following two forms: ● Xa, where a, or ● XBC, where B and C are elements of V - . Advantages: ● Parsers can use binary trees. ● Exact length of derivations is known: S A B A A B B aab B B bb

34. Normal Forms for Grammars Greibach Normal Form, in which all rules are of the following form: ● Xa, where a and  (V - )*. Advantages: ● Every derivation of a string s contains |s| rule applications. ● Greibach normal form grammars can easily be converted to pushdown automata with no - transitions. This is useful because such PDAs are guaranteed to halt.

35. Theorems: Normal Forms Exist Theorem: Given a CFG G, there exists an equivalent Chomsky normal form grammar GC such that: L(GC) = L(G) – {}. Proof: The proof is by construction. Theorem:Given a CFG G, there exists an equivalent Greibach normal form grammar GG such that: L(GG) = L(G) – {}. Proof: The proof is also by construction. Details of Chomsky conversion are complex but straightforward; I leave them for you to read in Chapter 11 and/or in the next 16 slides. Details of Greibach conversion are more complex but still straightforward; I leave them for you to read in Appendix D if you wish (not req'd).

36. The Price of Normal Forms EE + E E (E) Eid Converting to Chomsky normal form: EEE EPE ELE EE R Eid L ( R ) P + Conversion doesn’t change weak generative capacity but it may change strong generative capacity.