Context-sensitive languages. Context-sensitive languages. Context-sensitive and length-increasing grammars

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1 Context-sensitive languages Context-sensitive languages Definition (λ-separated grammar) A λ-rule is a rule of the form w λ. A grammar G = (N, T, P, S) is λ-free if P does not contain λ-rules. The grammar G is λ-separated if G is either λ-free or S λ is the only λ-rule in P where in the latter case the variable S must not occur on the right-hand side of any rule in P. Remark (λ-separated grammar) A λ-separated grammar generating a language L is essentially the same as a λ-free grammar generating the language L \ {λ}. Any λ-separated grammar generating L can be transformed into a λ-free grammar generating L \ {λ} by removing the rule S λ. Any λ-free grammar with start symbol S generating L can be transformed into a λ-separated grammar generating L {λ} by introducing a new start symbol S 0 and rules S 0 S and S 0 λ. Definition (Length-increasing and context-sensitive grammars) A grammar G = (N, T, P, S) is length-increasing if G is λ-separated and for every rule u v in P it holds that u v, with the possible exception of the rule S λ. The grammar G is context-sensitive if G is λ-separated and all rules in P, with the possible exception S λ, are of the form uxv uwv where X N, u, v (N T ), w (N T ) +. By definition, every context-sensitive grammar is length-increasing. Conversely, for every length-increasing grammar there is a context-sensitive grammar that generates the same language. Theorem (Length-increasing and context-sensitive grammars) Every length-increasing grammar G can be effectively transformed into a context-sensitive grammar G that generates the same language, i.e., L(G) = L(G ). Definition (Context-sensitive languages) A language L is context-sensitive if L = L(G) for some context-sensitive grammar G. Corollary A language L is context-sensitive if and only if L = L(G) for some length-increasing grammar G. Proof of the theorem. For any given length-increasing grammar G = (N, T, P, S), we construct a context-sensitive grammar G = (N, T, P, S) such that L(G) = L(G ). We can assume that λ / L(G) because otherwise, by the remark above, we can simply remove the rule S λ from G and finally add a new initial state S 0 and the rules S 0 S λ to G. Furthermore, we can assume that all rules in G are of the form v w where v, w N +, v w or X a where X N, a T. Otherwise, replace in all rules in P each terminal symbol a by a corresponding new variable Z a and add rules of the form a Z a.

2 Proof of the theorem, cont. Each rule in P is either of the form X 1 X m Y 1 Y n where m n ( ) and X 1,..., X m, Y 1,..., Y n N or of the form X a. In order to obtain P, we replace in P each rule ρ of the form ( ) where m > 1 by the rules X 1 X m Z ρ 1 X 2 X m, ρ Z 1 Z m ρ Z ρ 1 Z ρ m 1 Y m, Z ρ 1 Z ρ m 1 X m Z ρ 1 Z ρ m, Z ρ 1 Z ρ m Z ρ 1 Z ρ my m+1 Y n, Z ρ 1 Z ρ 2 Z ρ m 1 Z ρ m Z ρ 1 Z ρ 2 Z ρ m 1 Z ρ m, Z ρ 1 Y 2 Y m Y 1 Y m. Proof of the theorem, cont. By construction, for any derivation in G of a terminal word w there is a corresponding derivation of w in G, hence L(G) L(G ). In order to see that L(G ) L(G) holds, it suffices to observe that when deriving a terminal word in G, the new rules can only be applied such that the rules introduced when replacing a rule ρ are applied all and in the order shown above, hence any useful application of new rules amounts to simulating rules of G. The variables Z 1,... Z m, Z 1, Z m are mutually distinct new variables that differ from all variables in N and from all variables introduced while replacing rules different from ρ. Example (A context-sensitive language) The language L = {0 n 1 n 0 n : n 0} is context-sensitive. For a sketch of proof, we specify the rules of a length-increasing grammar that generates L: S Z a Z b Zc 010 λ, Zc Z c Z a Z b Zc, Z b Z a Z a Z b, Z c Z a Z a Z c, Z c Z b Z b Z c, Z a Z a 0 Z a, Zb Z b 1 Z b, Zc Z c 0 Z c, Z a Z b 0 Z b, Zb Z c 1 Z c, Zc Zc 00. The rule S 010 is required, alternatively, we may replace it by the rule Z b Zc 10. Theorem (Context-free versus context-sensitive languages) The context-free languages form a proper subclass of the class of context-sensitive languages. Proof: The language {0 n 1 n 0 n : n 0} is context-sensitive according to the example above but is not context-free, as can easily be shown by the pumping lemma for context-free languages. In order to see that all context-free languages are context-sensitive, recall that every context-free language is generated by a context-free grammar in Chomsky normal form. It then suffices to observe that by definition any context-free grammar in Chomsky normal form is a length-increasing grammar.

3 Definition (Pushdown automaton) A linear bounded automaton, or LBA for short, is a sixtupel M = (Q, Σ, Γ,, s, F ), Q is a finite set of states, Σ is the input alphabet, where Γ is the tape alphabet, where Σ Γ and {# A, # E } Γ \ Σ, is the transition relation or program of M, where (Q Γ) (Q Γ Move) and Move = {L, R, S}, s Q is the initial state, F Q is the set of accepting states. Similar to the case of PDAs, the relation is identified with a relation that contains tuples of the form (q, a, q, a, Z). Each such tuple in is referred to as an instruction of M. A linear bounded automata - has a single two-way read-write tape, i.e., the head can move in both directions and symbols on the tape can be overwritten, - the head can only access the region of the tape from the left end marker # A to the right end marker # E, - the end markers # A and # E cannot be overwritten, - on input w, initally the tape holds the word # A w# E and M is in the initial state, - the available instructions depend on the current state and the symbol currently read on the tape, - each instruction specifies the subsequent state, a symbol to be written at the current tape position, and a move, - a move is either to stay (S) at the current position, or to move one position left (L) or right (R). Let (Q Γ) (Q Γ Move) be the transition relation of an LBA. A single instruction (q, a, q, a, Z) in has the meaning that - in case q is the current state and a Γ is the symbol currently read on the tape, - then the instruction may be executed by going into state q, replacing the tape symbol currently read by a, and finally performing the move Z. LBAs are in general nondeterministic, i.e., for any given pair (q, a) the program may contain several (or no) instructions of the form (q, a, q, a, Z). The computation model LBA is equivalent to nondeterministic Turing machines that are restricted to space equal to the length of their input. Definition (Configuration of a linear bounded automaton) Let M = (Q, Σ, Γ,, s, F ) be an LBA. With inputs of length n, a configuration of M is a triple (q, # A u# E, i) Q Γ {0,..., n + 1} where u = n. The inital configuration on input w is (s, # A w# E, 0) A configuration (q, # A u# E, i) represents a situation where q is the current state of the computation, # A u# E is the tape content, i is the position of the head on the tape. Positions are numbered in the natural way. For u = n, the left and right end marker are at position 0 and n + 1, respectively.

4 Definition (Transition relation of an LBA) Let M = (Q, Σ, Γ,, s, F ) be an LBA. In order to define the transition relation M, let for all words u of length n over Γ, all states q and q in Q, and all i where 0 i n + 1 (q, # A u# E, i) M (q, # A u # E, i ) if for # A u# E = a 0 a n+1 there is an instruction (q, a i, q, a, Z) in such that { u a 0... a i 1 a a i+1 in case a i / {# A, # E } = u otherwise. max{0, i 1} in case Z = L, i = i in case Z = S, min{n + 1, i + 1} in case Z = R. Definition (Computations of an LBA) Let M = (Q, Σ, Γ,, s, F ) be an LBA. A stop configuration of M on inputs of length n is a configuration (q, # A u# E, i) such that for # A u# E = a 0 a n+1 there is no instruction of the form (q, a i,.,.,.) in. A partial computation of M of length t is a sequence C 0,..., C t of configurations such that C 0 M C1 M M Ct. We write C M,t C if there is such a computation where C = C 0 and C = C t. We write C M, C if C M,t C for some t 0. Definition (Language recognized by an LBA) Let M = (Q, Σ, Γ,, s, F ) be an LBA. A finite or terminating computation of M is a partial computation that starts with the initial configuration (s, # A w# E, 0) of some input w and ends with a stop configuration. An accepting computation is a finite computation where the state of the last configuration of the computation, the stop configuration, is an accepting state. The LBA M accepts an input w if there is some accepting computation that starts with the initial configuration of input w. The language accepted by the LBA M is LBAs and context-sensitive languages Remark (Tracks) The tape of an LBA can be partitioned into a constant number of tracks where each track essentially works like a separate tape, except that all tracks share a common head. In order to obtain k tracks each with alphabet Σ 0, the alphabet of the LBA is chosen as Σ = Σ k 0, that is, each symbol of the tape alphabet Σ has k components from Σ 0. For any j where 1 j k, the jth components of the symbols on the tape can be viewed as forming a separate tape with an inscription over Γ 0. L(M) = {w Σ : M accepts w}.

5 LBAs and context-sensitive languages Theorem (Context-sensitive grammars and LBAs) A language L is context-sensitive if and only if L is recognized by some LBA. Lemma 1 For every length-increasing grammar G there is an LBA M such that L(G) = L(M). Lemma 2 For every LBA M there is a length-increasing grammar G such that L(G) = L(M). LBAs and context-sensitive languages Lemma 1 For every length-increasing grammar G there is an LBA M such that L(G) = L(M). Sketch of proof. Let G = (N, T, P, S) be a length-increasing grammar. An LBA M that recognizes L(G) works as follows: On an input w of length n, initially w is copied to the first of two tracks, while on the second track the first symbol is equal to S and all other symbols are equal to a blank symbol / N T. On the second track, within the given space n all possible derivations in G are nondeterminisitically simulated. At any time, the simulation may be ended nondeterministically in order to check wether the derived sentential form is equal to w. The input w is accepted in some computation if and only if w could be derived during the corresponding simulation. LBAs and context-sensitive languages Lemma 2 For every LBA M there is a length-increasing grammar G such that L(G) = L(M). Sketch of proof. Let M = (Q, Σ, Γ,, s, F ) be an LBA. We can assume that M has two tracks and initially copies its input w to both tracks, resulting in an initial configuration with double input. Then M computes as usual using only the second track, while the copy of w on the first track is left untouched. A length-increasing grammar G generating L(M) works as follows. The rules of G allow to generate arbitrary initial configuration with double input, followed by simulations of arbitrary steps of M. On simulating an accepting computation, the simulated tape content can be transformed into the input w on the first track. End markers need to be incorporated into first and last symbol. Remark (Boolean functions and set-theoretical operations) Recall that every k-ary Boolean function α: {0, 1} k {0, 1} induces a k-ary set-theoretical operation L α. More precisely, for any given alphabet Σ and languages L 1,..., L K over Σ, we have ((L α (L 1,..., L k ))(w) = α(l 1 (w),..., L k (w)). For example, if we let α be equal to the ternary AND function, we have L α (L 1, L 2, L 3 ) = L 1 L 2 L 3, where as usual 0 and 1 represent false and true, respectively. Given any alphabet Σ, the set-theoretical operations of arity k are k = 0: the constant mappings with values and Σ, k = 1: identity L L and complement L L = Σ \ L, k = 2: binary union L 1 L 2, binary difference L 1 \ L 2, etc., k = 3: ternary union L 1 L 2 L 3, etc.

6 Theorem (Closure under complementation) The class of contextsensitive languages is closed under complementation. Sketch of proof. The theorem of Immerman and Szelepcsényi states that for every space-constructible function s where s(n) log n, the class of languages that can be recognized by nondeterministic Turing acceptors restricted to space s(n) is closed under complementation. The theorem above is then immediate as a special case because the function s(n) = n is space-constructible, the class of contex-sensitive languages coincides with the class of languages that can be recognized by LBAs, which in turn coincides with the class of languages that can be recognized by nondeterministic Turing acceptors restricted to space n. Theorem (Closure under union) The class of context-sensitive languages is closed under union. Proof. Given two context-sensitive languages L 1 over T 1 and L 2 over T 2, fix context-sensitive grammars G 1 = (N 1, T 1, S 1, P 1 ) and G 2 = (N 2, T 2, S 2, P 2 ) that generate L 1 and L 2, respectively. We can assume N 1 N 2 =. Then the grammar G = (N, T 1 T 2, S, P) where S / N 1 N 2, N = N 1 N 2 {S} and P = P 1 P 2 {S S 1 S 2 } is context-sensitive and generates the language L 1 L 2. In connection with the latter assertion, recall our convention that the left-hand side of each rule must contain some variable symbol, hence each derivation in G uses only rules from either P 1 or P 2. Theorem (Closure under set-theoretical operations) The class of context-sensitive languages is closed under set-theoretical operations. Proof: The class of context-sensitive languages is closed under 0-ary set-theoretical operations since it contains the empty set and Σ for any alphabet Σ. We have already seen that the class of context-sensitive languages is closed under complementation and binary union. All other set-theoretical operations can be expressed by the set-theoretical operations mentioned so far. For example, by the De Morgan s rules, binary intersection can be expressed by complementation and binary union. Theorem (Closure under concatenation and Kleene closure) The class of context-sensitive languages is closed under concatenation, and Kleene closure, i.e., (i) if L 1 and L 2 are context-sensitive, then L 1 L 2 is context-sensitive, (ii) if L is context-sensitive, then L is context-sensitive, Proof. We give the proof of assertion (ii) and omit the essentially identical considerations for assertion (i). We construct an LBA that recognizes L, which first guesses nondeterministically a partition of the input w of the form w = w 1 w t, and then simulates successively an LBA that recognizes L with inputs w 1,..., w t, and accepts w if all simulations were accepting.

7 Proof cont. More precisely, given a context-sensitive languages L pick some LBA M such that L = L(M). As outlined above, we construct an LBA M that recognizes L. On input w, first M scans w and marks first and last letters of subwords w 1,..., w t of w such that w = w 1 w t, Then M successively simulates M on the inputs w 1,..., w t with appropriately simulated left and right end markers. The input w is accepted by some computation of M if and only if in this computation all simulated computations accept. By construction, we have L(G) = L because M will on input w L accept for some guessed partition of w, w / L accept for no partition of w. Theorem (Closure under inverse homomorphisms) The class of context-sensitive languages is closed under inverse homomorphisms. Proof. Let L 2 be a context-sensitive language over some alphabet T 2 that is recognized by some LBA M 2. For some alphabet T 1, let h : T 1 T 2 be a homomorpism. For d = max a T1 h(a), we have for all w that h(w) d w. An LBA M 1 that recognizes L 1 = h 1 [L 2 ] works as follows: On input w, M 1 writes h(w) on at most d tracks. Then M 1 simulates the computations of M 2 on input h(w). A single computation of M 1 accepts if and only if the corresponding simulated computation of M 2 accepts. Theorem (Nonclosure under homomorphisms) Definition (Nonerasing homomorphism) A homomorphism h is nonerasing in case h(w) is always different from the empty word. Theorem (Closure under nonerasing homomorphisms) The class of context-free languages is closed under nonerasing homomorphisms. Proof. See exercises. The class of context-free languages is not closed under arbitrary homomorphisms. Proof. For a grammar G = (N, T, P, S) and a symbol # / N T, we say the word w = w 1 #w 2 # #w t represents a derivation in G in case w 1 = S, w t T, and w i G wi+1 for i = 1,..., t 1, i.e., w i can be transformed into w i+1 by applying a single rule in P. Let T = {ã: a T } be an alphabet of new, mutually distinct symbols that are different from the symbols in N and T.

8 Proof cont. For a word w = a 1 a n, let w = ã 1 ã n. Given a grammar G, consider the language L G of all words w = w 1 #w 2 # #w t that represent derivations in G and let L G = {w 1 #w 2 # # w t : w 1 #w 2 # #w t L G } the language of all words obtained from representations w L G by replacing in the terminal subword w t of w all symbols a by ã, where we assume that the symbols of the form ã are not in N T {#}. Given any grammar G, the languages L G and L G are both recognized by appropriate LBAs, hence are context-sensitive. We have L(G) = h[ L G ] for the homomorphism h that maps all symbols to the empty string except that h(ã) = a for all a T. So L G is context-sensitive for all grammars G, while we will see later that there are grammars G such that L(G) = h[ L G ] is not context-sensitive.