CS 155, Programming Paradigms Fall 2014, SJSU nondeterminism

1. CS 155, Programming ParadigmsFall 2014, SJSUnondeterminism Jeff Smith

2. NP and nondeterminism • The definition of NP depends on the notion of nondeterminism • And it’s easiest to define nondeterminism in terms of the state of a computation • here, the term state includes information like the values of variables and other memory contents • so the closest CS 154 concept is instantaneous description (ID) rather than state.

3. Models of nondeterminism • There are several useful models of nondeterministic computation. • In each model, we think of computation as proceeding from state to state. • All of these models are equivalent -- they describe the same class of (nondeterministic) algorithms.

4. Nondeterministic Turing machines • Nondeterministic Turing machines: a transition function gives the possible successors of a given ID • control may pass to any of the successors. • This model is what we’ll use to show that there’s a hardest problem in NP.

5. Unbounded parallelism and decision trees • In unbounded parallelism, each successor state is assigned to a newly allocated processor • In decision trees, states are represented as tree nodes, • where state T is a child of state U iff T can be reached from U by a single decision.

6. Nondeterministic time complexity • In each of the three models above, it's natural to define (nondeterministic) time complexity to be the length of a shortest accepting computation. • This is a natural concept in each model. • To define the length of the input, we use the length of the string that represents the instance.

7. Guessing and verifying • In the guessing and verifying model, a solution is guessed in a preliminary guessing phase • where each guess takes O(1) time, • and must select from a bounded set of possible choices. • The solution is then verified deterministically • This is the model we'll use most.

8. Verifying without guessing • Sometimes this model is taken to have only a guessing phase • The sequence of guesses is replaced by a bit string called the certificate • which is the input to the verification process • CLRS takes this approach, on p.1064 • note that they need to stipulate that the certificate has polynomial length • corresponding to a polynomial number of guesses

9. Time complexity of guessing and verifying • Time complexity in the guess-and-verify model is computed as always … • Except that if verification fails, the time complexity is irrelevant • formally, it’s taken to be O(1) in this case • note that this gives an asymmetry between success and failure

10. Defining the class NP • The class NP = {P | P has a nondeterministic polynomial-time solution algorithm}. • Its membership does not depend on which model of nondeterminism is used to define it. • a polynomial-time algorithm in one model takes polynomial time in every other model • So this definition of NP now gives us a precise definition of NP-completeness.

11. NP-completeness • The hardest problems in NP are the problems P such that Q α P for every Q in NP. • Such problems are called NP-complete. • It’s not obvious that such problems exist • but they do! • If an NP-complete problem is in P, then P=NP. • it’s not known whether this is the case • that is, we don’t know whether these apparently hard problems are actually hard