Pushdown Automata

Machines for Context-Free Languages

Wednesday, November 11, 2009 Reading: Sipser 2.2; Kozen 23

CS235 Languages and Automata

Department of Computer Science Wellesley College

Goals

• Complete the analogy: regular expression : finite automaton :: context-free grammar : ??? ??? = pushdown automaton • Study/design some nondeterministic pushdown automata (NPDAs). • Discuss the relationship between CFGs and NPDAs. • Discuss nondeterministic vs. deterministic PDAs.

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1 Finite Automata Review

A finite automaton accepts/rejects input tape strings that can be viewed as a t 1 t2 t3 t4 t5 t6 … read-onlyyp tape of sy mbols read by a tape head that can move only left to left-to-right, Finite-State right, as controlled by a finite-state read-only Control controller.

Comes in many flavors (Stoughton): FA = transitions between states are relations involving arbitrary strings. EFA = transitions are relations involving strings of length  1. NFA = transitions are relations involving strings of length 1. DFA = transitions are functions involving strings of length 1. Other authors define such machines in slightly different ways, sometimes with different names. Pushdown Automata 27-3

Pushdown Automata input tape t1 t2 t3 t4 t t6 … A pushdown automaton is a finite 5 automaton + a stack of symbols that left-to-right, can be pushed/popped on each Finite-State read-only transition. The finite-state Control controller can read only the top push/pop, symbol of the stack, which can can only read top affect the transition. sn top S Comes in many flavors (Sipser/Kozen): n-1 … Nondeterministic Pushdown Automata (Sipser’s PDA, Kozen’s NPDA) = Stoug hton’ s EFA + read top of stack and s2 push/pop stack symbols on each transition. s1 bottom Deterministic Pushdown Automata (DPDA) = Stoughton’s DFA + stack operations + end-of-input symbol .

Definitions differ slightly from author to author. Pushdown Automata 27-4

2 An NPDA for On1n Example Execution State Input Stack A A 000111 % B 000111 $ %, % → $ read no input and push B 00111 0$ bottom-of-stack marker $ B 0111 00$ B 0, % → 0 read 0 from input and push 0 onto stack B 111 000$ %, % → % transition to 1-reading state C 111 000$ without reading input C 11 00$ C 1, 0 → % read 1 from input C 1 0$ and pop 0 from stack C % $ %, $ → % read no input and pop $ from stack D % % (Note: not necessary to D pop $ since stack needn’t Each row of table is a be empty at end) machine configuration. * Note: Where we use %, % → $, Transitions go from one Sipser uses ,  → $. configuration to another. Pushdown Automata 27-5

Formally Defining NPDAs

Definitions differ among authors. This one is similar to Sipser’s: A Nondeterministic Pushdown Automaton (NPDA)* in a sextuple:

(1) A set of states Q (2) An input alphabet  (not containing the empty string symbol %) (3) A stack alphabet  (4) A start state s  Q (5) A set of final states F  Q (6) A transition relation T  (Q x ( U %) x ( U %)) x (Q x ( U %))

This is similar to Stoughton’s EFA except: • The input alphabet is explicit (rather than implicitly being Sym) •The stack alphabet is new • The transition relation allows pushing/popping a stack symbol (Stoughton’s transition relation T  (Q x ( U %) x Q). )

* Sipser calls it a Pushdown Automaton (PDA) Pushdown Automata 27-6

3 Defining Acceptance for NPDAs

A configuration is a triple in Q x *x*.

Let p,q in Q; t in ( U %); x, y in *; k,k’ in ( U %); and G in *.

We define the step relation, , on configurations: (p, ty, kG)  (q, y, k’G) iff ( (p, t, k), (q, k’) )  T A NPDA accepts a string x iff (s, x, %) * (p, %, G) (for some stack G; it needn’t be empty) where s is the start state and p  the final states F.

Example for 0n1n NPDA (A, 000111, %)  (B, 000111, $)  (B, 00111, 0$)  (B, 0111, 00$)  (B, 111, 000$)  (C, 111, 000$)  (C, 11, 00$)  (C, 1, 0$)  (C, %, $)  (D, %, %) Pushdown Automata 27-7

An NPDA for {Om1n | m > n} How do we complete the following NPDA to do the right thing? (Hint: stack needn’t be empty in accepting state.)

State Input Stack A A 000011 % %, % → $ B 000011 $ B 00011 0$ B 0, % → 0 B 0011 00$ B 011 000$ %, % → % B 11 0000$ C 1, 0 → % C 11 0000$ C 1 000$ C % 00$

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4 An NPDA for {w | w  {0,1}* and #0s = #1s}

Create an NPDA for this language.

State Input Stack 11000110 %

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An NPDA for {0i1j2k | i = j or i = k} Create an NPDA for this language (use nondeterminism!)

State Input Stack 0011122 %

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5 An NPDA for {wwR | w  {0,1}*} Create an NPDA for this language (use nondeterminism!)

State Input Stack 100001 %

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NPDAs are Equivalent to CFGs

It can be shown that:

(()1) A CFG can be converted to an NPDA. Context-Free I.e., every context-free language Languages (CFLs) (the language of some CFG) is the language accepted by some NPDA. CFGs NPDAs

(2) An NPDA can be converted to a CFG. Regular Languages I.e., the language accepted by an FAs NPDA is context-free. regular expressions EFAs NFAs For details, see Sipser 2.2 or Kozen 24-25. right-linear grammars DFAs

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6 Deterministic PDAs (DPDAs)

Like an NPDA except:

• Have a special symbol (  ) for the end of input. • Transition relation must be a function (no choices/guessing!) Effectively a DPDA = DFA + stack operations, except that some transitions might not read any input (and instead perform stack operations). For details, see Kozen Supplementary Lecture F.

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Where do DPDAs Fit In?

The language of a DPDA is a Context-Free Languages (CFLs) Deterministic Context-Free Language (DCFL). CFGs NPDAs

DCFLs are a proper subset of CFLs. Deterministic Context-Free Languages (DCFLs) I.e., there are CFLs that are DPDAs not DCFLS. E.g. {aibjck | i = j or i = k} Regular Languages

R FAs {ww | w in {0, 1}* } regular EFAs I.e. DPDAs are strictly less expressions powerful than NPDAs. NFAs right-linear DFAs Unlike CFLs, DCFLs are closed under grammars complement.

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