Algorithms & Models of Computation CS/ECE 374, Fall 2017 Undecidability II: More problems via reductions Lecture 21 Thursday, November 16, 2017 Sariel Har-Peled (UIUC) CS374 1 Fall 2017 1 / 5
Turing machines... TM = Turing machine = program. Sariel Har-Peled (UIUC) CS374 2 Fall 2017 2 / 5
Reminder: Undecidability Definition 1 Language L ⊆ Σ ∗ is undecidable if no program P , given w ∈ Σ ∗ as input, can always stop and output whether w ∈ L or w / ∈ L . (Usually defined using TM not programs. But equivalent. Sariel Har-Peled (UIUC) CS374 3 Fall 2017 3 / 5
Reminder: Undecidability Definition 1 Language L ⊆ Σ ∗ is undecidable if no program P , given w ∈ Σ ∗ as input, can always stop and output whether w ∈ L or w / ∈ L . (Usually defined using TM not programs. But equivalent. Sariel Har-Peled (UIUC) CS374 3 Fall 2017 3 / 5
Reminder: Undecidability Definition 1 Language L ⊆ Σ ∗ is undecidable if no program P , given w ∈ Σ ∗ as input, can always stop and output whether w ∈ L or w / ∈ L . (Usually defined using TM not programs. But equivalent. Sariel Har-Peled (UIUC) CS374 3 Fall 2017 3 / 5
Reminder: The following language is undecidable Decide if given a program M , and an input w , does M accepts w . Formally, the corresponding language is � � � A TM = � M , w � � M is a TM and M accepts w . � Definition 2 A decider for a language L , is a program (or a TM ) that always stops, and outputs for any input string w ∈ Σ ∗ whether or not w ∈ L . A language that has a decider is decidable . Turing proved the following: Theorem 3 A TM is undecidable. Sariel Har-Peled (UIUC) CS374 4 Fall 2017 4 / 5
Reminder: The following language is undecidable Decide if given a program M , and an input w , does M accepts w . Formally, the corresponding language is � � � A TM = � M , w � � M is a TM and M accepts w . � Definition 2 A decider for a language L , is a program (or a TM ) that always stops, and outputs for any input string w ∈ Σ ∗ whether or not w ∈ L . A language that has a decider is decidable . Turing proved the following: Theorem 3 A TM is undecidable. Sariel Har-Peled (UIUC) CS374 4 Fall 2017 4 / 5
Reminder: The following language is undecidable Decide if given a program M , and an input w , does M accepts w . Formally, the corresponding language is � � � A TM = � M , w � � M is a TM and M accepts w . � Definition 2 A decider for a language L , is a program (or a TM ) that always stops, and outputs for any input string w ∈ Σ ∗ whether or not w ∈ L . A language that has a decider is decidable . Turing proved the following: Theorem 3 A TM is undecidable. Sariel Har-Peled (UIUC) CS374 4 Fall 2017 4 / 5
Part I Reductions Sariel Har-Peled (UIUC) CS374 5 Fall 2017 5 / 5
Reduction Meta definition: Problem A reduces to problem B , if given a solution to B , then it implies a solution for A . Namely, we can solve B then we can solve A . We will done this by A = ⇒ B . Definition 4 oracle ORAC for language L is a function that receives as a word w , returns TRUE ⇐ ⇒ w ∈ L . Definition 5 A language X reduces to a language Y , if one can construct a TM decider for X using a given oracle ORAC Y for Y . We will denote this fact by X = ⇒ Y . Sariel Har-Peled (UIUC) CS374 6 Fall 2017 6 / 5
Reduction Meta definition: Problem A reduces to problem B , if given a solution to B , then it implies a solution for A . Namely, we can solve B then we can solve A . We will done this by A = ⇒ B . Definition 4 oracle ORAC for language L is a function that receives as a word w , returns TRUE ⇐ ⇒ w ∈ L . Definition 5 A language X reduces to a language Y , if one can construct a TM decider for X using a given oracle ORAC Y for Y . We will denote this fact by X = ⇒ Y . Sariel Har-Peled (UIUC) CS374 6 Fall 2017 6 / 5
Reduction Meta definition: Problem A reduces to problem B , if given a solution to B , then it implies a solution for A . Namely, we can solve B then we can solve A . We will done this by A = ⇒ B . Definition 4 oracle ORAC for language L is a function that receives as a word w , returns TRUE ⇐ ⇒ w ∈ L . Definition 5 A language X reduces to a language Y , if one can construct a TM decider for X using a given oracle ORAC Y for Y . We will denote this fact by X = ⇒ Y . Sariel Har-Peled (UIUC) CS374 6 Fall 2017 6 / 5
Reduction proof technique B : Problem/language for which we want to prove undecidable. 1 Proof via reduction. Result in a proof by contradiction. 2 L : language of B . 3 Assume L is decided by TM M . 4 Create a decider for known undecidable problem A using M . 5 Result in decider for A (i.e., A TM ). 6 Contradiction A is not decidable. 7 Thus, L must be not decidable. 8 Sariel Har-Peled (UIUC) CS374 7 Fall 2017 7 / 5
Reduction proof technique B : Problem/language for which we want to prove undecidable. 1 Proof via reduction. Result in a proof by contradiction. 2 L : language of B . 3 Assume L is decided by TM M . 4 Create a decider for known undecidable problem A using M . 5 Result in decider for A (i.e., A TM ). 6 Contradiction A is not decidable. 7 Thus, L must be not decidable. 8 Sariel Har-Peled (UIUC) CS374 7 Fall 2017 7 / 5
Reduction proof technique B : Problem/language for which we want to prove undecidable. 1 Proof via reduction. Result in a proof by contradiction. 2 L : language of B . 3 Assume L is decided by TM M . 4 Create a decider for known undecidable problem A using M . 5 Result in decider for A (i.e., A TM ). 6 Contradiction A is not decidable. 7 Thus, L must be not decidable. 8 Sariel Har-Peled (UIUC) CS374 7 Fall 2017 7 / 5
Reduction proof technique B : Problem/language for which we want to prove undecidable. 1 Proof via reduction. Result in a proof by contradiction. 2 L : language of B . 3 Assume L is decided by TM M . 4 Create a decider for known undecidable problem A using M . 5 Result in decider for A (i.e., A TM ). 6 Contradiction A is not decidable. 7 Thus, L must be not decidable. 8 Sariel Har-Peled (UIUC) CS374 7 Fall 2017 7 / 5
Reduction proof technique B : Problem/language for which we want to prove undecidable. 1 Proof via reduction. Result in a proof by contradiction. 2 L : language of B . 3 Assume L is decided by TM M . 4 Create a decider for known undecidable problem A using M . 5 Result in decider for A (i.e., A TM ). 6 Contradiction A is not decidable. 7 Thus, L must be not decidable. 8 Sariel Har-Peled (UIUC) CS374 7 Fall 2017 7 / 5
Reduction proof technique B : Problem/language for which we want to prove undecidable. 1 Proof via reduction. Result in a proof by contradiction. 2 L : language of B . 3 Assume L is decided by TM M . 4 Create a decider for known undecidable problem A using M . 5 Result in decider for A (i.e., A TM ). 6 Contradiction A is not decidable. 7 Thus, L must be not decidable. 8 Sariel Har-Peled (UIUC) CS374 7 Fall 2017 7 / 5
Reduction proof technique B : Problem/language for which we want to prove undecidable. 1 Proof via reduction. Result in a proof by contradiction. 2 L : language of B . 3 Assume L is decided by TM M . 4 Create a decider for known undecidable problem A using M . 5 Result in decider for A (i.e., A TM ). 6 Contradiction A is not decidable. 7 Thus, L must be not decidable. 8 Sariel Har-Peled (UIUC) CS374 7 Fall 2017 7 / 5
Reduction proof technique B : Problem/language for which we want to prove undecidable. 1 Proof via reduction. Result in a proof by contradiction. 2 L : language of B . 3 Assume L is decided by TM M . 4 Create a decider for known undecidable problem A using M . 5 Result in decider for A (i.e., A TM ). 6 Contradiction A is not decidable. 7 Thus, L must be not decidable. 8 Sariel Har-Peled (UIUC) CS374 7 Fall 2017 7 / 5
Reduction implies decidability Lemma 6 Let X and Y be two languages, and assume that X = ⇒ Y . If Y is decidable then X is decidable. Proof. Let T be a decider for Y (i.e., a program or a TM ). Since X reduces to Y , it follows that there is a procedure T X | Y (i.e., decider) for X that uses an oracle for Y as a subroutine. We replace the calls to this oracle in T X | Y by calls to T . The resulting program T X is a decider and its language is X . Thus X is decidable (or more formally TM decidable). Sariel Har-Peled (UIUC) CS374 8 Fall 2017 8 / 5
The countrapositive... Lemma 7 Let X and Y be two languages, and assume that X = ⇒ Y . If X is undecidable then Y is undecidable. Sariel Har-Peled (UIUC) CS374 9 Fall 2017 9 / 5
Part II Halting Sariel Har-Peled (UIUC) CS374 10 Fall 2017 10 / 5
The halting problem Language of all pairs � M , w � such that M halts on w : � � � A Halt = � M , w � � M is a TM and M stops on w . � Similar to language already known to be undecidable: � � � A TM = � M , w � � M is a TM and M accepts w . � Sariel Har-Peled (UIUC) CS374 11 Fall 2017 11 / 5
The halting problem Language of all pairs � M , w � such that M halts on w : � � � A Halt = � M , w � � M is a TM and M stops on w . � Similar to language already known to be undecidable: � � � A TM = � M , w � � M is a TM and M accepts w . � Sariel Har-Peled (UIUC) CS374 11 Fall 2017 11 / 5
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