1. CSE 105 theory of computation
Fall 2017

2. Today's learning goals Sipser 2.1
Test if a specific string can be generated by a given context-free grammar
Design a context-free grammar to generate a given language
Determine if a context-free grammar is ambiguous
Define push down automata
Trace the computation of a push down automaton

3. Context-free languages over Σ
Birds' eye view
All languages over Σ
Context-free languages over Σ
Regular languages over Σ
Finite languages over Σ

4. Can CFGs describe simple sets?
Designing a CFG
Can CFGs describe simple sets?
Building a CFG to describe the language { abba } V = {S} Σ = ? R = { S  abba } S
What's the set of terminals of this CFG?
{a,b}
V U S U Σ
{S, a, b}
{a,b, ε}
I don't know.

5. Is every regular language a CFL?
Approach 1: start with an arbitrary DFA M, build a CFG that generates L(M).
Approach 2: build CFGs for {a}, {ε}, {}; then show that the class of CFL is closed under the regular operations (union, concatenation, Kleene star).

6. Approach 1
Claim: Given any DFA M, there is a CFG whose language is L(M). Construction: Trace computation using variables to denote state Given M = (Q,Σ,δ,q0,F) a DFA, define the CFG V = { Si | qi is in Q } Σ R = { Si  aSj | δ(qi,a) = qj } U { Si  ε | qi is in F} S = S0 Then prove correctness…

7. Approach 2
Build CFGs for {a}, {ε}, {}; then show that the class of CFL is closed under the regular operations (union, concatenation, Kleene star).

8. Approach 2 If G1 = (V1, Σ, R1, S1) and G2 = (V2, Σ, R2, S2) are CFGs
and G1 describes L1, G2 describes L2, how can we combine the grammars so we describe L1 U L2 ?
G = (V1 U V2, Σ, R1 U R2, S1 U S2)
G = (V1 x V2, Σ, R1 x R2, (S1, S2) )
We might not always be able to: the class of CFG describable languages might not be closed under union.
I don't know.

9. Approach 2
If G1 = (V1, Σ, R1, S1) and G2 = (V2, Σ, R2, S2) are CFGs and G1 describes L1, G2 describes L2, how can we combine the grammars so we describe L1 U L2 ?

10. We know this set is not regular!
Designing a CFG
We know this set is not regular!
Building a CFG to describe the language { anbn | n ≥ 0 }

11. Designing a CFG Building a CFG to describe the language
{ anbn | n ≥ 0 }
One approach:
what is shortest string in the language?
how do we go from shorter strings to longer ones?

12. Designing a CFG
Building a CFG to describe the language { anbn | n ≥ 0 } V = { S } Σ = { a,b } R = S
Which rules would complete this CFG?
S  ε | ab
S  ε | aS | Sb
S  ε | aSb
We need another variable other than S.
I don't know.

13. Designing a CFG
Also not a regular set
Building a CFG to describe the language { 0n1m2n | n,m ≥ 0 } Hint: work from the outside in.

14. Designing a CFG
Also not a regular set
Building a CFG to describe the language { 0n1m2n | n,m ≥ 0 } Hint: work from the outside in. V = { S, T } Σ = { 0,1,2 } R = { S  0S2 | T | ε , T  1T | ε } S

15. CFGs in the wild
V = {E}, Σ = {1,+,x,(,)}, R = { E  E+E | ExE | (E) | 1 }, S=E
Describing well-formed arithmetic expressions
Which of the followings strings is generated by this CFG? E 11
1+1x1 ε
I don't know.

16. Derivations and parsing
E  E+E | ExE | (E) | 1 Lots of derivations for 1+1x1

17. Derivations and parsing
E  E+E | ExE | (E) | 1 Lots of derivations for 1+1x1

18. Derivations and parsing
E  E+E | ExE | (E) | 1 Lots of derivations for 1+1x1
A string is ambiguously derived in a CFG if it has more than one leftmost derivation i.e. more than one parsing tree

19. Recap: Context-free languages
Context-free grammar One-step derivation Derivation Language generated by grammar

20. An alternative … Sipser p. 109
NFA + stack

21. Pushdown automata Sipser p. 109
NFA + stack
At each step
1. Transition to new state based on current state, letter read, and top letter of stack.
2. (Possibly) push or pop a letter to (or from) top of stack

22. Pushdown automata Sipser p. 109
NFA + stack
Accept a string if
there is some sequence of states and some sequence of stack contents which processes the entire input string and ends in an accepting state.

23. State diagram for PDA Sipser p. 109
If hand-drawn or in Sipser State transition labelled a, b  c means "when machine reads an a from the input and the top symbol of the stack is a b, it may replace the b with a c."

24. State diagram for PDA Sipser p. 109
If hand-drawn or in Sipser State transition labelled a, b  c means "when machine reads an a from the input and the top symbol of the stack is a b, it may replace the b with a c."
What edge label would indicate "Read a 0, don't pop anything from stack, don’t push anything to the stack"?
0, ε  ε
ε, 0  ε
ε, ε  0
ε  ε, 0
I don't know.

25. Formal definition of PDA Sipser Def 2.13 p. 111