The rest of the TAPE is blank. Pushdown Automata •Like an FA, PDAs have a finite number of states •The difference is now the FA can interact with the stack Specifically, the action of an automaton depends on 1.the current state, 2.the symbol on the top of the stack, and 3.optionally the next symbol from the input An automaton can 1.update its state and A pushdown automaton, cf. Give pushdown automata that recognize the following languages. A stack (inﬁnite in 1 direction), initially blank. AnNFA transitionlookslikethis: q 2 (p;a) \$ p q a APDAadds(optional)stackelementstopopandpush: hq;Bi2 (p;a;A) \$ p q a;A=B p q INPUT POP PUSH a;A=B 7/31 Each transition is based on the current input symbol and the top of the stack, optionally pops the top of the stack, and optionally pushes new symbols onto the stack. Intuition: NPDA= NFA+ one stackfor memory. Push-down automaton are nondeterministic and can recognize context-free languages. Pushdown Automata: Introduction • Pushdown automaton (PDA) M = (K,Σ,Γ,∆,s,A) where K is a set of states Σ is an input alphabet Γ is a set of stack symbols s ∈ K is the start state A ⊆ K is a set of accepting states, and ∆ is a transition relation: One START state that has only out-edges. on expansion Use k symbols look ahead to decide about expansions LR(k): Deterministic right parsers Read the input from left to right Find rightmost derivation in reverse order Recursive functions in a computer program can be modelled by suitable grammatical rules. 4. Figure 1.2, is a ﬁnite state device equipp ed with a one-wa y input tap e and with a pushdown stack which is a lifo … 3. Pushdown Automata and Parser Deterministic Parsers LL(k): Deterministic left parsers Read the input from left to right Find leftmost derivation Take decisions as early as possible, i.e. 1 Push-down Automata A push-down automaton is a ﬁnite automaton with an additional last-in ﬁrst-out push-down stack; anything read from the stack is immediately de-stroyed. An alphabet Γ of stack symbols. An alphabet Σ of input symbols. An input TAPE (inﬁnite in 1 direction). A stack provides additional memory beyond the finite amount available. 6. As an example, cf. Stack remembers info about previous part of string E.g., anbn Deterministic PushDown Automata (DPDA) can accept some but not allof the CFLs. Pushdown Automata Pushdown automata are like non-deterministic finite automata, but have an extra component called a stack. Pushdown Automaton Apushdownautomaton(PDA)isanNFA withastack. Initially, the stack holds a special symbol Z 0 that indicates the bottom of the stack. 5. The input word starts in the leftmost cell. In par-ticular, after formally introducing push-down automata in Section 14.1, we introduce two notions of acceptance - by ﬁnal state and by empty Pushdown Automata A pushdown automaton (PDA) is a finite automaton equipped with a stack-based memory. Push-down automata are partway to a Turing machine. Nondeterministic Push-Down Automata A language is context freeiff some Nondeterministic PushDown Automata (NPDA) recognizes (accepts) it. A pushdown automaton, PDA, is a collection of 8 things: 1. Give both a drawing and 6-tuple speciﬁcation for each PDA. Push-down Automata and Context-free Grammars This chapter details the design of push-down automata (PDA) for vari-ous languages, the conversion of CFGs to PDAs, and vice versa. The stack allows pushdown automata to recognize some nonregular languages. A pushdown automaton is a 6-tuple where , , , and are ﬁnite sets, and: 1. is a set of states 2. is the input alphabet 3. is the stack alphabet 4. is the transition function 5. is the start state 6. is the set of accept states Pushdown Automata – p.13/25 Figure 1.1, the recursive function Hanoi, moving n disks from pin s to pin t using additional pin v can be represented by productions like Hstv(n) → Hsvt(n−1) mst Hvts(n−1) and Hstv(0) → λ —with terminal symbols mxy, x, y ∈ {s, t, v}. 2.
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