The Chomski Hierarchy

Transcription

1 The Chomski Hierarchy The Chomski hierarchy is named after Noam Chomski, who defined the following four classes of grammars as potential models for natural language. Type 0 Grammars: Unrestricted grammars that generate exactly the recursively enumerable languages. Equivalent to Turing Machines The Chomski Hierarchy Type 1 Grammars: Context-Sensitive grammars that generate exactly the context-sensitive languages. Equivalent to linear bounded automaton (def later) Type 2 Grammars: Context-free grammars that generate exactly the context-free languages. Equivalent to PDAs Type 3 Grammars: Regular grammars that generate exactly the regular languages. Equivalent to regular sets and finite automatons. 1

2 The Chomsky Hierarchy Type Languages (grammars) Regular Context-free Context-sensitive Recursively enumerable (unrestricted) Production Form in grammar A ab, A a (A,B V, a Σ) A α (A V, α (V U Σ)*) α β (α, β (V U Σ)*, β α ) α β (α, β (V U Σ)*, α contains a variable) Accepting device Finite automaton Pushdown automaton Linear-bounded automaton Turing Machine Unrestricted Grammars Definition: An unrestricted grammar (Type 0) is a quadruple G = (V, Σ, S, P) where: V is a finite set of variables Σ is a finite set of terminal symbols P is the finite set of productions. Each production is of the form α β, where α (V U Σ) + and β (VUΣ)* S V is the start symbol. V and Σ are assumed to be disjoint. 2

3 Unrestricted Grammars Type 1, 2, and 3 grammars can be viewed as type 0 grammars with certain restrictions. Unrestricted Grammars A grammar that generates {a i b i c i i 0} G = (V, Σ, S, P) where: V = {S, A, C} Σ = {a, b, c} P = { S aabc ε A aabc ε CbbC Cc cc } 3

4 Unrestricted Grammars A grammar that generates {a 2i i > 0} S Ca CB ACaB aac DB CB E ad AD ae AE Da AC Ea ε Context-Sensitive Grammars A context-sensitive grammar is a type 1 grammar G = (V, Σ, S, P) such that for every α β P, it holds that α β. Note that the null string can never be generated by a context-sensitive language. Every context-free language that does not contain a null string is context-sensitive. 4

5 Context-Sensitive Grammars L is a context-sensitive language iff L {ε} is accepted by a linear bounded automaton (a non-deterministic Turing machine that uses only the input part of the tape). We will skip the proof. From the above, it follows that every context-sensitive language is recursive. This requires some work to show. Regular Grammars A grammar G = (V, Σ, S, P) is regular iff every production is of the form B ac, or B a, where B, C V and a Σ. L is regular iff L {ε} is accepted by a regular grammar. 5

6 Decidability Decision Problems A decision problem is a yes/no type of problem. For example, the following problem is a decision problem: Given an integer p, is p prime? A solution to a decision problem is an algorithm (effective procedure) that gives the right answer for every input. Such a decision problem is decidable (or solvable) Decision Problems Decision problems can be viewed as languages. For example, the previous problem can be viewed as the language: {p p is the binary representation of a prime} Decidable problems correspond to which languages? 6

7 Decision Problems for FAs and Regular Languages Theorem: A DFA = {<B, w> B is a DFA that accepts input string w} is a decidable language. Theorem: A NFA = {<B, w> B is a NFA that accepts input string w} is a decidable language. Decision Problems for FAs and Regular Languages Theorem: A REX = {<R, w> R is a regular expression that generates string w} is a decidable language. Theorem: E DFA = {<A> A is a DFA and L(A) = } is a decidable language. 7

8 Decision Problems for FAs and Regular Languages Theorem: EQ DFA = {<A, B> A and B are DFAs and L(A) = L(B)} is a decidable language. Decision Problems for CFLs Theorem: A CFG = {<G, w> G is a CFG that generates string w} is a decidable language. Here 8

9 Decision Problems for CFLs Theorem: E CFG = {<G> G is a CFG and L(G) = } is a decidable language. Decidable Problems for CFLs EQ CFG = {<G, H> G and H are CFGs and L(G) = L(H)} Is this decidable? EQ DFA = {<A, B> A and B are DFAs and L(A) = L(B)} is a decidable language. But, EQ CFG is undecidable 9

10 Decision Problems for CFLs Theorem: Every CFL is a decidable language. Example Problems Let INFINITE DFA = {<A> A is a DFA and L(A) is an infinite language}. Show that is INFINITE DFA decidable. We have to develop an algorithm to decide this problem. 10

11 Example Problems Let INFINITE DFA = {<A> A is a DFA and L(A) is an infinite language}. Show that is INFINITE DFA decidable. I = On input <A> where A is a DFA: 1. Let k be the number of states A. 2. Construct a DFA D that accepts all strings of length k or more. 3. Construct a DFA M st L(M) = L(A) IL(D). 4. Test L(M) =, using the E DFA decider T from Theorem If T accepts, reject; of T rejects, accept. Example Problems Let Σ = {0,1}. Show that the problem of determining whether a CFG generates some string 1* is decidable. In other words, show that {<G> G is a DFG over {0,1} and 1* IL(G) }. This problem requires you to recognize that if C is a context free language and R is a regular language, then CI R is context free. You have to show this, but it has been done for you as the solution to problem On input <G>: 1. Construct CFG H st L(H) = 1* I L(G). 2. Test whether L(H) =, using the E CFG decider R from Theorem If R accepts, reject; if R rejects, accept. 11

12 Relationships between languages Regular Context-free Recursive (Decidable) Recursively enumerable (Turning-recognizable) Decidability Are all decision problems decidable? Given a non-empty alphabet Σ: How many languages over Σ are there? How many Turing machines with alphabet Σ are there? Choose from finite, countable infinite, or uncountable. 12