Friday Four Square! Today at 4:15PM, Outside Gates

Transcription

1 Pushdown Automata

2 Friday Four Square! Today at 4:15PM, Outside Gates

3 Announcements Problem Set 5 due right now Or Monday at 2:15PM with a late day. Problem Set 6 out, due next Friday, November 9. Covers context-free languages, CFGs, and PDAs. Midterm and Problem Set 4 should be graded by Monday.

4 Generation vs. Recognition We saw two approaches to describe regular languages: Build automata that accept precisely strings in language. Design regular expressions that describe precisely strings in language. Regular expressions generate all of strings in language. Useful for listing off all strings in language. Finite automata recognize all of strings in language. Useful for detecting wher a specific string is in language.

5 Context-Free Languages Yesterday, we saw context-free languages, which are those that can be generated by context-free grammars. Is re some way to build an automaton that can recognize context-free languages?

6 The Problem Finite automata accept precisely regular languages. We may need unbounded memory to recognize context-free languages. e.g. { 0n 1 n n N } requires unbounded counting. How do we build an automaton with finitely many states but unbounded memory?

7 The The finite-state finite-state control control acts acts as as a a finite finite memory. memory. The The input input tape tape holds holds input input string. string. A B C A... Memory Device We We can can add add an an infinite infinite memory memory device device finite-state finite-state control control can can use use to to store store information. information.

8 Adding Memory to Automata We can augment a finite automaton by adding in a memory device for automaton to store extra information. The finite automaton now can base its transition on both current symbol being read and values stored in memory. The finite automaton can issue commands to memory device whenever it makes a transition. e.g. add new data, change existing data, etc.

9 Stack-Based Memory Only top of stack is visible at any point in time. New symbols may be pushed onto stack, which cover up old stack top. The top symbol of stack may be popped, exposing symbol below it.

10 Pushdown Automata A pushdown automaton (PDA) is a finite automaton equipped with a stack-based memory. Each transition is based on current input symbol and top of stack, optionally pops top of stack, and optionally pushes new symbols onto stack. Initially, stack holds a special symbol Z 0 that indicates bottom of stack.

11 Our First PDA Consider language L = { w Σ* w is a string of balanced digits } over Σ = { 0, 1 } We can exploit stack to our advantage: Whenever we see a 0, push it onto stack. Whenever we see a 1, pop corresponding 0 from stack (or fail if not matched) When input is consumed, if stack is empty, accept.

12 A Simple Pushdown Automaton start 0, Z 0 0Z 0 0, , 0 ε ε, Z 0 ε To To find find an an applicable applicable transition, transition, match match current current input/stack input/stack pair. pair. A A transition transition of of form form a, a, b z Means Means If If current current input input symbol symbol is is a a and and current current stack stack symbol symbol is is b, b, n n follow follow this this transition, transition, pop pop b, b, and and push push string string z. z. Z

13 A Simple Pushdown Automaton start 0, Z 0 0Z 0 0, , 0 ε ε, Z 0 ε If If a a transition transition reads reads top top symbol symbol of of stack, stack, it it always always pops pops that that symbol symbol (though (though it it might might replace replace it) it)

14 A Simple Pushdown Automaton start 0, Z 0 0Z 0 0, , 0 ε ε, Z 0 ε Each Each transition transition n n pushes pushes some some (possibly (possibly empty) empty) string string back back onto onto stack. stack. Notice Notice that that leftmost leftmost symbol symbol is is pushed pushed onto onto top. top. 0 Z

15 A Simple Pushdown Automaton start 0, Z 0 0Z 0 0, , 0 ε ε, Z 0 ε We We now now push push string string ε ε onto onto stack, stack, which which adds adds no no new new characters. characters. This This essentially essentially means means pop pop stack. stack. 0 0 Z

16 A Simple Pushdown Automaton start 0, Z 0 0Z 0 0, , 0 ε ε, Z 0 ε This This transition transition can can be be taken taken at at any any time time Z Z 0 is 0 is atop atop stack, stack, but but we've we've nondeterministically nondeterministically guessed guessed that that this this would would be be a a good good time time to to take take it. it. Z

17 A Simple Pushdown Automaton start 0, Z 0 0Z 0 0, , 0 ε ε, Z 0 ε

18 Pushdown Automata Formally, a pushdown automaton is a nondeterministic machine defined by 7-tuple (Q, Σ, Γ, δ, q 0, Z 0, F), where Q is a finite set of states, Σ is an alphabet, Γ is stack alphabet of symbols that can be pushed on stack, δ : Q Σ ε Γ ε (Q Γ*) is transition function, where no tuple is mapped to an infinite set, q 0 Q is start state, Z 0 Γ is stack start symbol, and F Q is set of accepting states. The automaton accepts if it ends in an accepting state with no input remaining.

19 The Language of a PDA The language of a PDA is set of strings that PDA accepts: L(P) = { w Σ* P accepts w } If P is a PDA where ( L P) = L, we say that P recognizes L.

20 A Note on Terminology Finite automata are highly standardized. There are many equivalent but different definitions of PDAs. The one we will use is a slight variant on one described in Sipser. Sipser does not have a start stack symbol. Sipser does not allow transitions to push multiple symbols onto stack. Feel free to use eir this version or Sipser's; two are equivalent to one anor.

21 A PDA for Palindromes A palindrome is a string that is same forwards and backwards. Let Σ = {0, 1} and consider language PALINDROME = { w Σ* w is a palindrome }. How would we build a PDA for PALINDROME? Idea: Push first half of symbols on to stack, n verify that second half of symbols match. Nondeterministically guess when we've read half of symbols. This handles even-length strings; we'll see a cute trick to handle odd-length strings in a minute.

22 A PDA for Palindromes start 0, ε 0 1, ε 1 This transition indicates that transition does not pop anything from stack. It just pushes on a new symbol instead.

23 A PDA for Palindromes start Σ, ε Σ The The Σ Σ here here refers refers to to same same symbol symbol in in both both contexts. contexts. It It is is a a shorthand shorthand for for treat treat any any symbol symbol in in Σ Σ this this way way

24 A PDA for Palindromes Σ, ε Σ start ε, ε ε This This transition transition means means don't don't consume consume any any input, input, don't don't change change top top of of stack, stack, and and don't don't add add anything anything to to a a stack. stack. It's It's equivalent equivalent of of an an ε ε -transition -transition in in an an NFA. NFA.

25 A PDA for Palindromes start Σ, ε Σ Σ, Σ ε ε, ε ε Σ, ε ε ε, Z 0 ε This This transition transition lets lets us us consume consume one one character character before before we we start start matching matching what what we we just just saw. saw. This This lets lets us us match match odd-length odd-length palindromes palindromes

26 A Note on Nondeterminism In a PDA, if re are multiple nondeterministic choices, you cannot treat machine as being in multiple states at once. Each state might have its own stack associated with it. Instead, re are multiple parallel copies of machine running at once, each of which has its own stack.

27 A PDA for Arithmetic Let Σ = { int, +, *, (, ) } and consider language ARITH = { w Σ* w is a legal arithmetic expression } Examples: int + int * int ((int + int) * (int + int)) + (int) Can we build a PDA for ARITH?

28 A PDA for Arithmetic start (, ε ( ), ( ε int, ε ε ε, Z 0 ε +, ε ε *, ε ε

29 Why PDAs Matter Recall: A language is context-free iff re is some CFG that generates it. Important, non-obvious orem: A language is context-free iff re is some PDA that recognizes it. Need to prove two directions: If L is context-free, n re is a PDA for it. If re is a PDA for L, n L is context-free. Part (1) is absolutely beautiful and we'll see it in a second. Part (2) is brilliant, but a bit too involved for lecture (you should read this in Sipser).

30 From CFGs to PDAs Theorem: If G is a CFG for a language L, n re exists a PDA for L as well. Idea: Build a PDA that simulates expanding out CFG from start symbol to some particular string. Stack holds part of string we haven't matched yet.

31 From CFGs to PDAs Example: Let Σ = { 1, } and let GE = { 1 m 1 n m, n N m n } GE GE GE GE One CFG for GE is following: S 1S1 1S How would we build a PDA for GE?

32 From CFGs to PDAs S 1S1 S 1S S ε, S 1S ε, S 1S1 ε, S Σ, Σ ε start ε, ε S ε, Z 0 Z 0

33 From CFGs to PDAs Make three states: start, parsing, and accepting. There is a transition ε, ε S from start to parsing. Corresponds to starting off with start symbol S. There is a transition ε, A ω from parsing to itself for each production A ω. Corresponds to predicting which production to use. There is a transition Σ, Σ ε from parsing to itself. Corresponds to matching a character of input. There is a transition ε, Z 0 Z 0 from parsing to accepting. Corresponds to completely matching input.

34 From CFGs to PDAs The PDA constructed this way is called a predict/match parser. Each step eir predicts which production to use or matches some symbol of input.

35 From PDAs to CFGs The or direction of proof (converting a PDA to a CFG) is much harder. Intuitively, create a CFG representing paths between states in PDA. Lots of tricky details, but a marvelous proof. It's just too large to fit into margins of this slide. Read Sipser for more details.

36 Regular and Context-Free Languages Theorem: Any regular language is context-free. Proof Sketch: Let L be any regular language and consider a DFA D for L. Then we can convert D into a PDA for L by converting any transition on a symbol a into a transition a, ε ε that ignores stack. This new PDA accepts L, so L is context-free. -ish

37 Refining Context-Free Languages

38 NPDAs and DPDAs With finite automata, we considered both deterministic (DFAs) and nondeterministic (NFAs) automata. So far, we've only seen nondeterministic PDAs (or NPDAs). What about deterministic PDAs (DPDAs)?

39 DPDAs A deterministic pushdown automaton is a PDA with extra property that For each state in PDA, and for any combination of a current input symbol and a current stack symbol, re is at most one transition defined. In or words, re is at most one legal sequence of transitions that can be followed for any input. This does not preclude ε-transitions, as long as re is never a conflict between following ε-transition or some or transition. However, re can be at most one ε-transition that could be followed at any one time. This does not preclude automaton dying from having no transitions defined; DPDAs can have undefined transitions.

40 Is this a DPDA? start 0, Z 0 0Z 0 0, , 0 ε ε, Z 0 ε

41 Is this a DPDA? This This ε-transition ε-transition is is allowable allowable because because no no or or transitions transitions in in this this state state use use input input symbol symbol 0 0 start 0, ε 0 0, , 0 ε ε, Z 0 Z 0 This This ε-transition ε-transition is is allowable allowable because because no no or or transitions transitions in in this this state state use use stack stack symbol symbol Z 0. 0.

42 Why DPDAs Matter Because DPDAs are deterministic, y can be simulated efficiently: Keep track of top of stack. Store an action/goto table that says what operations to perform on stack and what state to enter on each input/stack pair. Loop over input, processing input/stack pairs until automaton rejects or ends in an accepting state with all input consumed. If we can find a DPDA for a CFL, n we can recognize strings in that language efficiently.

43 If we can find a DPDA for a CFL, n we can recognize strings in that language efficiently. Can we guarantee that we can always find a DPDA for a CFL?

44 The Power of Nondeterminism When dealing with finite automata, re is no difference in power of NFAs and DFAs. However, when dealing with PDAs, re are CFLs that can be recognized by NPDAs that cannot be recognized by DPDAs. Simple example: The language of palindromes. How do you know when you've read half string? NPDAs are more powerful than DPDAs.

45 Deterministic CFLs A context-free language L is called a deterministic context-free language (DCFL) if re is some DPDA that recognizes L. Not all CFLs are DCFLs, though many important ones are. Balanced parenses, most programming languages, etc. Why Why are are all all regular regular languages languages DCFLs? DCFLs? Regular Languages DCFLs CFLs

46 Summary Automata can be augmented with a memory storage to increase ir power. PDAs are finite automata equipped with a stack. PDAs accept precisely context-free languages: Any CFG can be converted to a PDA. Any PDA can be converted to a CFG. Deterministic PDAs are strictly weaker than nondeterministic PDAs.

47 Next Time The Limits of CFLs A New Pumping Lemma Non-Closure Properties of CFLs Turing Machines An extremely powerful computing device......that is almost impossible to program.