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1 CPS Mathematical Foundations of CS Dr. S. Rodger Section: Recursively Enumerable Languages èhandoutè Deænition: A language L is recursively enumerable if there exists a TM M such that L=LèMè. if w 2 L? if w 62 L? recursively enumerable context-free regular Deænition: A language L is recursive if there exists a TM M such that L=LèMè and M halts on every w2 æ +. Enumeration procedure for recursive To enumerate all w2 æ + in a recursive language L: æ Let M be a TM that recognizes L, L = LèMè. æ Construct 2-tape TM M' Tape 1 will enumerate the strings in æ + Tape 2 will enumerate the strings in L. í On tape 1 generate the next string v in æ + í simulate M on v if M accepts v, then write v on tape 2. Enumeration procedure for recursively enumerable To enumerate all w2 æ + in a recursively enumerable language L: Repeat forever æ Generate next string èsuppose k strings have been generated: w 1 ;w 2 ; :::; w k è æ Run M for one step on w k Run M for two steps on w k,1.... Run M for k steps on w 1. If any of the strings are accepted then write them to tape 2. Theorem For any nonempty æ, there exist that are not recursively enumerable. 1

2 æ A language is a subset of æ æ. The set of all over æ is Theorem There exists a recursively enumerable language L such that ç L is not recursively enumerable. æ Let æ = fag Enumerate all TM's over æ: a aa aaa aaaa aaaaa... LèM 1 è LèM 2 è LèM 3 è LèM 4 è LèM 5 è The next two theorems in conjunction with the previous theorem will show that there are some that are recursively enumerable, but not recursive. Theorem If L and ç L are both RE, then L is recursive. æ There exists an M 1 such that M 1 can enumerate all elements in L. There exists an M 2 such that M 2 can enumerate all elements in ç L. To determine if a string w is in L or not in L perform the following algorithm: 2

3 Theorem: If L is recursive, then ç L is recursive. æ L is recursive, then there exists a TM M such that M can determine if w is in L or w is not in L. M outputs a 1 if a string w is in L, and outputs a 0 if a string w is not in L. Construct TM M' that does the following. M' ærst simulates TM M. If TM M halts with a 1, then M' erases the 1 and writes a 0. If TM M halts with a 0, then M' erases the 0 and writes a 1. Hierarchy of Languages: all recursively enumerable recursive context-free regular Deænition A grammar G=èV,T,R,Sè is unrestricted if all productions are of the form u! v where u 2èVëTè + and v 2èVëTè æ Example: Let G=èfS,A,Xg,fa,bg,R,Sè, R= S! baaax baa! aba AX! æ Example Find an unrestricted grammar G s.t. LèGè=fa n b n c n jné0g G=èV,T,R,Sè V=fS,A,B,D,E,Xg 3

4 T=fa,b,cg R= 1è S! AX 2è A! aabc 3è A! abbc 4è Bb! bb 5è Bc! D 6è Dc! cd 7è Db! bd 8è DX! EXc 9è BX! æ 10è ce! Ec 11è be! Eb 12è ae! ab There are some rules missing in the grammar. To derive string aaabbbccc, use productions 1,2 and 3 to generate a string that has the correct number of a's b's and c's. The a's will all be together, but the b's and c's will be intertwined. S è AX è aabcx è aaabcbcx è aaabbcbcbcx Theorem If G is an unrestricted grammar, then LèGè is recursively enumerable. æ List all strings that can be derived in one step. List all strings that can be derived in two steps. Theorem If L is recursively enumerable, then there exists an unrestricted grammar G such that L=LèGè. æ L is recursively enumerable. è there exists a TM M such that LèMè=L. M=èK; æ;,;æ;q 0 ;B;Fè q 0 w æ `x 1 q f x 2 for some q f 2F, x 1 ;x 2 2, æ 4

5 Construct an unrestricted grammar G s.t. LèGè=LèMè. S è æ w Three steps 1. S è æ B:::Bèxq f yb:::b with x,y2, æ for every possible combination 2. B:::Bèxq f yb:::b èb:::bèq æ 0 wb:::b 3. B:::Bèq 0 wb:::b èw æ Deænition A grammar G is context-sensitive if all productions are of the form x! y where x; y 2 èv ë T è + and jxj é jyj Deænition L is context-sensitive ècslè if there exists a context-sensitive grammar G such that L=LèGè or L=LèGè ëfæg. Theorem For every CSL L not including æ, 9 an LBA M s.t. L=LèMè. Theorem If L is accepted by an LBA M, then 9 CSG G s.t. LèMè=LèGè. Theorem Every context-sensitive language L is recursive. Theorem There exists a recursive language that is not CSL. 5

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