CSE200. Computability and Complexity. Daniele Micciancio. Fall 2008 UCSD

Transcription

1 Computability and Complexity UCSD Fall 2008

2 Outline Reductions 1 Reductions

3 Summary Reductions Last lecture: We proved that the language Diag = { P : P( P ) {0, }} is undecidable Technique: diagonalization Remark: Diag was defined to make diagonalization work Today: More natural examples of undecidable languages Proofs by reduction

4 More undecidable languages An interesting undecidable language: Loop = {[ P L]: P(L) = }?!!?!?! A script on this slide is not responding. Do you want to stop it? How can we prove that Loop is undecidable? Diagonalization seems tailored to study problems specifically designed to make it work. Instead of proving the undecidability of Loop from scratch, we prove it by comparing it to Diag Reductions: technical tool to compare the computational hardness of different problems

5 Reductions A reduction from A to B (typically denoted A B) is a method to show that A is at most as hard as B Properties of reductions If A B and B is easy, then A is also easy If A B and B C, then A C. Remarks: Equivalently: If A B and A is hard, then B is also hard. (Example: we want to prove Diag Loop.) You can establish a reduction relation regardless of the (known) solvability of either problem. Different kinds of reductions can be defined, depending on the notion of solvability we are interested in.

6 Map Reductions Reductions Definition A reduction from A to B is a computable function f such that L.L A f (L) B. Think of reductions f : A B as a way to translate questions regarding problem A to questions regarding problem B. If you can translate Arabic into Chinese, and you can translate Chinese into English, then you can also translate Arabic into English. If you can translate Chinese into English, and you can read English, then you can read Chinese. If you can translate Arabic into Chinese, and you cannot read Arabic, then you cannot read Chinese. If you can translate Arabic into Chinese, and you cannot read Chinese, then...

7 Using reductions Reductions Theorem If A B and B is decidable, then A is decidable Proof. Let {F 1 ;... F n } be a WHILE program computing the reduction A B, and {P 1 ;... ; P m } be a WHILE program deciding B. Then {F 1 ;... ; F n ; P 1 ;... ; P m } is a WHILE program deciding A. (See whiteboard for details.) Corollary If A B and A is undecidable, then B is undecidable.

8 Reducing Diag to Loop Theorem Loop is undecidable. Proof. By reduction from Diag. On input Diag instance M, the reduction outputs a pair [ P M ] where P is the program M; Y=1 while X do { X=0; Y=0 } Y:=not(X); X:=0 while Y do {} Loop unless Y is 0 Details on the next slide.

9 Proof details Reductions The function f : M [ P M ] is computable. It is computed by the program: T = [[LET #1 [LIST []]] [WHILE #0 [[LET #0 [LIST []]] [LET #1 [LIST []]]]] [WHILE #1 []]; APPEND(X,Y) X = X [] X = T [] The function f correctly reduces Diag to Loop Assume M Diag (i.e., M( M ) {0, }). Is [ P M ] Loop? YES, because P( M ) =. Assume M / Diag (i.e., M( M ) = 1). Is [ P M ] / Loop? YES, because P( M ) = 0.

10 What about the Halting problem You probably heard of the Halting problem : Halting Problem {[ P L] : P(L) } Notice the similarity with Loop = {[ P L] : P(L) = }. Question Can you give a reduction from Diag or Loop to Halting? No, at least using our current definition of reduction. Still, Halting is undecidable and can be proved using Turing reductions.

11 Turing reductions Reductions Definition A Turing reduction is a program written in an extension of the WHILE language with a special query expression : E ::= L V head V tail V V V Query(V ) Remarks Intuitively, Query(V ) represents a remote procedure call. The code to compute Query(V ) is not run locally. The program simply transmits the content of V to an external device, and and evaluates Query(V ) to whatever value is returned from it. A reduction is a finite sequence of commands, just like normal WHILE programs. Reductions can be represented as lists, just like ordinary WHILE programs by adding one more constant QUERY to the expression types.

12 Running Turing reductions Unlike map reductions, Turing reduction cannot be directly executed on your computer: you also need an external device to handle the remote procedure calls. We are not concerned with the complexity and internal workings of this external device. (Say, we use a service provided by SDSC). All you care about is the local computation performed on your computer. Still, we have some expectations on how our queries will be answered. E.g., assume the external service tests whether the query L belongs to the language Halt, and returns either 0 or 1. Can you solve the Loop problem using this external service?

13 Turing Reduction from Loop to Halt Consider the reduction R(X) Y=Query(X) X=1; while Y do {X=0; Y=0} Assuming Query(X) properly evaluates to X Halt, then If X Loop, then Query(X) = 0 and R(X) accepts. If X / Loop, then Query(X) = 1 and R(X) rejects. Conclusion: The decision problem (Halt) solved by Query(X) must be undecidable!

14 Properties of Turing reductions Given a reduction R and a language A, the execution of R with oracle access to A (typically denoted R A ) is defined in the usual way by setting Query(L) = 1 if L A, and Query(L) = 0 otherwise. R is a Turing reduction from A to B (written A T B) if for all X A, R B (X) = 1 for all X / A, R B (X) = 0. (In particular, R B always terminates.) Theorem If A T B and B is decidable, then A is also decidable. Corollary If A T B and A is undecidable, then B is also decidable.

15 Final remarks about reductions Reductions are real (programs): They exists independently from the (often hypothetical) oracle you intend to run them with In principle, the same reductions R can be run with different oracles R A, R B to solve different problems. However, this is rarely useful, and each reduction R is designed with a specific oracle A in mind. Just like programs, reductions can be represented as lists, and manipulated by other programs. Theorem If A T B and B T C, then A T C. In fact, one can say more: there is a computable function that on input two reductions [ F G ], outputs a third reduction H, such that if Lang(F B ) = A and Lang(G C ) = B, then Lang(H C ) = A.