CSCI 4325 / 6339 Theory of Computation Zhixiang Chen

Chapter 9 Intractability

Topics Hierarchy Theorems  Time Hierarchy  Space Hierarchy Relativation Circuit Complexity

Space Contractible Functions Definition.

Space Hierarchy Theorem Theorem Proof.

Construction of A

Corollaries

Time Constructible Functions Definition. Examples:

Time Hierarchy Theorem Theorem. Proof. Construct such an A via diagonalization.

Construction of A

Corollaries

Relativation We have shown that diagonalization can be used to establish both the space hierarchy and the time hierarchy. Can this technique be used to separate NP from P? Relativation shows strong evidence that diagonalization cannot be used to this.

Oracle Turing Machines Definition.  An oracle for a language A is a device that is capable of reporting whether any string w is a member of A.  An oracle TM is a modified TM that has the additional capability of querying an oracle. Whenever writes a string on a special oracle tape it is informed whether that string is in A or not in a single step.  Let be the class of all languages decidable with a polynomial time oracle TM that uses oracle A.  Let be the class of all languages decidable with a polynomial time oracle NTM that uses oracle A.

Easy Examples

Limits of Diagonalization Theorem. Proof.

Circuit Complexity Definition. A Boolean Circuit is a collection of gates and inputs connected by wires. Cycles are not permitted. Gates are of three forms: AND gates, OR gates, and NOT gates. inputs OR NOT AND

Examples A Boolean circuit Computing of a Boolean Circuit

Computing n-Input Parity Function

Circuit family, size and depth

Circuit Complexity of a Language Definition. Example.

Time Complexity vs. Circuit Complexity It is obvious that the circuit complexity of a language is the related to its time complexity. The following result also shows that the time complexity of a language is related its circuit complexity. Theorem.  This theorem gives an approach to proving that P is not equal to NP: Prove some language in NP has more than polynomial circuit complexity.

Proof of The Theorem. Idea. Use circuit to simulate the computation of a TM M on an input w of size n in p(n) time.

CIRCUIT-SAT A Boolean Circuit is satisfiable if some setting of the inputs causes the circuit to output 1. Theorem.