Lecture 7 Undecidability cont.

1. Lecture 7Undecidability cont. Jan Maluszynski, IDA, 2007 http://www.ida.liu.se/~janma Jan Maluszynski - HT 2007

2. Outline (Sipser 5.1,5.3, 6.2) • Reducibility • Examples of undecidable problems • Mapping reducibility • Decidability of Logical Theories Jan Maluszynski - HT 2007

3. Reduction techniques For proving B is undecidable • Assume B is decidable • Find an undecidable problem A that can be decided using the decision procedure for B A reducible to B Example: HALTTM = {<M,w>| M is a TM and M halts on w} We show: ATM = {<M,w>| M is a TM and M accepts w} is reducible to HALTTM Jan Maluszynski - HT 2007

4. Reduction techniques example Assume TM R decides HALTTM = On input <M,w> run R on it: • If R rejects – reject • If R accepts ie. M halts on w: simulate M on w • If M accepts w – accept • If M rejects w - reject Thus with R we could decide ATM = {<M,w>| M is a TM and M accepts w} But we know ATM is undecidable Hence R cannot exist and HALTTM is undecidable Jan Maluszynski - HT 2007

5. ETM (emptiness) is undecidable ETM = {<M>| M is a TM and L(M) is empty} Assume R decides ETM. Reduce ATM to ETM . Given M and w construct M1 that on any input x : • If x = w simulates M on w • Otherwise rejects x. Run R on <M1>: M1 is rejected iff M accepts w. Thus with R we could decide ATM = {<M,w>| M is a TM and M accepts w} But we know ATM is undecidable Hence R cannot exist and ETM is undecidable Jan Maluszynski - HT 2007

6. Rice’s theorem Let P be any nontrivial property of the language of a Turing machine M. Nontrivial: it contains some but not all TM descriptions. Theorem: {<M>| L(M) satisfies P} is undecidable e.g given M it is undecidable if L(M) is: • empty, • regular, • Context-free …. Jan Maluszynski - HT 2007

7. Mapping reducibility A function f: *  * is computable if some TM M on every input w halts with f(w) on its tape Language A is mapping reducible to B written AmB if there is a computable function f s.that for every w wA  f(w)B f is called reduction of A to B. Jan Maluszynski - HT 2007

8. Mapping reducibility cont. Theorem: If AmB and A is undecidable then B is undecidable Example: f(<M>) = <M,M1> where M1 rejects all inputs reduces emptiness problem to TM equivalence Jan Maluszynski - HT 2007

9. Mapping reducibility cont. Theorem: If AmB and A is not Turing-recognizable then B is not Turing-recognizable Remark: AmB iff complement(A) m complement(B) To prove B not Turing-recognizable show: A mcomplement(B) for some undecidable A Jan Maluszynski - HT 2007

10. TM equivalence is not T-recognizable Reduce ATM to (negation) EQTM For given <M,w> construct TMs: (M1 rejects all inputs) M3 accepts all inputs M2: On any input simulates M on w accepts if M accepts w • <M1,M2> equivalent iff M does not accept w • <M3,M2> equivalent iff M accepts w Hence: by (1) EQTM not Turing-recognizable by (2) complement of EQTM not Turing-recognizable Jan Maluszynski - HT 2007

11. More examples of undecidable languages Hilbert’s tenth problem: Does a given polynomial with integer coefficients has an integral root? Stated early 1900 proved undecidable 1970 The language Th(N,+,) of true closed arithmetic formulae interpreted on natural numbers e.g. xyz[x=y+z] Jan Maluszynski - HT 2007