Theory of Computation 5 Combining Languages Frank Stephan Department of Computer Science Department of Mathematics National University of Singapore [email protected] Theory of Computation 5 Combining Languages – p. 1 Repetition 1 If (Q, Σ, δ, s, F) is a non-deterministic finite automaton (nfa) then δ has a set of values (not always single value), that is, for q ∈ Q and a ∈ Σ there can be several p ∈ Q with p ∈ δ(q, a). A run of an nfa on a word a1 a2 . . . an is a sequence q0 q1 q2 . . . qn ∈ Q∗ such that q0 = s and qm+1 ∈ δ(qm , am+1 ) for all m < n. If qn ∈ F then the run is “accepting” else the run is “rejecting”. The nfa accepts a word w iff it has an accepting run on w; this is also the case if there exist other rejecting runs. Theory of Computation 5 Combining Languages – p. 2 Repetition 2 The language {w : some letter appears twice} has an nfa with n + 2 states while a dfa needs 2n + 1 states; here for n = 4, where n = |Σ|. 0,1,2,3 0,1,2,3 ∅ # start 3 1 0 {0} 1,2,3 0 2 3 2 1 {1} {2} {3} 0,2,3 0,1,3 0,1,2 Theory of Computation 5 Combining Languages – p. 3 Repetition 3 Given an nfa, one let for given state q and symbol a the set δ(q, a) denote all states q′ to which the nfa can transit from q on symbol a. Theorem 4.5 [Büchi; Rabin and Scott] For each nfa (Q, Σ, δ, s, F) with n = |Q| states, there is an equivalent dfa ({Q′ : Q′ ⊆ Q}, Σ, δ ′ , {s}, F′ ) with 2n states such that F′ = {Q′ ⊆ Q : Q′ ∩SF 6= ∅} and ∀Q′ ⊆ Q ∀a ∈ Σ [δ ′ (Q′ , a) = q′ ∈Q δ(q′ , a) = {q′′ ∈ Q : ∃q′ ∈ Q′ [q′′ ∈ δ(q′ , a)]}]. As the number of states is often overshooting, it is good to minimise the resulting automaton with the algorithm of Myhill and Nerode. Theory of Computation 5 Combining Languages – p. 4 Repetition 4 The following statements are all equivalent to “L is regular”: (a) L is generated by a regular expression; (b) L is generated by a regular grammar; (c) L is recognised by a determinisitic finite automaton; (d) L is recognised by a non-determinisitic finite automaton; (e) L and Σ∗ − L both satisfy the Block Pumping Lemma; (f) L satsifies Jaffe’s Matching Pumping Lemma; (g) L has only finitely many derivatives. Theory of Computation 5 Combining Languages – p. 5 Product Automata Let (Q1 , Σ, δ1 , s1 , F1 ) and (Q2 , Σ, δ2 , s2 , F2 ) be dfas which recognise L1 and L2 , respectively. Consider (Q1 × Q2 , Σ, δ1 × δ2 , (s1 , s2 ), F) with (δ1 × δ2 )((q1 , q2 ), a) = (δ1 (q1 , a), δ2 (q2 , a)). This automaton is called a product automaton and one can choose F such that it recognises the union or intersection or difference of the respective languages. Union: F = F1 × Q2 ∪ Q1 × F2 ; Intersection: F = F1 × F2 = F1 × Q2 ∩ Q1 × F2 ; Difference: F = F1 × (Q2 − F2 ); Symmetric Difference: F = F1 × (Q2 − F2 ) ∪ (Q1 − F1 ) × F2 . Theory of Computation 5 Combining Languages – p. 6 Example For a = 1, 2, let automaton ({s, t}, {0, 1, 2}, δa , s, {s}) recognise when there is an even number of a; if input b equals a then state is changed else state remains unchanged. Quiz: Which Boolean combination does this product automaton recognise? 0 0 1 (s, s) start 2 0 1 2 (t, s) (s, t) 2 1 1 (t, t) 2 0 Theory of Computation 5 Combining Languages – p. 7 Kleene Star Assume (Q, Σ, δ, s, F) is an nfa recognising L. Now L∗ is recognised by (Q ∪ {s′ }, Σ, δ ′ , s′ , {s′ } ∪ F) where δ ′ (s′ , a) = δ(s, a) and δ ′ (p, a) = δ(p, a) for p ∈ Q − F and δ ′ (p, a) = δ(p, a) ∪ δ(s, a) for p ∈ F. 0 0 s start 0 start 0 s 1 1 t s′ 1 0 1 t 1 Theory of Computation 5 Combining Languages – p. 8 Concatenation Assume (Q1 , Σ, δ1 , s1 , F1 ) and (Q2 , Σ, δ2 , s2 , F2 ) are nfas recognising L1 and L2 with Q1 ∩ Q2 = ∅ and assume ε∈ / L2 . Now (Q1 ∪ Q2 , Σ, δ, s1 , F2 ) recognises L1 · L2 where (p, a, q) ∈ δ whenever (p, a, q) ∈ δ1 ∪ δ2 or (p ∈ F1 and (s2 , a, q) ∈ δ2 ). If L2 contains ε then one can consider the union of L1 and L1 · (L2 − {ε}). Theory of Computation 5 Combining Languages – p. 9 Example L1 · L2 with L1 = {00, 11}∗ and L2 = 2∗ 1+ 0+ . q1 0 start s2 2 0 1 r1 1 1 s1 1 2 q2 1 0 r2 0 Theory of Computation 5 Combining Languages – p. 10 Exercise 5.3 The previous slides give upper bounds on the size of the dfa for a union, intersection, difference and symmetric difference as n2 states, provided that the original two dfas have at most n states. Give the corresponding bounds for nfas: If L and H are recognised by nfas having at most n states each, how many states does one need at most for an nfa recognising (a) the union L ∪ H, (b) the intersection L ∩ H, (c) the difference L − H and (d) the symmetric difference (L − H) ∪ (H − L)? Give the bounds in terms of “linear”, “quadratic” and “exponential”. Explain your bounds. Theory of Computation 5 Combining Languages – p. 11 Sample Automata Exercise 5.4 Let Σ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Construct a (not necessarily complete) dfa recognising the language Σ · {aa : a ∈ Σ} ∩ {aaaaa : a ∈ Σ}. It is not needed to give a full table for the dfa, but a general schema and an explanation how it works. Exercise 5.5 Make an nfa for the intersection of the following languages: {0, 1, 2}∗ · {001} · {0, 1, 2}∗ · {001} · {0, 1, 2}∗ ; {001, 0001, 2}∗ ; {0, 1, 2}∗ · {00120001} · {0, 1, 2}∗ . Exercise 5.6 Make an nfa for the union L0 ∪ L1 ∪ L2 with La = {0, 1, 2}∗ · {aa} · {0, 1, 2}∗ · {aa} · {0, 1, 2}∗ for a ∈ {0, 1, 2}. Theory of Computation 5 Combining Languages – p. 12 Exercise 5.7 Consider two context-free grammars with terminals Σ, disjoint non-terminals N1 and N2 , start symbols S1 ∈ N1 and S2 ∈ N2 and rule sets P1 and P2 which generate L and H, respectively. Explain how to form from these a new context-free grammar for (a) L ∪ H, (b) L · H and (c) L∗ . Write down the context-free grammars for {0n 12n : n ∈ N} and {0n 13n : n ∈ N} and form the grammars for the the union, concatenation and star explicitly. Theory of Computation 5 Combining Languages – p. 13 Example 5.8 The language {0}∗ · {1n 2n : n ∈ N} is context-free. Grammar ({S, T}, {0, 1, 2}, P, S) with P be given by S → 0S|T|ε and T → 1T2|ε. The language {0n 1n : n ∈ N} · 2∗ is context-free. L = {0n 1n 2n : n ∈ N} is not context-free but the intersection of the two above. The complement of L is the union of {0n 1m 2k : n < k}, {0n 1m 2k : n > k}, {0n 1m 2k : m < k}, {0n 1m 2k : m > k}, {0n 1m 2k : n < m}, {0n 1m 2k : n > m} and {0, 1, 2}∗ · {10, 20, 21} · {0, 1, 2}∗ . Each of these languages is context-free. Grammar for the first of them: S → 0S2|S2|T2, T → 1T|ε. The union is also context-free. Hence L has a context-free complement. Theory of Computation 5 Combining Languages – p. 14 Context-Free Intersects Regular Theorem 5.9 If L is context-free and H is regular then L ∩ H is context-free. Construction. Let (N, Σ, P, S) be a context-free grammar generating L with every rule being either A → w or A → BC with A, B, C ∈ N and w ∈ Σ∗ . Let (Q, Σ, δ, s, F) be a dfa recognising H. / Q × N × Q and make the following new grammar Let S′ ∈ (Q × N × Q ∪ {S′ }, Σ, R, S′ ) with rules R: S′ → (s, S, q) for all q ∈ F; (p, A, q) → (p, B, r)(r, C, q) for all rules A → BC in P and all p, q, r ∈ Q; (p, A, q) → w for all rules A → w in P with δ(p, w) = q. Theory of Computation 5 Combining Languages – p. 15 Exercises 5.10 and 5.11 Recall that the language L of all words which contain as many 0s as 1s is context-free; a grammar for it is ({S}, {0, 1}, {S → SS|ε|0S1|1S0}, S). Exercise 5.10 Construct a context-free grammar for L ∩ (001+ )∗ . Exercise 5.11 Construct a context-free grammar for L ∩ 0∗ 1∗ 0∗ 1∗ . Theory of Computation 5 Combining Languages – p. 16 Context-Sensitive and Concatenation Let L1 and L2 be context-sensitive languages not containing ε. Let (N1 , Σ, P1 , S1 ) and (N2 , Σ, P2 , S2 ) be two context-senstive grammers generating L1 and L2 , respectively, where N1 ∩ N2 = ∅ and where each rule l → r satisfies |l| ≤ |r| and l ∈ N+ e for the respective e ∈ {1, 2}. / N 1 ∪ N 2 ∪ Σ. Let S ∈ Now (N1 ∪ N2 ∪ {S}, Σ, P1 ∪ P2 ∪ {S → S1 S2 }, S) generates L1 · L2 . If v ∈ L1 and w ∈ L2 then S ⇒ S1 S2 ⇒∗ vS2 ⇒∗ vw. Furthermore, the first rule has to be S ⇒ S1 S2 and from then onwards, each rule has on the left side either l ∈ N+ 1 so that it applies to the part generated from S1 or it has in the left side l ∈ N+ 2 so that l is in the part of the word generated from S2 . Hence every intermediate word z in the derivation is of the form xy = z with S1 ⇒∗ x and S2 ⇒∗ y. Theory of Computation 5 Combining Languages – p. 17 Context-Sensitive and Kleene-star Let (N1 , Σ, P1 , S1 ) and (N2 , Σ, P2 , S2 ) be context-sensitive grammars for L − {ε} with N1 ∩ N2 = ∅ and all rules l → r + satisfying |l| ≤ |r| and l ∈ N+ or l ∈ N 1 2 , respectively. Let S, S′ be symbols not in N1 ∪ N2 ∪ Σ. Now consider (N1 ∪ N2 ∪ {S, S′ }, Σ, P, S) where P contains the rules S → S′ |ε and S′ → S1 S2 S′ | S1 S2 | S1 plus all rules in P1 ∪ P2 . This grammar generates L∗ . Theory of Computation 5 Combining Languages – p. 18 Context-Sensitive and Intersection Theorem. The intersection of two context-sensitive languages is context-sensitive. Construction. Let (Nk , Σ, Pk , S) be grammars for L1 and L2 . Now make a new non-terminal set N = (N1 ∪ Σ ∪ {#}) × (N2 ∪ Σ ∪ {#}) S with start symbol S and following types of rules: (a) Rules to generate and manage space; (b) Rules to generate a word v in the upper row; (c) Rules to generate a word w in the lower row; (d) Rules to convert a string from N into v provided that the upper components and lower components of the string are both v. Theory of Computation 5 Combining Languages – p. 19 Type of Rules (a): and S # S A # for producing space; → S # S B C A B A B → # C for space management. C # → # B A C (b) and (c): For each rule in P1 , for example, for AB → CDE ∈ P1 , and all symbols F, G, H, . . . in N2 , one C D E A B # has the corresponding rule F G H → F G H . So rules in P1 are simulated in the upper half and rules in P2 are simulated in the lower half and they use up # if the left side is shorter than the right one. a (d): Each rule a → a for a ∈ Σ is there to convert a a matching pair a from Σ × Σ (a nonterminal) to a (a terminal). Theory of Computation 5 Combining Languages – p. 20 n n n Grammar for 0 1 2 with n > 0 Grammar L1 : S → S2|0S1|01. Grammar L2 : S → 0S|1S2|12. Grammar for Intersection. A, B, C stand for any members of {S, 0, 1, 2, #}. A N = { B : A, B ∈ {S, 0, 1, 2, #}}. S S # Rules: S → S # ; A B # A A B A # B C → B C ; C # → # C ; S 1 2 S # # 0 S S # A B → A B; A B C → A B C ; 1 0 S # → A B; A B A B C A B A B A B C S 2 ; S ; S # # → 1 S # → 0 A B A B 2 ; S # → 1 0 1 2 0 → 0; 1 → 1; 2 → 2. Theory of Computation 5 Combining Languages – p. 21 Deriving 001122 S S S S S S 0 S 0 0 0 0 0 0 ⇒∗ # # # # S # 0 S 0 0 0 0 S # # # # # S 2 # # # # ⇒ S # # # # # S # # # # # ∗ # # # 2 S 2 # # # 2 # # # # ⇒ S # # # # # ⇒ 2 2 # # 2 0 S 1 # 2 # # # # ⇒ S # # # # # ⇒ 1 1 2 2 2 2 # 1 0 0 # # # # ⇒ S # # # # # ⇒ 2 2 1 2 2 0 0 1 1 1 # # # # ⇒ 0 0 S # # # ⇒ 0 0 1 1 2 2 1 1 2 2 1 S 2 # ⇒ 0 0 1 S # 2 ⇒ ∗ 1 1 2 2 1 1 2 2 ⇒ 001122. ⇒∗ Theory of Computation 5 Combining Languages – p. 22 Exercises 5.14 and 5.17 Exercise 5.14 Construct a context-sensitive grammar for {0n 1n 2n : n ∈ N}∗ . Exercise 5.17 Consider the language L = {00} · {0, 1, 2, 3}∗ ∪ {1, 2, 3} · {0, 1, 2, 3}∗ ∪ {0, 1, 2, 3}∗ · {02, 03, 13, 10, 20, 30, 21, 31, 32} · {0, 1, 2, 3}∗ ∪ {ε} ∪ {01n 2n 3n : n ∈ N}. Which versions of the Pumping Lemma does it satisfy: • Regular Pumping Lemma (with / without bounds); • Context-Free Pumping Lemma (with / without bounds); • Block Pumping Lemma (for regular languages)? Determine the exact position of L in the Chomsky hierarchy. Theory of Computation 5 Combining Languages – p. 23 Mirror Images Define (a1 a2 . . . an )mi = an . . . a2 a1 as the mirror image of a string. It follows from the definition of context-free and context-sensitive, that if L is context-free / context-sensitive so is Lmi . This can be achieved by replacing every rule l → r by lmi → rmi . For example, the mirror image of the language of the words 0n 13n+3 is given by language of the words 13n+3 0n . While L is generated by a context-free grammar with one non-terminal S and rules S → 0S111 | 111, Lmi is then generated by a similar grammar with the rules S → 111S0 | 111. Theory of Computation 5 Combining Languages – p. 24 Exercise 5.18 Recall that xmi is the mirror image of x, so (01001)mi = 10010. Furthermore, Lmi = {xmi : x ∈ L}. Show the following two statements: (a) If an nfa with n states recognises L then there is also an nfa with up to n + 1 states recognising Lmi . (b) Find the smallest nfas which recognise L = 0∗ (1∗ ∪ 2∗ ) as well as Lmi . Theory of Computation 5 Combining Languages – p. 25 Palindromes The members of the language {x ∈ Σ∗ : x = xmi } are called palindromes. A palindrome is a word or phrase which looks the same from both directions. An example is the German name “OTTO”; furthermore, when ignoring spaces and punctuation marks, a famous palindrome is the phrase “A man, a plan, a canal: Panama.” originating from the time when the canal in Panama was built. The grammar with the rules S → aSa|aa|a|ε with a ranging over all members of Σ generates all palindromes; so for Σ = {0, 1, 2} the rules of the grammar would be S → 0S0 | 1S1 | 2S2 | 00 | 11 | 22 | 0 | 1 | 2 | ε. The set of palindromes is not regular. Theory of Computation 5 Combining Languages – p. 26 Exercises Exercise 5.20 Let w ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}∗ be a palindrome of even length and n be its decimal value. Prove that n is a multiple of 11. Note that it is essential that the length is even, as for odd length there are counter examples (like 111 and 202). Exercise 5.21 Given a context-free grammar for a language L, is there also one for L ∩ Lmi ? If so, explain how to construct the grammar; if not, provide a counter example where L is context-free but L ∩ Lmi is not. Exercise 5.22 Is the following statement true or false? Prove your answer: Given a language L, the language L ∩ Lmi equals to {w ∈ L : w is a palindrome}. Theory of Computation 5 Combining Languages – p. 27 Next Week’s Midterm Examination Topics Defining and proving using structural induction Making and analysing finite automata Converting regular languages from one form into another form, Deterministic versus non-deterministic finite automata, Bounds on number of states Pumping lemmas: Usage for proofs; Properties Combining finite automata Basic properties of context-free grammars: Making of such grammars, Usage of pumping lemma for context-free languages Revise lecture notes; Try exercises and compare with solutions by fellow students Theory of Computation 5 Combining Languages – p. 28 Example of Induction ε <ll 0 <ll 1 <ll 00 <ll 01 <ll 10 <ll 11 <ll 000 <ll . . .; use this length-lexicographical order <ll to define sw(reg exp): sw(∅) = ∞; sw({w1 , . . . , wn }) = minll {w1 , . . . , wn }; ( sw(σ) if sw(τ ) = ∞; sw(σ ∪ τ ) = sw(τ ) if sw(σ) = ∞; minll {sw(σ), sw(τ )} otherwise; ( ∞ if sw(σ) = ∞ sw(σ · τ ) = or sw(τ ) = ∞; sw(σ) · sw(τ ) otherwise; sw(σ ∗ ) = ε. Prove by structural induction that whenever σ generates a nonempty language then sw(σ) is a shortest word. Theory of Computation 5 Combining Languages – p. 29

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