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Chapter 7: Q49P (page 275)


Let f:N→Nbe any function where fn=onlogn. Show that TIMEfn contains only the regular languages.

Short Answer

Expert verified

Thus, it is only a regular language. It is accepted by the TIMEfn.

Step by step solution

01

To Regular Language

A language is a collection of strings made up of characters or symbols from a specific alphabet.

A regular language is one that may be stated using regular expressions, finite automata, or state machines, both deterministic and non-deterministic.

02

To Explain and remove the form

Suppose language is A={0k1k|k≥0}∈TIME(nlogn)

  • If w is not of the form, reject M(w).
  • Rep till all w's have been crossed out.
  • Reject ifparityf0's≠parityf1's.
  • Remove every other 0 and 1 from the equation.
  • If all other bits are crossed out, accept.

A language is a collection of strings made up of characters or symbols from a specific alphabet.

Thus, it is only a regular language. It is accepted by the TIMEfn.

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Most popular questions from this chapter

For a cnf-formula ∅ with m variables and cclauses O(cm) , show that you can construct in polynomial time an NFA with states that accept all nonsatisfying assignments, represented as Boolean strings of length m. Conclude that P≠NP implies that NFAs cannot be minimized in polynomial time.

Let A⊆1* be any unary language. Show that if A is NP-complete, then P = NP. (Hint: Consider a polynomial time reduction f from SATto A. For a formula ϕ, let ϕ0100be the reduced formula where variables x1, x2, x3, and x4 inϕ are set to the values 0, 1, 0, and 0, respectively. What happens when you apply f to all of these exponentially many reduced formulas?)

A 2cnf-formula is an AND of clauses, where each clause is an OR of at most two literals. Let . Show that2SAT={ϕ|ϕ is a satisfiable 2CNF formula}. Show that 2SAT∈P.

Let UNARY−SSUM be the subset sum problem in which all numbers are represented in unary. Why does the NP− completeness proof for SUBSET−SUM fail to show UNARY−SSUM is NP− complete? Show that UNARY−SSUM∈P

Let CONNECTED={<G>|Gisaconnectedundirectedgraph}.Analyse the algorithm given on page 185 to show that this language is in .
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