Subject Name FORMAL LANGUAGES AND AUTOMATA THEORY Subject

Subject Name: FORMAL LANGUAGES AND AUTOMATA THEORY Subject Code: 10 CS 56 Prepared By: Mrs. Pramela Devi, Mrs. Annapoorani, Mrs. Madhusmitha Department: CSE Date: 25. 09. 2014 1/10/2022

UNIT 8 : Undecidability • • A language that is not recursively enumerable An undecidable problem that is RE Post’s Correspondence problem Other undecidable problems 1/10/2022

1. A language that is not recursively enumerable Any TM for a Recursive language is going to look like this: w M “accept” “reject” Any TM for a Recursively Enumerable (RE) language is going to look like this “accept” w 1/10/2022 M

1. 1 Binary-Strings from TM’s • • • We shall restrict ourselves to TM’s with input alphabet {0, 1}. Assign positive integers to the three classes of elements involved in moves: 1. States: q 1(start state), q 2 (final state), q 3, … 2. Symbols X 1 (0), X 2 (1), X 3 (blank), X 4, … 3. Directions D 1 (L) and D 2 (R). Suppose δ(qi, Xj) = (qk, Xl, Dm). Represent this rule by string 0 i 10 j 10 k 10 l 10 m. Key point: since integers i, j, … are all > 0, there cannot be two consecutive 1’s in these strings. 1/10/2022

1. 2 Enumerating TM’s and Binary Strings • • • Recall we can convert binary strings to integers by prepending a 1 and treating the resulting string as a base 2 integer. Thus, it makes sense to talk about “the i-th binary string” and about “the i-th Turing machine. ” Note: if i makes no sense as a TM, assume the i-th TM accepts nothing. 1/10/2022

1. 3 Table of Acceptance String j 1 2 3 4 5 6. . . TM i 1/10/2022 1 2 3 4 5 6. . . x x = 0 means the i-th TM does not accept the j-th string; 1 means it does.

1. 4 Diagonalization Argument • Whenever we have a table like the on the previous slide, we can diagonalizable it. • That is, construct a sequence D by complementing each bit along the major diagonal. Formally, D = a 1 a 2…, where ai = 0 if the (i, i) table entry is 1, and vice-versa • Could D be a row (representing the language accepted by a TM) of the table? • Suppose it were the j-th row. • But D disagrees with the j-th row at the j-th column. • Thus D is not a row. • Consider the diagonalization language Ld = {w | w is the i-th string, and the i-th TM does not accept w}. • We have shown that Ld is not a recursively enumerable language; i. e. , it has no TM. 1/10/2022

1. 5 A Problem About Turing Machines • • • We can think of the language Ld as a problem. “Does this TM not accept its own code? ” Aside: We could also think of it as a problem about binary strings. – Do you see how to phrase it? A problem is decidable if there is an algorithm to answer it. – Recall: An “algorithm, ” formally, is a TM that halts on all inputs, accepted or not. – Put another way, “decidable problem” = “recursive language. ” Otherwise, the problem is undecidable 1/10/2022

1. 6 Relationship between languages Not recursively enumerable languages Ld Decidable problems = Recursive languages Recursively enumerable languages 1/10/2022

2. An undecidable problem that is RE Complements of Recursive Languages Theorem: If L is a recursive language, L is also recursive. Proof: Let M be a TM for L that always halt. We can construct another TM M from M for L that always halts as follows: M Accept M Input 1/10/2022 Reject

2. 1 Complements of RE Languages Theorem: If both a language L and its complement L are RE, L is recursive. Proof: Let M 1 and M 2 be TM for L and L respectively. We can construct a TM M from M 1 and M 2 for L that always halt as follows: M M 1 Input 1/10/2022 M 2 Accept Reject

2. 2 From the Abstract to the Real • • • While the fact that Ld is undecidable is interesting intellectually, it doesn’t impact the real world directly. We first shall develop some TM-related problems that are undecidable, but our goal is to use theory to show some real problems are undecidable Can a particular line of code in a program ever be executed? Is a given context-free grammar ambiguous? Do two given CFG’s generate the same language? 1/10/2022

2. 3 The Universal Language An example of a recursively enumerable, but not recursive language is the language Lu of a universal Turing machine. That is, the UTM takes as input the code for some TM M and some binary string w and accepts if and only if M accepts w. Inputs are of the form. Code for M 111 w Note: A valid TM code never has 111, so we can split M from w. The UTM must accept its input if and only if M is a valid TM code and that TM accepts w. The UTM will have several tapes. Tape 1 holds the input M 111 w Tape 2 holds the tape of M. Mark the current head position of M. Tape 3 holds the state of M. Step 1: The UTM checks that M is a valid code for a TM. E. g. , all moves have five components, no two moves have the same state/symbol as first two components. If M is not valid, its language is empty, so the UTM immediately halts without accepting. Step 2: The UTM examines M to see how many of its own tape squares it needs to represent one symbol of M. Step 3: Initialize Tape 2 to represent the tape of M with input w, and initialize Tape 3 to hold the start state. Step 4: Simulate M. Look for a move on Tape 1 that matches the state on Tape 3 and the tape symbol under the head on Tape 2. If found, change the symbol and move the head marker on Tape 2 and change the State on Tape 3. If M accepts, the UTM also accepts. 1/10/2022

2. 4 Lu is Recursively Enumerable, but not Recursive • • We designed a TM for Lu, so it is surely RE. Suppose it were recursive; that is, we could design a UTM U that always halted. Then we could also design an algorithm for Ld, as follows. Given input w, we can decide if it is in Ld by the following steps. 1. Check that w is a valid TM code. If not, then its language is empty, so w is in Ld. 2. If valid, use the hypothetical algorithm to decide whether w 111 w is in Lu. 3. If so, then w is not in Ld; else it is. But we already know there is no algorithm for Ld. Thus, our assumption that there was an algorithm for Lu is wrong. Lu is RE, but not recursive. 1/10/2022

2. 5 Undecidability of PCP To show that MPCP is undecidable, we will reduce the universal language problem (ULP) to MPCP: If MPCP can be solved, ULP can also be solved. Since we have already shown that ULP is un-decidable, MPCP must also be undecidable. Universal Language Problem (ULP) 1/10/2022 MPCP

2. 6 CFG’s from PCP • • • We are going to prove that the ambiguity problem (is a given CFG ambiguous? ) is undecidable. As with PCP instances, CFG instances must be coded to have a finite alphabet. Let a followed by a binary integer i represent the i-th terminal. Let A followed by a binary integer i represent the i-th variable. Let A 1 be the start symbol. Symbols ->, comma, and ε represent themselves. Example: S -> 0 S 1 | A, A -> c is represented by A 1 ->a 1 A 1 a 10, A 1 ->A 10, A 10 ->a 11 Consider a PCP instance with k pairs. i-th pair is (wi, xi). Assume index symbols a 1, …, ak are not in the alphabet of the PCP instance. The list language for w 1, …, wk has a CFG with productions A -> wi. Aai and A -> wiai for all i = 1, 2, …, k. 1/10/2022

2. 7 Reduction of PCP to the Ambiguity Problem • • Given a PCP instance, construct grammars for the two list languages, with variables A and B. Add productions S -> A | B. The resulting grammar is ambiguous if and only if there is a solution to the PCP instance. If the two derivations begin with the same first step, say S -> A, then the sequence of index symbols uniquely determines which productions are used. – Each except the last would be the one with A in the middle and that index symbol at the end. – The last is the same, but no A in the middle 1/10/2022 .

Example: S =>A=>*… 2321 S w 1 w 2 w 3 1/10/2022 A A A 1 2 3