Ambiguity in Context-Free Grammars, Introduction to Pushdown Automata

Transcription

1 Ambiguity in Context-Free Grammars, Introduction to Pushdown Automata Martin Fränzle Informatics and Mathematical Modelling The Technical University of Denmark Context-free languages II p.1/25

2 What you ll learn 1. Context-free grammars: What s ambiguity Why it is a problem How to avoid it How to build useful CFGs 2. Pushdown automata: Machines recognizing CFLs Operational approach towards CFLs Tools for CFLs Context-free languages II p.2/25

3 Ambiguity in CFGs Context-free languages II p.3/25

4 Pragmatics The use cases of regular languages and context-free languages are subtly different: For RLs, one is primarily interested in language membership: finding strings in a text, classifying words/phrases in a text, With CFLs, one is often also interested in the structural information conveyed by a parse tree: if BE then else if BE then if BE then if BE then else Context-free languages II p.4/25

5 A classical ambiguity Given if then else if then print true false what shall if false then if false then print true else print false output? Context-free languages II p.5/25

6 A classical ambiguity Given if then else if then print true false what shall if false then if false then print true else print false output? if BE then else if BE then if BE then if BE then else Context-free languages II p.5/25

7 A classical ambiguity resolved Given if if true then then false else print print what shall if false then if false then print true else print false output? if BE then else if BE then if BE then if BE then else Context-free languages II p.6/25

8 Ambiguity Def: A CFG is called ambiguous iff there is (at least) two different parse trees wrt.. s.t. has N.B. It is not the existence of multiple derivations for some string that renders ambiguous! Context-free languages II p.7/25

9 Removing ambiguity There is no algorithm for resolving ambiguity (in the sense of automatically deriving an unambiguous grammar from a given grammar). Context-free languages II p.8/25

10 Removing ambiguity There is no algorithm for resolving ambiguity (in the sense of automatically deriving an unambiguous grammar from a given grammar). There is not even an algorithm for finding out whether a given CFG is ambiguous. Context-free languages II p.8/25

11 Removing ambiguity There is no algorithm for resolving ambiguity (in the sense of automatically deriving an unambiguous grammar from a given grammar). There is not even an algorithm for finding out whether a given CFG is ambiguous. However, there are standard techniques for writing an unambiguous grammar that help in most cases. Context-free languages II p.8/25

12 Context-free languages II p.9/25 Expression grammars Ambiguous

13 Expression grammars Ambiguous Unambiguous Context-free languages II p.9/25

14 Derivations vs. ambiguity Let be a grammar and. Thm: The following statements are equivalent: has more than one parse tree has more than one leftmost derivation has more than one rightmost derivation. Prf. Leftmost (rightmost, resp.) derivations can be understood as a canonical way of traversing a parse tree and are thus in one-to-one correspondence to parse trees. Context-free languages II p.10/25

15 Inherent ambiguity Def: A CFL is called inherently ambiguous iff all CFGs are ambiguous. with N.B. The mere existence of one ambiguous CFG for is not sufficient to render inherently ambiguous. (In fact, any non-empty CFL has an ambiguous CFG.) All the previous examples had an unambiguous CFG and are thus not inherently ambiguous. Context-free languages II p.11/25

16 An inherently ambiguous language The language is inherently ambiguous. Context-free languages II p.12/25

17 An inherently ambiguous language The language is inherently ambiguous. An example grammar: Context-free languages II p.12/25

18 Pushdown Automata Context-free languages II p.13/25

19 Idea of pushdown automaton Take an -NFA and 1. add a stack for unbounded storage, yet with access limited to last-in-first-out ; 2. make transitions dependent on input symbol and stack top; 3. add side effect on stack to transitions: transitions replace stack top by an arbitrary string, thus being able to pop stack (replacement by ), preserve stack (replace stack top by ), push onto stack (replace stack top by ), pop and then push (replace stack top by ). Context-free languages II p.14/25

20 Formal definition A pushdown automaton (PDA) is a seven-tuple with being a finite set of states, being the input alphabet, i.e. the finite set of input symbols, being the stack alphabet, fin implies that symbol is (or arbitrary in case replaced by. being the start state, being the transition function, can move from to when the input ) and the stack top is. is then being the start symbol on the stack (i.e., execution starts with the stack containing exactely one item, namely ), being the set of accepting states. Context-free languages II p.15/25

21 Instantaneous descriptions (IDs) An instantaneous description (ID) is a triple, where denotes the current state, denotes the remaining input, denotes the stack contents (top of the stack first letter of ). IDs describe configurations of PDA computations. Context-free languages II p.16/25

22 Instantaneous descriptions (IDs) An instantaneous description (ID) is a triple, where denotes the current state, denotes the remaining input, denotes the stack contents (top of the stack first letter of ). IDs describe configurations of PDA computations. is a relation between IDs of PDA formalizing moves of : iff Context-free languages II p.16/25

23 Translation invariance Thm: If then is a PDA and and implies I.e., PDA behaviours are preserved under stack extension and/or input extension. Prf: It is easy to see from the definition that implies The theorem follows by induction on the number of steps. Context-free languages II p.17/25

24 Translation invariance Thm: If is a PDA and then implies I.e., PDA behaviour is independent from trailing input. N.B. The same is not true wrt. trailing stack contents, as the PDA might well pop off, inspect, and then restore some part of the stack contents. Prf: It is easy to see from the definition that implies The theorem follows by induction on the number of steps. Context-free languages II p.18/25

25 Language of a PDA Def: The language of a PDA is def This form of language acceptance is often called acceptance by final state. Context-free languages II p.19/25

26 Language of a PDA is of a PDA Def: The language def This form of language acceptance is often called acceptance by final state. is of a PDA Def: The language def This form of language acceptance is often called acceptance by empty stack.. N.B. In general, Context-free languages II p.19/25

27 Expressiveness of the acceptance criteria Thm: If for some pushdown automaton then there is a pushdown automaton with. (Given, the construction of is completely mechanic.) Both forms of acceptance are thus equally expressive. Context-free languages II p.20/25

28 Expressiveness of the acceptance criteria Thm: If for some pushdown automaton then there is a pushdown automaton with. (Given, the construction of is completely mechanic.) Thm: If for some pushdown automaton then there is a pushdown automaton with. (Given, the construction of is completely mechanic.) Both forms of acceptance are thus equally expressive. Context-free languages II p.20/25

29 PDAs vs. CFGs Definable by a CFG Accepted by a PDA by empty stack Accepted by a PDA by final state Claim: PDAs and CFGs define the same languages. Context-free languages II p.21/25

30 Proof of the claim Have already shown equivalence between being accepted by some PDA by empty stack being accepted by some PDA by final state. uffices to show equivalence of being definable by a CFG to one of the two forms of PDA definability. We will show equivalence of CFG-definabilty to being accepted by some PDA by empty stack. Context-free languages II p.22/25

31 From CFG to PDA Thm: If for some contextfree grammar then there is a pushdown automaton with. (Given, the construction of is completely mechanic.) Context-free languages II p.23/25

32 From CFG to PDA for some contextfree grammar then there is a pushdown automaton with. (Given, the construction of is completely mechanic.) Thm: If can be, a corresponding 1-state PDA Prf: (Essence of) Given defined by for for otherwise. and any Then show that for any lm iff Context-free languages II p.23/25

33 From PDA to CFG Thm: If for some pushdown automaton then there is a contextfree grammar with. (Given, the construction of is completely mechanic.) Context-free languages II p.24/25

34 From PDA to CFG for some pushdown automaton then there is a contextfree grammar with. (Given, the construction of is completely mechanic.) Thm: If, a corresponding CFG can be defined by Prf: Given, and any, any Then show that for any iff Context-free languages II p.24/25

35 tack symbols can simulate states Cor: For each PDA such that there is a PDA. with Prf: 1. Convert to an equivalent CFG previous theorem, 2. convert to a one-state PDA second-last theorem. using the construction of the using the construction of the Context-free languages II p.25/25

36 tack symbols can simulate states Cor: For each PDA such that there is a PDA. with Prf: 1. Convert to an equivalent CFG previous theorem, 2. convert to a one-state PDA second-last theorem. using the construction of the using the construction of the Prize is blowup in the size of the stack alphabet! Context-free languages II p.25/25