Observe that A is a context-free language so it is also Turing-decidable. Thus
Typical approach to show L is undecidable via reduction from ATM to L. Suppose L is decidable. Universe ? = {?M?
4 mars 2019 Reducibility: a tool to show undecidability ... reduce ATM to HALTTM ... Turing machine M on input w
by creating a decider S of ATM: some Turing machine M on every input w
5.5 Show that ATM is not mapping reducible to ETM. I will be using the ¬ symbol ATM is Turing recognizable but not co-Turing recognizable. The TM below.
28 nov. 2006 Lets do this reduction more formally now and give a computable function that shows ATM ?m ETM. First
assumption to show that ATM is decidable contradicting Theorem Example 6.19 shows that ETM is Turing reducible to ATM. Computability Theory.
HALTTM is not Turing-decidable. Proof: We will reduce ATM to HALTTM. Assume TM R decides HALTTM. We construct TM S that decides ATM as follows: On
Since f is a mapping reduction ATM ? EQTM it is also a mapping reduction ATM ? EQTM. Hence
What we've called a reduction up until now is also called a Turing reduction. Theorem Show that ETM ?m EQTM by giving a TM T that computes the mapping.
One can often show that a language Lis undecidableby showing that if L is decidable then so is ATM We reduceATM to the language L ATM ? L We showed:ATM ? HALT TM Mapping Reductions ??* is a computable functionif there is a Turing machine M that halts with just f(w) written on its tape for every input w A language A is mapping reducible to
show that some other problem already known to be undecidable reduces to UNDECIDABLE PROBLEMS FROM LANGUAGE THEORY We have already established the undecidability of ATM the problem of mining whether a Turing machine accepts a given input Let's consider a problem HALT TM the problem of determining whether a Turing machine
2 Show that A TM is not mapping reducible to E TM In other words show that no computable function reduces A TM to E TM (Hint: Use a proof by contradiction and facts you already know about A TM and E TM ) Answer: Suppose for a contradiction that A TM m E TM via reduction f This means that w 2A TM if and only if f(w) 2E TM which is
Intuitively for recognizability we allow our TM to run forever on inputs that are not inthe language while for decidability we require that the TM halt on every input Now recall our two most important results: Theorem 1 There exist (uncountably many!) languages which are not Turing-recognizable
Show ATMis reducible to AH(Theorem 5 1 in text) Suppose Ais decidable Hthere’s a decider MHfor A H Then we can construct a decider DTMfor ATM: On input run M on If MHrejects then REJ (this takes care of M looping on w) If MHaccepts then simulate M on w until M halts If M accepts then ACC input ; else REJ
We show A TM reduces to E TM Proof: Assume E TM is decidable with TM R We show then that A TM is decidable a contradiction • Using R we construct a TM M ATM that decides A TM: M ATM = Correctness: M ATM is a decider since R is and accepts iff L(X) is nonempty iff M accepts w • But A TM is undecidable a
Answer: We need to show there is a Turing machine that recognizes ETM, the com- plement of ETM. Let s1,s2,s3,... be a list of all strings in ??. For a given Turing machine M, we want to determine if any of the strings s1,s2,s3,... is accepted by M.
Turing machine M that halts with just f(w) written on its tape, for every input w. language A is mapping reducibleto language B, written A ?mB, if there is a computable f : ?* ??*, such that for every w, ?A ?f(w) ?B is called a mapping reduction (or many8one reduction) from A to BLet f : ?* ??* be a computable functionsuch that w ?A ?f(w) ?B
The original password authentication method combined with the biometric technology for identification in the ATM machine has improved the security of the transaction with increasing emphasizes given to security automated personal identification needs to be added to the traditional ATM system to overcome its disadvantages.
The term refers to low-level operators within a criminal organization who are assigned high-risk jobs, such as installing ATM skimmers and otherwise physically tampering with cash machines.