计算理论笔记（二）CFL PDA

Context-Free Language(CFL)

• Generator: Context-Free Generator(CFG)
• Recoganizer: PushDown Automata(PDA)
• Pumping Theorem for CFL
• Example:
  • $S\to aSb,S\to A,A\to c$
  • $a,b$: terminal
  • $S,A$: non-terminal
  • $S$: start symbol

A CFL is a 4-tuple $G=(V,\Sigma ,R,S)$

• $V$ is a set of symbols
• $\Sigma \subseteq V$ is the set of terminals, $V-\Sigma$ is the set of non-terminals
• $R$ is the set of rules, a finite subset of $(V-\Sigma )\times V^{\ast }$
• $S\in V-\Sigma$ is the start symbol

Derives in one step: $xAy\underset{G}{\Rightarrow }xuy$ if $(A,u)\in R$, for $x,y,u\in V^{\ast }$ and $A\in V-\Sigma$

• $x{\underset{G}{\Rightarrow }}^{\ast }y$ if $x=y$ or $\exists w_0,w_1,\dots ,w_n$ satisfying:
  • $x=w_0$
  • $w_{i-1}\underset{G}{\Rightarrow }{w}_i,i\in [1,n]$
  • $y=w_n$
• Called derivation from $w_0$ to $w_n$ of length $n$

$G$ generates a string $w\in {\Sigma }^{\ast }$ if $S{\underset{G}{\Rightarrow }}^{\ast }w$

• L(G)={w∈Σ∗:GL(G)=\{w\in\Sigma^*:GL(G)={w∈Σ∗:G generates $w\}$, $G$ generates $L(G)$
• A language is context-free if some CFG generates it

Exercise: Show that {w∈{a,b}∗:w=wR}\{w\in\{a,b\}^*:w=w^R\}{w∈{a,b}∗:w=wR} is a CFL

• $|w|=0,w=e=w^R$, i.e. $S\to e$
• $|w|=1,w=a=b$, i.e. $S\to a$ and $S\to b$
• $|w|\ge 2,w=aua=bub,u=u^R$(by induction), i.e. $S\to aSa$ and $S\to bSb$
• Conclusively, $S\to e|a|b|aSa|bSb$
• $w\in L(G)\to w=w^R$: by induction on # steps of derivation of $w$
• $w=w^R\to w\in L(G)$: by induction on length of $w$

Exercise: Show that {w∈{(,)}∗:w\{w\in\{(,)\}^*:w{w∈{(,)}∗:w is a string of balanced parentheses $\}$ is a CFL

• Derivation is not unique: leftmost or rightmost

Parse Tree

• Internal nodes: non-terminals
• Leaves: terminals or $e$, and $e$ must be the only child of its parent
• Edges: spatial and temporal order
• Yields: reconstruct a string from its parse tree
• A CFG $G$ is ambiguous if $\exists w\in L(G)$, there exists two or more parse trees with the same root $S$ yielding $w$
• Fact: Some CFL is inherently ambiguous. That is, every CFG that generates this language is ambiguous
  • For {aibjck:i=j∨j=k}\{a^ib^jc^k:i=j\lor j=k\}{aibjck:i=j∨j=k}, ambiguity occurs on $a^4{b}^4{c}^4$

Theorem: Every regular language is context-free

• Idea: DFA $M=(K,\Sigma ,\delta ,s,F)\to$ CFG $G=(V,\Sigma ,R,S)$, s.t. $L(G)=L(M)$
• $V=K\cup \Sigma$
• R={q→ap:δ(q,a)=p}∪{q→e:q∈F}R=\{q\to ap:\delta(q,a)=p\}\cup\{q\to e:q\in F\}R={q→ap:δ(q,a)=p}∪{q→e:q∈F}
• $S=s$
• $\forall w\in {\Sigma }^{\ast },(s,w){⊢}_M^{\ast }(q,e)$ for some $q\in F⟺s{\underset{G}{\Rightarrow }}^{\ast }w$
  • Claim: $\forall w\in {\Sigma }^{\ast },\forall p,q\in K,(p,w){⊢}_M^{\ast }(q,e)⟺p{\underset{G}{\Rightarrow }}^{\ast }wq$
  • Therefore $s{\underset{G}{\Rightarrow }}^{\ast }wq\underset{G}{\Rightarrow }w$
• Proof:
  • sufficiency by induction on # steps of computations
  • Base case: # steps $=0$, $w=e,p=q,p{\underset{G}{\Rightarrow }}^{\ast }wq$ (trivial)
  • Inductive case: holds for # steps $=k$, then for # steps $=k+1$
    • $w=au,a\in \Sigma ,u\in {\Sigma }^{\ast },(p,w){⊢}_M(p^{\prime },u){⊢}_M^{\ast }(q,e)$ if $\delta (p,a)=p^{\prime }$
    • Let $ p\to ap’\in R$,therefore $p\underset{G}{\Rightarrow }a{p}^{\prime }{\underset{G}{\Rightarrow }}^{\ast }auq$, with hypothesis $p^{\prime }{\underset{G}{\Rightarrow }}^{\ast }uq$
  • necessity by # steps of derivation(omitted)

PushDown Automata(PDA)

PDA(NFA+Stack) is a 6-tuple $P=(K,\Sigma ,\Gamma ,\Delta ,s,F)$

• $K$ is a set of states
• $\Sigma$ is a set of input symbols
• $\Gamma$ is a set of stack symbols
• $s\in K$ is the initial state
• $F\subseteq K$ is the set of final states
• $\Delta$, transition relation, is a finite subset of (K×(Σ∪{e})×Γ∗)×(K×Γ∗)(K\times(\Sigma\cup\{e\})\times \Gamma^*)\times(K\times\Gamma^*)(K×(Σ∪{e})×Γ∗)×(K×Γ∗)

A configuration for $P$ is a member of $K\times {\Sigma }^{\ast }\times {\Gamma }^{\ast }$

Yields in one step: $(p,x,\alpha ){⊢}_P(q,y,\beta )$, if $\exists ((p,x,\gamma ),(q,\eta ))\in \Delta$ where $x=ay,\alpha =\gamma \tau ,\beta =\eta \tau$, for some $\tau \in {\Gamma }^{\ast }$

• $(p,x,\alpha ){⊢}_P^{\ast }(q,y,\beta )$ if $(p,x,\alpha )=(q,y,\beta )$ or $\exists (p_0,x_0,{\alpha }_0),\dots ,(p_n,x_n,{\alpha }_n)$ satisfying:
  • $(p,x,\alpha )=(p_0,x_0,{\alpha }_0)$
  • $(p_{i-1}{x}_{i-1}{\alpha }_{i-1}){⊢}_P(p_i,x_i,{\alpha }_i),\forall i\in [1,n]$
  • $(q,y,\beta )=(p_n,x_n,{\alpha }_n)$

$P$ accepts $w\in {\Sigma }^{\ast }$ if $(s,w,e){⊢}_P^{\ast }(q,e,e)$, for some $q\in F$

$L(P)$: a set of all $w\in {\Sigma }^{\ast }$ accepted by $P$, uniquely

Exercise: Design a PDA to accept L={wwR:w∈{a,b}∗}L=\{ww^ R:w\in\{a,b\}^*\}L={wwR:w∈{a,b}∗}

• K={l,r}K=\{l,r\}K={l,r}
• Σ={a,b}\Sigma=\{a,b\}Σ={a,b}
• Γ={a,b}\Gamma=\{a,b\}Γ={a,b}
• $s=l$
• F={r}F=\{r\}F={r}
• Δ={((l,a,e),(l,a)),((l,b,e),(l,b)),((l,a,a),(l,e)),((l,b,b),(l,e)),((l,e,e),(l,e))}\Delta=\begin{aligned}\{&((l,a,e),(l,a)),((l,b,e),(l,b)),\\ &((l,a,a),(l,e)),((l,b,b),(l,e)),\\&((l,e,e),(l,e))\}\end{aligned}Δ={​((l,a,e),(l,a)),((l,b,e),(l,b)),((l,a,a),(l,e)),((l,b,b),(l,e)),((l,e,e),(l,e))}​

Exercise: Design a PDA to accept {w∈{0,1}∗:\{w\in\{0,1\}^*:{w∈{0,1}∗: # of $0$’s equals to # of $1$’s in $w\}$

• K={s,q,f}K=\{s,q,f\}K={s,q,f}
• Σ={0,1}\Sigma=\{0,1\}Σ={0,1}
• Γ={0,1,@}\Gamma=\{0,1,@\}Γ={0,1,@}
• $s=l$
• F={f}F=\{f\}F={f}
• Δ={((s,e,e),(q,@)),((q,e,@),(f,e)),((q,0,@),(q,0@)),((q,1,@),(q,1,@)),((q,0,0),(q,00)),((q,0,1),(q,e)),((q,1,1),(q,11)),((q,1,0),(q,e))}\Delta=\begin{aligned}\{&((s,e,e),(q,@)),((q,e,@),(f,e)),\\ &((q,0,@),(q,0@)),((q,1,@),(q,1,@)),\\ &((q,0,0),(q,00)),((q,0,1),(q,e)),\\ &((q,1,1),(q,11)),((q,1,0),(q,e))\}\end{aligned}Δ={​((s,e,e),(q,@)),((q,e,@),(f,e)),((q,0,@),(q,0@)),((q,1,@),(q,1,@)),((q,0,0),(q,00)),((q,0,1),(q,e)),((q,1,1),(q,11)),((q,1,0),(q,e))}​

Theorem

• For each CFG $G$, there exists a PDA $P$ s.t. $L(P)=L(G)$
• For each PDA $P$, there exists a CFG $G$ s.t. $L(G)=L(P)$
• $G\to P$, idea:
  • non-deterministic generates a string in stack using $G$
  • compare it to input
  • accept input if they match
• Construction:
  • $G=(V,\Sigma ,R,S)\to P=(K,\Sigma ,\Gamma ,\Delta ,s,F)$
  • K={p,q}K=\{p,q\}K={p,q}
  • $\Gamma =V$
  • $s=p$
  • F={q}F=\{q\}F={q}
  • Δ={((p,e,e),(q,s)),((q,e,A),(q,u)),∀(A,u)∈R((q,a,a),(q,e)),∀a∈Σ}\Delta=\begin{aligned}\{&((p,e,e),(q,s)),\\ &((q,e,A),(q,u)),\forall(A,u)\in R\\ &((q,a,a),(q,e)),\forall a\in\Sigma\}\end{aligned}Δ={​((p,e,e),(q,s)),((q,e,A),(q,u)),∀(A,u)∈R((q,a,a),(q,e)),∀a∈Σ}​
• Prove $L(G)=L(P)$:
  • $\forall w\in {\Sigma }^{\ast }$, $s{\Rightarrow }^{\ast }w$ if $(p,w,e){⊢}_P(q,w,s){⊢}_P^{\ast }(q,e,e)$
• Claim: $\forall w\in {\Sigma }^{\ast }$ and ∀α∈((V−Σ)V∗)∪{e},S⇒L∗αw\forall\alpha\in((V-\Sigma)V^*)\cup\{e\},S\xRightarrow{L}^*\alpha w∀α∈((V−Σ)V∗)∪{e},SL​∗αw iff $(q,w,S){⊢}_P^{\ast }(q,e,\alpha )$
• Prove sufficiency by induction
  • Base case: # steps $=0$, $S{\stackrel{L}{\Rightarrow }}^{\ast }w\alpha$
    • $\Rightarrow S=w\alpha$
    • $\Rightarrow w=e,\alpha =S$
    • $\Rightarrow (q,w,S){⊢}_P^{\ast }(q,e,\alpha )$
  • Inductive case: holds for # steps $=k$, then for # steps $=k+1$
    • $s\stackrel{L}{\Rightarrow }{u}_1\stackrel{L}{\Rightarrow }{u}_2\stackrel{L}{\Rightarrow }\cdots {\stackrel{L}{\Rightarrow }}^{\ast }{u}_k\stackrel{L}{\Rightarrow }\alpha$
    • Let $A$ be the leftmost non-terminal in $u_k$
    • $u_k=xA\beta$ where $x\in {\Sigma }^{\ast }$ and $\beta \in V^{\ast }$,
    • $w\alpha =x\gamma \beta$ for some $(A,\gamma )\in R$
    • $\exists y\in {\Sigma }^{\ast }$ s.t. $w=xy$, therefore $y\alpha =\gamma \beta$
    • $S{\stackrel{L}{\Rightarrow }}^{\ast }{u}_k$
      • $\Rightarrow (q,x,S){⊢}_P^{\ast }(q,e,A\beta ){⊢}_P(q,e,\gamma \beta )$
      • $\Rightarrow (q,xy,S){⊢}_P^{\ast }(q,y,\gamma \beta )$
      • $\Rightarrow (q,w,S){⊢}_P^{\ast }(q,y,y\alpha ){⊢}_P^{\ast }(q,e,\alpha )$
• Prove necessity by induction on # type-2 transition
  • Base case: # steps of type-2 $=0$, $(q,w,S){⊢}_P^{\ast }(q,e,\alpha )$
    • $\Rightarrow w=e,\alpha =S$
    • $\Rightarrow S{\stackrel{L}{\Rightarrow }}^{\ast }w\alpha$
  • Inductive case: holds for # steps of type-2 $=k$, then for # steps $=k+1$
    • Let $((q,e,A),(q,\gamma ))$ be the last type-2 transition
    • $\exists y\in {\Sigma }^{\ast },\exists \beta \in V^{\ast },(q,w,S){⊢}_P^{\ast }(q,y,A\beta ){⊢}_P(q,y,\gamma \beta ){⊢}_P^{\ast }(q,e,\alpha )$
    • $(q,w,S){⊢}_P^{\ast }(q,y,A\beta )$
      • $\Rightarrow \exists x\in {\Sigma }^{\ast },w=xy,(q,xy,S){⊢}_P^{\ast }(q,y,A\beta )$
      • $\Rightarrow (q,x,S){⊢}_P^{\ast }(q,e,A\beta )$
      • $\Rightarrow S{\stackrel{L}{\Rightarrow }}^{\ast }xA\beta$
    • $(q,y,A\beta ){⊢}_P(q,y,\gamma \beta )$
      • $\Rightarrow (A,\gamma )\in R$
    • $(q,y,\gamma \beta ){⊢}_P^{\ast }(q,e,\alpha )$
      • $\Rightarrow \exists \alpha ,y\alpha =\gamma \beta$
    • Therefore, $S{\stackrel{L}{\Rightarrow }}^{\ast }xA\beta \stackrel{L}{\Rightarrow }x\gamma \beta {\stackrel{L}{\Rightarrow }}^{\ast }xy\alpha (=w\alpha )$
• $P\to G$, via simple PDA
• A PDA $P=(K,\Sigma ,\Gamma ,\Delta ,s,F)$ is simple if:
  • $|F|=1$
  • $\forall ((q,a,\alpha ),(q,\beta ))\in \Delta$:
    • either $\alpha =e,|\beta |=1$ (push)
    • or $|\alpha |=1,\beta =e$ (pop)
• Lemma: Every PDA has a equivalent simple PDA
• Proof:
  • Let $P=(K,\Sigma ,\Gamma ,\Delta ,s,F)$ be a PDA
  • If $|F|>1$:
    • create a new state $f$
    • for $\forall q\in F$, create a new transition $((q,e,e),(f,e))$
    • replace $F$ with {f}\{f\}{f}
  • Violations:
  • For each $((q,a,\alpha ),(q,\beta ))$ with $|\alpha |\ge 1$ and $|\beta |\ge 1$:
    • create a new state $r$
    • replace with $((p,a,\alpha ),(r,e))$ and $((r,e,e),(q,\beta ))$, respectively
  • For each $((q,a,\alpha ),(q,e))$ with $\alpha =c_1{c}_2\dots c_k$ for some $k\ge 2$:
    • create $k-1$ new states $r_1,r_2,\dots ,r_{k-1}$
    • replace with $((p,a,c_1),(r_1,e))$ and $((r_i,e,c_2),(r_{i+1},e)),i\in [1,k-1)$ and $((r_{k-1},e,c_k),(q,e))$, respectively
  • For each $((q,a,e),(q,\beta ))$ with $\beta =c_1{c}_2\dots c_k$ for some $k\ge 2$:
    • create $k-1$ new states $r_1,r_2,\dots ,r_{k-1}$
    • replace with $((p,a,e),(r_1,c_k))$ and $((r_i,e,e),(r_{i+1},c_{k-i})),i\in [1,k-1)$ and $((r_{k-1},e,e),(q,c_1))$, respectively
  • For each $((p,a,e),(q,e))\in \Delta$
    • create a new symbol $@$
    • replace with $((p,a,e),(r,@))$ and $((r,e,@),(q,e))$
• Lemma: Every simple PDA has a equivalent CFG
• Proof:
  • Let $P=(K,\Sigma ,\Gamma ,\Delta ,s,F)$ be a simple PDA
  • Construct $G=(V,\Sigma ,R,S)$ s.t. $L(G)=L(P)$
    • Non-terminals: {Apq:p,q∈K}\{A_{pq}:p,q\in K\}{Apq​:p,q∈K}
      • $A_{pq}$ represents {w∈Σ∗:(p,w,e)⊢P∗(q,e,e)}\{w\in\Sigma^*:(p,w,e)\vdash_P^*(q,e,e)\}{w∈Σ∗:(p,w,e)⊢P∗​(q,e,e)}
    • $S=A_{sf}$
    • R={Apq→aArtb:((p,a,e),(r,u)),((t,b,u),(q,e))∈Δ}∪{Apq→AprArq:r∈K}∪{App→e:p∈K}R=\{A_{pq}\to aA_{rt}b:((p,a,e),(r,u)),((t,b,u),(q,e))\in\Delta\}\cup\{A_{pq}\to A_{pr}A_{rq}:r\in K\}\cup\{A_{pp}\to e:p\in K\}R={Apq​→aArt​b:((p,a,e),(r,u)),((t,b,u),(q,e))∈Δ}∪{Apq​→Apr​Arq​:r∈K}∪{App​→e:p∈K}
      • $w=avb$ for some $v\in {\Sigma }^{\ast }$ with $(r,v,e){⊢}_P^{\ast }(t,e,e)$
  • Claim: $A_{pq}{\Rightarrow }^{\ast }w\in {\Sigma }^{\ast }$ iff $(p,w,e){⊢}_P^{\ast }(q,e,e)$ (proof omitted)
• Collary: A language is context-free iff it is accepted by some PDA