Complexity and Computability Theory I

1. Complexity and Computability Theory I Lecture #11 Instructor: Rina Zviel-Girshin Lea Epstein

2. Overview • Models and limitations • Examples • Nondeterministic PDA • Examples • Equivalence of PDA’s models Rina Zviel-Girshin @ASC

3. Finite automata • A finite automaton has a huge limitation. • It can count only by changing states. • So the highest number FA can remember is not more than number of it's states. • This limitation confined the class of languages that FA can recognize to a rather small category - that of regular languages. Rina Zviel-Girshin @ASC

4. Turing machine • Infinite memory can allow us recognition of larger class of languages. • If we put no constrains on the use of this memory, i.e. the automaton can read and write in any location in the memory, then we have the most potent automaton - Turing machine. Rina Zviel-Girshin @ASC

5. Power of TM • We claim that Turing machine (TM) can perform any computation the most powerful computer can. Rina Zviel-Girshin @ASC

6. The stack model • If we add to FA an unlimited memory but limit the use of it we get a machine which is stronger than FA but less powerful than TM. • Our limitation - the memory will be a stack. • We can read/write only at the top of the stack, obliterating the previous content. Rina Zviel-Girshin @ASC

7. The stack model (cont.) • We still have states to remember the previous contents (if we wish to do so). • But as the number of states is finite we can remember only finite contents while our stack is infinite. Rina Zviel-Girshin @ASC

8. Pushdown automata • Informally a pushdown automata (PDA) is a FA+stack. • To remember or to count we write to the stack and read afterwards when we need the information. Rina Zviel-Girshin @ASC

9. Example • For example the following language can be recognized by PDA but not FA. L = {aibi | i>0} • The basic idea of recognition: • remembering a's by pushing it to the stack • comparing the number of b's to the number of a's by popping one a for each b. Rina Zviel-Girshin @ASC

10. Example (cont.) • If the stack was emptied before the end of input - we had more b's than a's. • If the stack was not emptied at the end of input - we had more a's than b's. • The automaton will accept a word if at exactly the end of input the stack is emptied for the first time. Rina Zviel-Girshin @ASC

11. Example (cont.) • It gets stuck and rejects the word if the two events - • end of input and • the first time stack becomes empty - are not simultaneous. Rina Zviel-Girshin @ASC

12. Acceptance by PDA • There are several models of PDA with different acceptance policies. • All are equivalent. • acceptance by an empty stack • acceptance using accepting states • acceptance by an empty stack and accepting states Rina Zviel-Girshin @ASC

13. Acceptance by an empty stack • The automaton will accept a word if exactly at the end of input the stack is emptied for the first time. • All states are the same, there are no accepting states. Rina Zviel-Girshin @ASC

14. Acceptance using accepting states • The automaton will accept a word if at exactly the end of input the automaton is in accepting state. • Whether the stack is empty or not at that time has no relevance. Rina Zviel-Girshin @ASC

15. Acceptance by an empty stack and accepting states • The automaton will accept a word if three events happen simultaneously: • the end of input • the stack is emptied for the first time • automaton is in accepting state. Rina Zviel-Girshin @ASC

16. Transition function in PDA • There are two differences between the transition functions of PDA and FA. • In PDA we move using • A state, • A letter of input and • A letter of stack • The transition function output is a state and a letter or string to be written into the stack. Rina Zviel-Girshin @ASC

17. Formal definition of PDA • A pushdown automaton is a 6-tuple (Q, , , , q0, F) where: • Q – a finite set of states •  - a finite input alphabet •  - a finite stack alphabet • : Q  (U)  P(Q *) • q0  Q the start state • F – a finite set of accepting states. Rina Zviel-Girshin @ASC

18. The possible stack transformations • Consider different possibilities of transition function: • (q,,X) = (q', Y) • where (q - current state, - input letter, X - top of the stack) = (q' - next state, Y - string to write into the stack) Rina Zviel-Girshin @ASC

19. Popping the stack • Popping the stack: • (q,,X) = (q', ) • write an empty word to the stack, i.e. deleting the X at the top of the stack and moving the top pointer one cell down. Rina Zviel-Girshin @ASC

20. No change • No change: • (q,,X) = (q', X) • write the same letter to the stack, i.e. not changing the X at the top of the stack and not moving the top pointer Rina Zviel-Girshin @ASC

21. Add to the stack • Add to the stack: • (q,,X) = (q', W1..WnX) • write the same letter to the stack and above it write a string, i.e. not changing the X at the top of the stack, but adding any number of letters above in the same one transition and moving the top pointer as needed Rina Zviel-Girshin @ASC

22. Change the stack • Change the stack: • (q,,X) = (q', W1..Wn) • delete the letter X in the stack and instead write a string, i.e. changing the X at the top of the stack into any number of letters above in the same one transition and moving the top pointer as needed Rina Zviel-Girshin @ASC

23. Bottom of the stack symbol • The acceptance model we will use will be by emptying the stack. • In this model of acceptance the PDA is stuck immediately if the stack is empty. • Therefore the PDA can't start with an empty stack. Rina Zviel-Girshin @ASC

24. Bottom of the stack symbol • Therefore we will assume that at the commencement of the computation, the stack already includes one symbol - a special "bottom of the stack symbol". • There are many possible notations for the symbol, such as $, ┤. • We will use the letter Z for the "bottom of the stack symbol". Rina Zviel-Girshin @ASC

25. Example Construct a PDA to recognize: L = {aibi | i>0} We will use the empty stack model. Rina Zviel-Girshin @ASC

26. Example • Construct a PDA to recognize: L = {wcwR |w{a,b}*} • We will use the empty stack model. Rina Zviel-Girshin @ASC

27. Example L = {aibj | i >0 and j=i+5} • Construct a PDA to recognize: • We will use the empty stack model. Rina Zviel-Girshin @ASC

28. Deterministic PDA • A PDA is called deterministic if”f it holds the following two conditions: • for each qQ, a{} and X • |(q,a,X)|  1 • for each qQ and X • if (q,,X) • then for each (q, ,X)= Rina Zviel-Girshin @ASC

29. Deterministic PDA Informally • A PDA is called non-deterministic if there exist at least two transitions from the current state q and stack top X: • one with  (-transition) and • another with . Rina Zviel-Girshin @ASC

30. Example of ND PDA • Construct a PDA to recognize: L = {wwR |w{a,b}*} We will use the empty stack model Rina Zviel-Girshin @ASC

31. Example of ND PDA • Construct a PDA to recognize: L = {aibjci+j | i,j0} We will use the empty stack model. The basic idea: • push a’s and b’s to the stack and than pop the stack top symbol for each c letter. Rina Zviel-Girshin @ASC

32. L = {aibjci+j | i,j0} Rina Zviel-Girshin @ASC

33. Example • Construct a PDA to recognize: L = {aibj | i,j>0} • We will use a finite state model. Rina Zviel-Girshin @ASC

34. Equivalent form of acceptance • The empty stack PDA equivalent to the final state PDA The basic idea: • We will use the empty stack PDA model during the computation and on the last stage we will produce -transition to the final state. Rina Zviel-Girshin @ASC

35. Algorithm • Given PDA=(Q, , , , q0, ∅) • Convert it to FS PDA=(QU{s,f}, , U{Y}, , s, F={f}) 1.Create a new start state s and a new final state f. 2.Create a new bottom of the stack symbol: Y. Rina Zviel-Girshin @ASC

36. Algorithm(cont.) 3.Connect a new start state s to the old start state q0 by an edge labeled ,Y; ZY that means (s,,Y)=(q0,ZY) 4.Connect each state of the PDA to the new final state f and label the edges with the expression ,Y; Y. Rina Zviel-Girshin @ASC

37. L={a} Given an empty stack automaton: Construct an equivalent final-state automaton. Rina Zviel-Girshin @ASC

38. L={a} The resulting finite-state automaton: Rina Zviel-Girshin @ASC

39. Equivalent form of acceptance • The final-state PDA is equivalent to the empty stack PDA The basic idea: • We will use a finite state PDA model during the computation and on the last stage we will produce -transition to empty the stack state - by popping everything from the stack. Rina Zviel-Girshin @ASC

40. Algorithm • Given PDA=(Q, , , , q0, F) • Convert it to ESPDA=(QU{s,f}, , U{Y}, , s, ∅) 1. Create a new start state s and a new “empty state” f. 2. Create a new bottom of the stack symbol: Y. Rina Zviel-Girshin @ASC

41. Algorithm(cont.) 3. Connect a new start state s to the old start state q0 by an edge labeled ,Y; ZY that means (s,,Y)=(q0,ZY) 4. Connect each final state of the PDA to the new empty state f and label the edges with the expression ,A;  where A means any symbol of the stack, including Z and Y. 5. Add new edges from f to f labeled as in step 4: (f,,A)=(f, ) for every A in . Rina Zviel-Girshin @ASC

42. L={a} Given a finite-state automaton: Construct an equivalent empty stack automaton. Rina Zviel-Girshin @ASC

43. L={a} The resulting automaton: Rina Zviel-Girshin @ASC

44. Any Questions? Rina Zviel-Girshin @ASC