1Computer Sciences Department

Book: INTRODUCTION TO THE THEORY OF COMPUTATION, SECOND EDITION, by: MICHAEL SIPSER Reference 3Computer Sciences Department

Reducibility 4 Text book Pages 187– 199

 Reduction  Proof : Undecidable Computer Sciences Dep55artment5 Objectives

Reducibility  A reduction is a way of converting one problem to another problem in such a way that a solution to the second problem can be used to solve the first problem.  Example: find your way around a new city – using a map.  Reducibility always involves two problems, which we call A and B. If A reduces to B, we can use a solution to B to solve A. 6

Examples of Reducibility  Sorting a list of numbers (ascending or descending) reduces to the pair of problems:  Find the smallest number in a list.  Swap two numbers in a list. 7

Reducibility (cont’d)  Reducibility also occurs in mathematical problems.  For example, the problem of measuring the area of a rectangle reduces to the problem of measuring its length and width. 8

 How could we reduce the problem of finding the area of the hexagon to simpler problems? Computer Sciences Department9 Geometric Example 1- partition the hexagon into a collection of rectangles and triangles. 2- find the areas of rectangles and triangles.

Reducibility (cont’d)  Reducibility plays an important role in classifying problems by decidability and later in complexity theory as well.  When A is reducible to B, solving A cannot be harder than solving B because a solution to B gives a solution to A.  In terms of computability theory, if A is reducible to B and B is decidable, A also is decidable.  If A is undecidable and reducible to B, B is undecidable. 10

 Theorem  If A is reducible to B and B is decidable, then A is decidable  Proof.  Let R be a Turing machine that reduces A to B.  Let D B be a decider of B. Computer Sciences Department11 Reducibility and Decidability

If A is undecidable and A is reducible to B, then B is undecidable. ??????????? Computer Sciences Department12 Theorem

UNDECIDABLE PROBLEMS FROM LANGUAGE THEORY 13 Define the language HALT TM to be

UNDECIDABLE PROBLEMS FROM LANGUAGE THEORY 14 PROOF IDEA

 Proof:  reduce A TM to HALT TM.  Suppose that HALT TM is decidable.  Let D H be a decider for HALT TM.  build a decider D A for A TM. Computer Sciences Department15 Explanation - A TM

UNDECIDABLE PROBLEMS FROM LANGUAGE THEORY 16 PROOF IDEA Define the language E TM to be

Computer Sciences Department17 Explanation - E TM Let CONVERT be a Turing machine that will read the pair and construct the Turing machine M w. Proof

UNDECIDABLE PROBLEMS FROM LANGUAGE THEORY 18

UNDECIDABLE PROBLEMS FROM LANGUAGE THEORY 19

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Computer Sciences Department21

UNDECIDABLE PROBLEMS FROM LANGUAGE THEORY  context-free language can be shown to be undecidable with similar proofs. 22

Thus, we could build a decider for all previous (E TM, Regular TM,…..etc, which we know to be impossible. Computer Sciences Department23 Build = Impossible

REDUCTIONS VIA COMPUTATION HISTORIES (read only) 24

A SIMPLE UNDECIDABLE PROBLEM  The phenomenon of undecidability is not confined to problems concerning automata  a collection of dominos, each containing two strings, one on each side. An individual domino looks like 25

Collection of dominos - match  a collection of dominos looks like 26 The task is to make a list of these dominos (repetitions permitted) so that the string we get by reading off the symbols on the top is the same as the string of symbols on the bottom. This list is called a match

Match may be possible  Reading off the top string we get abcaaabc, which is the same as reading off the bottom. We can also depict this match by deforming the dominos so that the corresponding symbols from top and bottom line up. 27

Match may not be possible  cannot contain a match because every top string is longer than the corresponding bottom string 28