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Understanding Reducibility: Solving Problems through Effective Problem Transformation

Reducibility is a concept that allows the transformation of one problem (A) into another (B) such that a solution for B can be utilized to resolve A. This principle is fundamental in areas like mathematical problem solving, where measuring a rectangle's area can reduce to finding its length and width. Reducibility is crucial for classifying problems by decidability and complexity. If A is reducible to B, then A cannot be harder than B. This concept extends to computability theory, highlighting connections between decidable and undecidable problems.

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Understanding Reducibility: Solving Problems through Effective Problem Transformation

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  1. Reducibility

  2. Text book Pages 187– 199

  3. Reducibility • A reductionis 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.

  4. 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.

  5. 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.

  6. UNDECIDABLE PROBLEMS FROMLANGUAGE THEORY

  7. UNDECIDABLE PROBLEMS FROMLANGUAGE THEORY PROOF IDEA

  8. UNDECIDABLE PROBLEMS FROMLANGUAGE THEORY PROOF IDEA

  9. UNDECIDABLE PROBLEMS FROMLANGUAGE THEORY

  10. UNDECIDABLE PROBLEMS FROMLANGUAGE THEORY

  11. UNDECIDABLE PROBLEMS FROMLANGUAGE THEORY • context-free language can be shown to be undecidable with similar proofs.

  12. REDUCTIONS VIA COMPUTATION HISTORIES (read only)

  13. 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

  14. Collection of dominos - match • a collection of dominos looks like 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

  15. 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.

  16. Match may not be possible • cannot contain a match because every top string is longer than the corresponding bottom string

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