How to Prove a Language Is Not Recognizable

We will see examples soon. To prove that a given language is non-Turing-recognizable.

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Can prove formally number of remaining symbols at each step Showing language is Turing recognizable but not decidable is harder 15 Many equivalent variants of TM TM that can stay put on tape for a given transition TM with multiple tapes TM with non-deterministic transitions Can select convenient alternative for current problem 16.

. Build a decider for it. Accept exactly when w accepts. Let L L 1 L 2 Σ such that L L 1 L 2 and L 2 is decidable.

How do we show that a language is TR. Alin Tomescu 6840 Theory of Computation Fall 2013 taught by Prof. One important result is that L is decidable if and only if L is recognizable and co-recognizable.

Requires reasoning about all possible TMs. Build a Recognizer for L. 2 Reductions Note that the key to this proof is to use the fact that A TMis not Turing-decidable.

How do we show that a language is TD. If not prove using reduction. If 𝐽 is undecidable and 𝐽𝑚𝐽 then both 𝐽 and 𝐽 are not Turing-recognizable.

And so far I believe theres three main ways to prove a language is not regular. Lf pm The language of M is finite Is Lf recognizable. A language is Turing recognizable aka recursively enumerable if theres a Turing machine that accepts words that are part of the language and either rejects or doesnt halt on words that are not part of the language.

Pushing as first then bs. Decidable languages are closed under complementation but recognizable languages are not. Does decidable imply recognizable.

It must either reject or loop on any string not in the language. D M M rejects input M where M is a Turing machine and M is a binary encoding of M and it is already assumed M is neither decidable nor recognizable. On input run MH on.

If is not decidable then or is not Turing-recognizable. But not all languages are regular. Build a decider or recognizer for it.

This known non-Turing-recognizable language can be any language for which. To demonstrate that a language is Turing recognizable you need only demonstrate that such a machine exists. I need to prove a language L is not Turing-recognizable by giving a reduction from the following language.

Prove that the language it recognizes is equal to the given language and that the algorithm halts on all inputs. PROVE that there is no Decider for l. Now we will not be able to.

We know X does not. Prove that if L is not recognizable then L 1 is not recognizable. We provide a reduction from ATM to S ATM m.

Decidable versus Recognizable Languages A language is Turing-recognizable if there is a Turing machine M such that LM L ¼For all strings in L M halts in state q ACC ¼For strings not in L M may either halt in q REJ or loop forever A language is decidable if there is a decider Turing machine M that halts on all inputs such that LM L. It must either reject or loop on any string not in the language. To prove that a given language is Turing-recognizable.

AH TM M halts on input w Need to show AH is undecidable We know ATM TM M accepts w is undecidable Show ATM is reducible to AH Theorem 51 in text Suppose AH is decidable theres a decider MH for AH Then we can construct a decider DTM for ATM. L1 and L2 recognizable INTERLACEL1 L2 recognizable. Language M is recognizable but not decidable.

If it answers YES the word is in the language if it answers NO the word is not in the language. To prove that a given language is Turing-recognizable. To prove a language is undecidable need to show there is no Turing Machine that can decide the language.

Im kinda sure that Lf can not be recognized but I aint sure how to prove it. One can easily prove that M is the decision algorithm for INTERLACEL1 L2 thus the language is decidable. It answers YES or NO.

There are three equivalent definitions of a recursively enumerable also recognizable language. For the second claim suppose that M would be decidable. When presented with If not reject.

Section 42 of the textbook extends this to a theorem giving that a language Ais decidable if and only if its both Turing-recognizable and co-Turing-recognizable. Regular languages are those that can be recognized by some DFA and I made some examples of that in the first Theory post. Complement would be decidable as well as the intersection of its.

Prove that its complement is Turing-recognizable. Construct a mapping reduction from another language already known to be non-Turing- recognizable to the given language. Being decidable means you can build an automatic process that takes a word as an input such that.

A point to remember is counting and comparison could only be done with the top of stack and not with bottom of stack in Push Down Automata hence a language exhibiting a characteristic that involves comparison with bottom of stack is not a context free language. How do we show that language L is TR and not TD. Languages can be classified as either regular or non-regular.

Im having a really hard time understanding some of these concepts. A language is Turing recognizable aka recursively enumerable if theres a Turing machine that accepts words that are part of the language and either rejects or doesnt halt on words that are not part of the language. Construct an algorithm that accepts exactly those strings that are in the language.

We did not need to examine a supposed Turing machine that recognizes A. Proof By definition since we know L 2 is decidable then there exists a Turing machine M over Σ such that for some input w L 2 M either accepts or reject w. Otherwise build a NDTM which recognizes it.

Michael Sipser 2 Turing-unrecognizability If 𝑚 and is not T-recognizable then is not Turing-recognizable by mapping-reducibility to unrecognizable language. For the first claim construct an algorithm recognizing M as follows. Example 1 L is not context free.

Proving Undecidability 6 Proof by Reduction 1. Either do both of these. LM M means that the only string the M accepts is its own description M hopefully you can see now that S is the language of TMs that accept only their own descriptions.

Given the following language. To demonstrate that a language is Turing recognizable you need only demonstrate that such a machine exists. Prove that its complement is undecidable.

Showing that a Language is DecidableRecognizable. Construct an algorithm that accepts exactly those strings that are in the language.

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