1 Module 25 Decision problems about regular languages –Basic problems are solvable halting, accepting, and emptiness problems –Solvability of other problems answer-preserving input transformations to basic problems

2 Programs In this unit, our programs are the following three types of objects –FSA’s –NFA’s –regular expressions Previously, they were C ++ programs –Review those topics after mastering today’s examples

3 Basic Decision Problems (and algorithms for solving them)

4 Halting Problem * Input –FSA M –Input string x to M Question –Does M halt on x? Give an algorithm for solving the FSA halting problem.

5 Accepting Problem * Input –FSA M –Input string x to M Question –Is x in L(M)? Give an algorithm ACCEPT for solving the accepting problem.

6 Empty Language Problem * Input –FSA M Question –Is L(M)={}? Give an algorithm for solving the empty language problem. –Don’t look ahead to the next slide.

7 Algorithms for solving empty language problem * Algorithm 1 –View FSA M as a directed graph (nodes, arcs) –See if any accepting node is reachable from the start node Algorithm 2 –Let n be the number of states in FSA M –Run ACCEPT(M,x) for all input strings of length < n –If any are accepted THEN no ELSE yes Why is algorithm 2 correct?

8 Solving Other Problems (using answer-preserving input transformations)

9 Complement Empty Problem * Input –FSA M Question –Is (L(M)) c = {}? Show how to use an answer-preserving input transformation to help solve this problem –Show that the Complement Empty problem transforms to the Empty Language problem –Don’t look at next two slides

10 Algorithm Description Convert input FSA M into an FSA M’ such that L(M’) = (L(M)) c –We do this by applying the algorithm which we used to show that LFSA is closed under complement Feed FSA M’ into algorithm which solves the empty language problem If that algorithm returns yes THEN yes ELSE no

11 Input Transformation Illustrated Algorithm for solving empty language problem FSA M Complement Construction FSA M’ Yes/No Algorithm for complement empty problem The complement construction algorithm is the answer-pres. input transformation. If M is a yes input instance of CE, then M’ is a yes input instance of EL. If M is a no input instance of CE, then M’ is a no input instance of EL.

12 NFA Empty Problem * Input –NFA M Question –Is L(M)={}? Show how to use answer-preserving input transformations to help solve this problem –Show that the NFA Empty problem transforms to the Empty Language problem

13 Input Transformation * Yes/No Algorithm for NFA empty problem

14 Equal Problem * Input –FSA’s M 1 and M 2 Question –Is L(M 1 ) = L(M 2 )? Show how to use answer-preserving input transformations to solve this problem –Try and transform this problem to the empty language problem –If L(M 1 ) = L(M 2 ), then what combination of L(M 1 ) and L(M 2 ) must be empty?

15 Input Transformation Illustrated Yes/No Algorithm for Equal problem

16 Summary Decision problems with programs as inputs Basic problems –You need to develop algorithms from scratch based on properties of FSA’s Solving new problems –You need to figure out how to combine the various algorithms we have seen in this unit to solve the given problem