Test GR2: Różnice pomiędzy wersjami

Aktualna wersja na dzień 22:16, 11 wrz 2023

$ϕ$	$ψ$	$ψ \Rightarrow ϕ$	$(ϕ \Rightarrow (ψ \Rightarrow ϕ))$
0	0	1	1
0	1	0	1
1	0	1	1
1	1	1	1

Parser nie mógł rozpoznać (błąd składni): {\displaystyle \left| x \right|\ = \left\{ \begin{array}{rll} x & \text{ gdy }, x\geq 0 \\ -x & \text{ w przeciwnym przypadku}. \end{array}}

Parser nie mógł rozpoznać (błąd składni): {\displaystyle w_3 &\rightarrow bv_3v_2w_3v_1v_3v_3v_2\ |\ aw_3v_1v_3v_3v_2\ |\ bv_3v_2v_1v_3v_3v_2 \\ \begin{array}{lll} & & |\ av_1v_3v_3v_2\ |\ bv_3v_3v_2 \end{array}}

oraz

Parser nie mógł rozpoznać (błąd składni): {\displaystyle w_3 &\rightarrow bv_3v_2w_3v_1v_3v_3v_2w_3\ |\ aw_3v_1v_3v_3v_2w_3\ |\ bv_3v_2v_1v_3v_3v_2w_3 \\ \begin{array}{lll} & & |\ av_1v_3v_3v_2w_3\ |\ bv_3v_3v_2w_3 \end{array}}

Ostatecznie, gramatyka w postaci Greibach ma postać:

Parser nie mógł rozpoznać (błąd składni): {\displaystyle v_1 &\rightarrow bv_3v_2w_3v_1v_3\ |\ aw_3v_1v_3\ |\ bv_3v_2v_1v_3\ |\ av_1v_3\ |\ bv_3 \\ v_2 &\rightarrow bv_3v_2w_3v_1\ |\ aw_3v_1\ |\ bv_3v_2v_1\ |\ av_1\ |\ b \\ v_3 &\rightarrow bv_3v_2w_3\ |\ aw_3\ |\ bv_3v_2\ |\ a \\ w_3 &\rightarrow bv_3v_2w_3v_1v_3v_3v_2\ |\ aw_3v_1v_3v_3v_2\ |\ bv_3v_2v_1v_3v_3v_2 \\ \begin{array}{lll} & & |\ av_1v_3v_3v_2\ |\ bv_3v_3v_2 bv_3v_2w_3v_1v_3v_3v_2w_3 \\ & & |\ aw_3v_1v_3v_3v_2w_3\ |\ bv_3v_2v_1v_3v_3v_2w_3\ \\ & & |\ av_1v_3v_3v_2w_3\ |\ bv_3v_3v_2w_3 \end{array}}

\begin{array}{ccccc} K r o k & d o d a n e d o S^{'} & o k r e s l e n i e f^{'} & d o d a n e d o T^{'} \\ 0 & {q_{0}} & \emptyset \\ 1 & {q_{0}, q_{3}} & f^{'} ({q_{0}}, a) = {q_{0}, q_{3}} & \emptyset \\ {q_{0}, q_{1}} & f^{'} ({q_{0}}, b) = {q_{0}, q_{1}} \\ 2 & {q_{0}, q_{3}, q_{4}} & f^{'} ({q_{0}, q_{3}}, a) = {q_{0}, q_{3}, q_{4}} & {q_{0}, q_{3}, q_{4}} \\ f^{'} ({q_{0}, q_{3}}, b) = {q_{0}, q_{1}} \\ f^{'} ({q_{0}, q_{1}}, a) = {q_{0}, q_{3}} \\ {q_{0}, q_{1}, q_{2}} & f^{'} ({q_{0}, q_{1}}, b) = {q_{0}, q_{1}, q_{2}} & {q_{0}, q_{1}, q_{2}} \\ 3 & f^{'} ({q_{0}, q_{3}, q_{4}}, a) = {q_{0}, q_{3}, q_{4}} \\ {q_{0}, q_{1}, q_{4}} & f^{'} ({q_{0}, q_{3}, q_{4}}, b) = {q_{0}, q_{1}, q_{4}} & {q_{0}, q_{1}, q_{4}} \\ {q_{0}, q_{2}, q_{3}} & f^{'} ({q_{0}, q_{1}, q_{2}}, a) = {q_{0}, q_{2}, q_{3}} & {q_{0}, q_{2}, q_{3}} \\ f^{'} ({q_{0}, q_{1}, q_{2}}, b) = {q_{0}, q_{1}, q_{2}} \\ 4 & f^{'} ({q_{0}, q_{1}, q_{4}}, a) = {q_{0}, q_{3}, q_{4}} \\ {q_{0}, q_{1}, q_{2}, q_{4}}, & f^{'} ({q_{0}, q_{1}, q_{4}}, b) = {q_{0}, q_{1}, q_{2}, q_{4}} & {q_{0}, q_{1}, q_{2}, q_{4}} \\ {q_{0}, q_{2}, q_{3}, q_{4}} & f^{'} ({q_{0}, q_{2}, q_{3}}, a) = {q_{0}, q_{2}, q_{3}, q_{4}} & {q_{0}, q_{2}, q_{3}, q_{4}} \\ f^{'} ({q_{0}, q_{2}, q_{3}}, b) = {q_{0}, q_{1}, q_{2}} \\ 5 & f^{'} ({q_{0}, q_{1}, q_{2}, q_{4}}, a) = {q_{0}, q_{2}, q_{3}, q_{4}} \\ f^{'} ({q_{0}, q_{1}, q_{2}, q_{4}}, b) = {q_{0}, q_{1}, q_{2}, q_{4}} \\ f^{'} ({q_{0}, q_{2}, q_{3}, q_{4}}, a) = {q_{0}, q_{2}, q_{3}, q_{4}} \\ f^{'} ({q_{0}, q_{2}, q_{3}, q_{4}}, b) = {q_{0}, q_{1}, q_{2}, q_{4}} \end{array}

\begin{array}{ccccc} (s_{0}, ♯) \mapsto (s_{A}, ♯, 0) & (s_{0}, 0) \mapsto (r_{0}, ♯, 1) & (s_{0}, 1) \mapsto (r_{1}, ♯, 1) \\ (r_{0}, ♯) \mapsto (s_{R}, ♯, 0) & (r_{0}, 0) \mapsto ({r_{0}}^{'}, 0, 1) & (r_{0}, 1) \mapsto ({r_{0}}^{'}, 1, 1) \\ ({r_{0}}^{'}, ♯) \mapsto (q_{0}, ♯, - 1) & ({r_{0}}^{'}, 0) \mapsto ({r_{0}}^{'}, 0, 1) & ({r_{0}}^{'}, 1) \mapsto ({r_{0}}^{'}, 1, 1) \\ (q_{0}, 0) \mapsto (l, ♯, - 1) & (q_{0}, 1) \mapsto (s_{R}, ♯, - 1) \\ (r_{1}, ♯) \mapsto (s_{R}, ♯, 0) & (r_{1}, 0) \mapsto ({r_{1}}^{'}, 0, 1) & (r_{1}, 1) \mapsto ({r_{1}}^{'}, 1, 1) \\ ({r_{1}}^{'}, ♯) \mapsto (q_{1}, ♯, - 1) & ({r_{1}}^{'}, 0) \mapsto ({r_{1}}^{'}, 0, 1) & ({r_{1}}^{'}, 1) \mapsto ({r_{1}}^{'}, 1, 1) \\ (q_{1}, 0) \mapsto (s_{R}, ♯, 0) & (q_{1}, 1) \mapsto (l, ♯, - 1) \\ (l, ♯) \mapsto (s_{0}, ♯, 1) & (l, 0) \mapsto (l, 0, - 1) & (l, 1) \mapsto (l, 1, - 1) \\ (s_{R}, ♯) \mapsto (s_{R}, ♯, 0) \\ (s_{A}, ♯) \mapsto (s_{A}, ♯, 0) \end{array}

\begin{array}{ccccc} (s_{0}, ♯) \mapsto (s_{R}, ♯, 0) & (s_{1}, ♢) \mapsto (s 1, ♢, 1) \\ (s_{0}, 0) \mapsto (s_{1}, ♣, 1) & (s_{1}, 0) \mapsto (s_{2}, ♢, 1) \\ (s_{1}, ♯) \mapsto (s_{A}, ♯, 0) \\ (s_{2}, ♢) \mapsto (s_{2}, ♢, 1) & (s_{3}, 0) \mapsto (s_{2}, ♢, 1) \\ (s_{2}, ♯) \mapsto (s_{4}, ♯, - 1) & (s_{3}, ♢) \mapsto (s_{3}, ♢, 1) \\ (s_{2}, 0) \mapsto (s_{3}, 0, 1) & (s_{3}, ♯) \mapsto (s_{R}, ♯, 0) \\ (s_{4}, 0) \mapsto (s_{4}, 0, - 1) \\ (s_{4}, ♢) \mapsto (s_{4}, ♢, - 1) \\ (s_{4}, ♣) \mapsto (s_{2}, ♣, 1) \\ (s_{A}, ♯) \mapsto (s_{A}, ♯, 0) & (s_{R}, ♯) \mapsto (s_{R}, ♯, 0) \end{array}

\begin{array}{ccccc} s_{0} & s_{1} & s_{2} \\ τ_{𝒜} (1) & s_{0} & s_{1} & s_{2} \\ τ_{𝒜} (a) & s_{1} & s_{2} & s_{2} \\ τ_{𝒜} (b) & s_{0} & s_{0} & s_{0} \\ τ_{𝒜} (a^{2}) & s_{2} & s_{2} & s_{2} \\ τ_{𝒜} (a b) & s_{0} & s_{0} & s_{2} \\ τ_{𝒜} (b a) & s_{1} & s_{1} & s_{1} \\ τ_{𝒜} (b^{2}) & s_{0} & s_{0} & s_{0} \\ τ_{𝒜} (a b a) & s_{1} & s_{1} & s_{2} \\ . . . & . . . & . . . & . . . \end{array}

\begin{array}{ccccc} f & s_{0} & s_{1} & s_{2} & s_{3} \\ a & s_{1} & s_{2} & s_{0} & s_{2} \\ b & s_{3} & s_{2} & s_{2} & s_{2} \end{array}

Algorytm Minimalizuj2 - algorytm minimalizacji automatu wykorzystujący stabilizujący się ciąg relacji

  1  Wejście:  $𝒜 = (S, A, f, s_{0}, T)$  - automat taki, że  $L = L (𝒜)$ .
  2  Wyjście: automat minimalny  $𝒜^{'} = (S^{'}, A^{'}, f^{'}, {s_{0}}^{'}, T^{'})$  dla  $𝒜$ .
  3   ${\overline{ρ}}_{1} \leftarrow \approx_{𝒜}$ ;
  4   $i \leftarrow 1$ ;
  5  repeat
  6    Parser nie mógł rozpoznać (nieznana funkcja „\slash”): {\displaystyle \slash \slash}
 oblicz  ${\overline{ρ}}_{i}$ :  $s_{1} {\overline{ρ}}_{i} s_{2} \Leftrightarrow (s_{1} {\overline{ρ}}_{i - 1} s_{2}) \land (\forall a \in A f (s_{1}, a) {\overline{ρ}}_{i - 1} f (s_{2}, a))$ ;
  7     $i \leftarrow i + 1$ ;
  8    empty $({\overline{ρ}}_{i})$  
  9    for each  $(s_{1}, s_{2}) \in S \times S$  do
 10      flag $\leftarrow$ true;
 11      for each  $a \in A$  
 12        if not  $f (s_{1}, a) {\overline{ρ}}_{i - 1} f (s_{2}, a)$  then
 13          flag $\leftarrow$ false;
 14        end if
 15      end for
 16      if flag=true and  $s_{1} {\overline{ρ}}_{i - 1} s_{2}$  then
 17         ${\overline{ρ}}_{i} \leftarrow {\overline{ρ}}_{i} \cup {(s_{1}, s_{2})}$ ;
 18      end if
 19    end for
 20  until  ${\overline{ρ}}_{i} = {\overline{ρ}}_{i - 1}$ 
 21  Parser nie mógł rozpoznać (nieznana funkcja „\slash”): {\displaystyle S' \leftarrow S \slash \overline{\rho}_i}
;
 22  for each Parser nie mógł rozpoznać (nieznana funkcja „\slash”): {\displaystyle [s]_{\overline{\rho}_i} \in S \slash \overline{\rho}_i}
 do
 23    for each  $a \in A$  do
 24       $f^{'} ([s]_{{\overline{ρ}}_{i}}, a) \leftarrow [f (s, a)]_{{\overline{ρ}}_{i}}$ ;
 25    end for
 26  end for
 27   ${s_{0}}^{'} \leftarrow [s_{0}]_{{\overline{ρ}}_{i}}$ ;
 28   $T^{'} \leftarrow {[t]_{{\overline{ρ}}_{i}} : t \in T}$ ;
 29  return  $𝒜^{'} = (S^{'}, A, f^{'}, {s_{0}}^{'}, T^{'})$ ;

Uzupelnij tytul
	$s_{0}$ \|\| $s_{1}$ \|\| $s_{2}$
$τ_{𝒜} (1)$ \|\| $s_{0}$ \|\| $s_{1}$ \|\| $s_{2}$
$τ_{𝒜} (a)$ \|\| $s_{1}$ \|\| $s_{2}$ \|\| $s_{2}$
$τ_{𝒜} (b)$ \|\| $s_{0}$ \|\| $s_{0}$ \|\| $s_{0}$
$τ_{𝒜} (a^{2})$ \|\| $s_{2}$ \|\| $s_{2}$ \|\| $s_{2}$
$τ_{𝒜} (a b)$ \|\| $s_{0}$ \|\| $s_{0}$ \|\| $s_{2}$
$τ_{𝒜} (b a)$ \|\| $s_{1}$ \|\| $s_{1}$ \|\| $s_{1}$
$τ_{𝒜} (b^{2})$ \|\| $s_{0}$ \|\| $s_{0}$ \|\| $s_{0}$
$τ_{𝒜} (a b a)$ \|\| $s_{1}$ \|\| $s_{1}$ \|\| $s_{2}$
... \|\| ... \|\| ... \|\| ...

$\begin{array}{lll} b) \lim_{x \to 2^{+}} (x - 2) e^{\frac{1}{x - 2}} & = & \lim_{x \to 2^{+}} \frac{e^{\frac{1}{x - 2}}}{(x - 2)^{- 1}} \begin{array}{c} [\frac{\infty}{\infty}] \\ = \\ H \end{array} \lim_{x \to 2^{+}} \frac{- (x - 2)^{- 2} e^{\frac{1}{x - 2}}}{- (x - 2)^{- 2}} = \\ = & \lim_{x \to 2^{+}} e^{\frac{1}{x - 2}} = + \infty; \end{array}$

 alalalalaa

alala

	Złożoność czasowa	Złożoność pamięciowa
Maszyna dodająca	$f (0) = 1$ $f (1) = 3$ $f (n) = n + 3; n \geq 2$	$f (0) = 2$ $f (1) = 3$ $f (n) = n + 1; n \geq 2$
Maszyna rozpoznająca $w w^{\leftarrow}$	$f (n) = 6 + 8 + \dots + (n + 3) + 2; n = 2 k + 1$ $f (n) = 5 + 7 + \dots + (n + 3) + 1; n = 2 k$	$f (n) = n + 1$

$\Rightarrow$	0	1	...	...
Cell1	Cell2

$\Rightarrow$	0	1
0	1	1
1	0	1

$p$	$\neg p$
0	1
1	0

$\land$	0	1
0	0	0
1	0	1

$\lor$	0	1
0	0	1
1	1	1

$p$	$q$	$p \land q$	$\neg (p \land q)$	$\neg p$	$\neg q$	$\neg p \lor \neg q$
0	0	0	1	1	1	1
0	1	0	1	1	0	1
1	0	0	1	0	1	1
1	1	1	0	0	0	0

$p$	$q$	$r$	$(p \land q)$	$(p \land r)$	$(q \land \neg r)$	$(p \land r) \lor (q \land \neg r)$	$(p \land q) \Rightarrow ((p \land r) \lor (q \land \neg r))$
0	0	0	0	0	0	0	1
0	0	1	0	0	0	0	1
0	1	0	0	0	1	1	1
0	1	1	0	0	0	0	1
1	0	0	0	0	0	0	1
1	0	1	0	1	0	1	1
1	1	0	1	0	1	1	1
1	1	1	1	1	0	1	1

Numer funkcji	$p = 0$ $q = 0$	$p = 0$ $q = 1$	$p = 1$ $q = 0$	$p = 1$ $q = 1$
0	0	0	0	0	$F$
1	0	0	0	1	$\land$
2	0	0	1	0	$\neg (p \Rightarrow q)$
3	0	0	1	1	$p$
4	0	1	0	0	$\neg (q \Rightarrow p)$
5	0	1	0	1	$q$
6	0	1	1	0	$X O R$
7	0	1	1	1	$\lor$
8	1	0	0	0	$N O R$
9	1	0	0	1	$\Leftrightarrow$
10	1	0	1	0	$\neg q$
11	1	0	1	1	$q \Rightarrow p$
12	1	1	0	0	$\neg p$
13	1	1	0	1	$p \Rightarrow q$
14	1	1	1	0	$N A N D$
15	1	1	1	1	$T$

$p$	$q$	$r$	$(p \leftrightarrow q)$	$(p \leftrightarrow q) \leftrightarrow r$	$(q \leftrightarrow r)$	$p \leftrightarrow (q \leftrightarrow r)$
0	0	0	1	0	1	0
0	0	1	1	1	0	1
0	1	0	0	1	0	1
0	1	1	0	0	1	0
1	0	0	0	1	1	1
1	0	1	0	0	0	0
1	1	0	1	0	0	0
1	1	1	1	1	1	1

$p$	$q$	$r$	$\circ (p, q, r)$
0	0	0	0
0	0	1	1
0	1	0	1
0	1	1	0
1	0	0	1
1	0	1	0
1	1	0	0
1	1	1	1

$p$	$q$	$r$	$f_{p \to_{(} q \to r)}$
0	0	0	1
0	0	1	1
0	1	0	1
0	1	1	1
1	0	0	1
1	0	1	1
1	1	0	0
1	1	1	1

$\Rightarrow$	0	1	2
0	2	2	2
1	0	2	2
2	0	1	2

$\hat{σ} (f (t_{0}, . ., t_{n})) = I (f) (\hat{σ} (t_{0}), . ., \hat{σ} (t_{n}))$

X^{*} = {1} \cup X \cup X^{2} \cup . . . \cup X^{n} \cup \dots = ⋃_{i = 0}^{\infty} X^{i}

Nagroda Goedla
Zobacz Nagroda Goedla]]

Nagroda Turinga
Zobacz Nagroda Turinga

Nagroda Knutha
Zobacz Nagroda Knutha

g (C) = {\begin{cases} C \cup {f (C^{'})} C \end{cases}

Parser nie mógł rozpoznać (błąd składni): {\displaystyle g(C)=\left\{\begin{align} C\cup \{f(C')\}\\C\end{align} \right}

Parser nie mógł rozpoznać (błąd składni): {\displaystyle c\forall d\; c\in C \land d\in C \land c\sqsubseteq d\implies c\sqsubseteq' d, (C,\sqsubseteq) \preccurlyeq (C',\sqsubseteq') \iff C\subset C' \land \left\{\begin{align} \forall c \forall d\; &(c\in C\land d\in C) \implies (c\sqsubseteq d \iff c\sqsubseteq' d) \text{ oraz }\\ \forall c \forall d\; &(c\in C\land d\in C'\setminus C) \implies c\sqsubseteq' d \end{align} \right}

$h (0, a) = f (a)$ dla każdego Parser nie mógł rozpoznać (błąd składni): {\displaystyle a \in A \\h(n', a) = g(h(n, a), n, a)} dla każdego $a \in A$ i $n \in ℕ$

$e (0, a) = f (a)$ dla każdego Parser nie mógł rozpoznać (błąd składni): {\displaystyle a \in A \\ e(g(n, a), n, a)} dla każdego $a \in A$ i $n \in m$

	$s_{0}$ \|\| $s_{1}$ \|\| $s_{2}$
$τ_{𝒜} (1)$ \|\| $s_{0}$ \|\| $s_{1}$ \|\| $s_{2}$
$τ_{𝒜} (a)$ \|\| $s_{1}$ \|\| $s_{2}$ \|\| $s_{2}$
$τ_{𝒜} (b)$ \|\| $s_{0}$ \|\| $s_{0}$ \|\| $s_{0}$
$τ_{𝒜} (a^{2})$ \|\| $s_{2}$ \|\| $s_{2}$ \|\| $s_{2}$
$τ_{𝒜} (a b)$ \|\| $s_{0}$ \|\| $s_{0}$ \|\| $s_{2}$
$τ_{𝒜} (b a)$ \|\| $s_{1}$ \|\| $s_{1}$ \|\| $s_{1}$
$τ_{𝒜} (b^{2})$ \|\| $s_{0}$ \|\| $s_{0}$ \|\| $s_{0}$
$τ_{𝒜} (a b a)$ \|\| $s_{1}$ \|\| $s_{1}$ \|\| $s_{2}$
... \|\| ... \|\| ... \|\| ...

Test GR2: Różnice pomiędzy wersjami

Aktualna wersja na dzień 22:16, 11 wrz 2023

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@@ Linia 1: / Linia 1: @@
+{| border="1" cellspacing="0"
+! <math>\phi</math>!! <math>\psi</math>!! <math>\psi \Rightarrow \phi</math>!! <math>(\phi \Rightarrow (\psi \Rightarrow \phi))</math>!!
+|-
+| &nbsp;0&nbsp;|| &nbsp;0&nbsp;|| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+|-
+| &nbsp;0&nbsp;|| &nbsp;1&nbsp;|| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;0&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+|-
+| &nbsp;1&nbsp;|| &nbsp;0&nbsp;|| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+|-
+| &nbsp;1&nbsp;|| &nbsp;1&nbsp;|| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
+|}
+<center><math>\left| x \right|\ = \left\{ \begin{array}{rll} x & \text{ gdy }, x\geq 0 \\ -x & \text{ w przeciwnym przypadku}.
+\end{array}</math></center>
+<center><math>w_3 &\rightarrow  bv_3v_2w_3v_1v_3v_3v_2\ |\
+aw_3v_1v_3v_3v_2\ |\ bv_3v_2v_1v_3v_3v_2 \\
+\begin{array}{lll} & & |\ av_1v_3v_3v_2\ |\ bv_3v_3v_2
+\end{array}</math></center>
+oraz
+<center><math>w_3 &\rightarrow  bv_3v_2w_3v_1v_3v_3v_2w_3\ |\
+aw_3v_1v_3v_3v_2w_3\ |\ bv_3v_2v_1v_3v_3v_2w_3 \\
+\begin{array}{lll} & & |\ av_1v_3v_3v_2w_3\ |\ bv_3v_3v_2w_3
+\end{array}</math></center>
+Ostatecznie, gramatyka w postaci Greibach ma postać:
+<center><math> v_1 &\rightarrow  bv_3v_2w_3v_1v_3\ |\ aw_3v_1v_3\ |\ bv_3v_2v_1v_3\ |\ av_1v_3\ |\ bv_3 \\
+ v_2 &\rightarrow  bv_3v_2w_3v_1\ |\ aw_3v_1\ |\ bv_3v_2v_1\ |\ av_1\ |\ b \\
+ v_3 &\rightarrow  bv_3v_2w_3\ |\ aw_3\ |\ bv_3v_2\ |\ a
+\\
+ w_3 &\rightarrow  bv_3v_2w_3v_1v_3v_3v_2\ |\
+aw_3v_1v_3v_3v_2\ |\ bv_3v_2v_1v_3v_3v_2 \\
+\begin{array}{lll} & & |\ av_1v_3v_3v_2\ |\ bv_3v_3v_2 bv_3v_2w_3v_1v_3v_3v_2w_3 \\
+ & & |\ aw_3v_1v_3v_3v_2w_3\ |\ bv_3v_2v_1v_3v_3v_2w_3\ \\
+ & & |\ av_1v_3v_3v_2w_3\ |\ bv_3v_3v_2w_3
+\end{array}</math></center>
+<center><math>\begin{array} {c|c|c|c|c|} Krok & dodane\quad do\quad S' & okreslenie\quad f' & dodane\quad do\quad T'\\
+\hline 0 & \{q_0\} &  & \emptyset\\
+\hline 1 & \{q_0,q_3\} & f'(\{q_0\},a)=\{q_0,q_3\} & \emptyset\\
+\hline  & \{q_0,q_1\} & f'(\{q_0\},b)=\{q_0,q_1\} & \\
+\hline 2 & \{q_0,q_3,q_4\} & f'(\{q_0,q_3\},a)=\{q_0,q_3,q_4\} & \{q_0,q_3,q_4\}\\
+\hline  &  & f'(\{q_0,q_3\},b)=\{q_0,q_1\} & \\
+\hline  &  & f'(\{q_0,q_1\},a)=\{q_0,q_3\} & \\
+\hline  & \{q_0,q_1,q_2\} & f'(\{q_0,q_1\},b)=\{q_0,q_1,q_2\} & \{q_0,q_1,q_2\}\\
+\hline 3 &  & f'(\{q_0,q_3,q_4\},a)=\{q_0,q_3,q_4\} & \\
+\hline  & \{q_0,q_1,q_4\} & f'(\{q_0,q_3,q_4\},b)=\{q_0,q_1,q_4\} & \{q_0,q_1,q_4\}\\
+\hline  & \{q_0,q_2,q_3\} & f'(\{q_0,q_1,q_2\},a)=\{q_0,q_2,q_3\} & \{q_0,q_2,q_3\}\\
+\hline  &  & f'(\{q_0,q_1,q_2\},b)=\{q_0,q_1,q_2\} & \\
+\hline 4 &  & f'(\{q_0,q_1,q_4\},a)=\{q_0,q_3,q_4\} & \\
+\hline  & \{q_0,q_1,q_2,q_4\}, & f'(\{q_0,q_1,q_4\},b)=\{q_0,q_1,q_2,q_4\} & \{q_0,q_1,q_2,q_4\}\\
+\hline  & \{q_0,q_2,q_3,q_4\} & f'(\{q_0,q_2,q_3\},a)=\{q_0,q_2,q_3,q_4\} & \{q_0,q_2,q_3,q_4\}\\
+\hline  &  & f'(\{q_0,q_2,q_3\},b)=\{q_0,q_1,q_2\} & \\
+\hline 5 &  & f'(\{q_0,q_1,q_2,q_4\},a)=\{q_0,q_2,q_3,q_4\} & \\
+\hline  &  & f'(\{q_0,q_1,q_2,q_4\},b)=\{q_0,q_1,q_2,q_4\} & \\
+\hline  &  & f'(\{q_0,q_2,q_3,q_4\},a)=\{q_0,q_2,q_3,q_4\} & \\
+\hline  &  & f'(\{q_0,q_2,q_3,q_4\},b)=\{q_0,q_1,q_2,q_4\} & \\
+\hline \end{array} </math></center>
+<center><math>\begin{array} {c|c|c|c|c|} (s_0,\sharp)\mapsto (s_A,\sharp,0) & (s_0,0)\mapsto (r_0,\sharp,1) & (s_0,1)\mapsto (r_1,\sharp,1)\\
+\hline (r_0,\sharp)\mapsto (s_R,\sharp,0) & (r_0,0)\mapsto (r_0',0,1) & (r_0,1)\mapsto (r_0',1,1)\\
+\hline (r_0',\sharp)\mapsto (q_0,\sharp,-1) & (r_0',0)\mapsto (r_0',0,1) & (r_0',1)\mapsto (r_0',1,1)\\
+\hline  & (q_0,0)\mapsto (l,\sharp,-1) & (q_0,1)\mapsto (s_R,\sharp,-1)\\
+\hline (r_1,\sharp)\mapsto (s_R,\sharp,0) & (r_1,0)\mapsto (r_1',0,1) & (r_1,1)\mapsto (r_1',1,1)\\
+\hline (r_1',\sharp)\mapsto (q_1,\sharp,-1) & (r_1',0)\mapsto (r_1',0,1) & (r_1',1)\mapsto (r_1',1,1)\\
+\hline  & (q_1,0)\mapsto (s_R,\sharp,0) & (q_1,1)\mapsto (l,\sharp,-1)\\
+\hline (l,\sharp)\mapsto (s_0,\sharp,1) & (l,0)\mapsto (l,0,-1) & (l,1)\mapsto (l,1,-1)\\
+\hline (s_R,\sharp)\mapsto (s_R,\sharp,0) &  & \\
+\hline (s_A,\sharp)\mapsto (s_A,\sharp,0) &  & \\
+\hline \end{array} </math></center>
+<center><math>\begin{array} {c|c|c|c|c|} (s_0,\sharp)\mapsto (s_R,\sharp,0) & (s_1,\diamondsuit)\mapsto(s1,\diamondsuit,1)\\
+\hline (s_0,0)\mapsto(s_1,\clubsuit,1) & (s_1,0) \mapsto(s_2,\diamondsuit,1)\\
+\hline  & (s_1,\sharp)\mapsto(s_A,\sharp,0)\\
+\hline (s_2,\diamondsuit)\mapsto(s_2,\diamondsuit,1) & (s_3,0)\mapsto(s_2,\diamondsuit,1)\\
+\hline (s_2,\sharp)\mapsto(s_4,\sharp,-1) & (s_3,\diamondsuit)\mapsto(s_3,\diamondsuit,1)\\
+\hline (s_2,0)\mapsto(s_3,0,1) & (s_3,\sharp)\mapsto(s_R,\sharp,0)\\
+\hline (s_4,0)\mapsto(s_4,0,-1) & \\
+\hline (s_4,\diamondsuit)\mapsto(s_4,\diamondsuit,-1) & \\
+\hline (s_4,\clubsuit)\mapsto(s_2,\clubsuit,1) & \\
+\hline (s_A,\sharp)\mapsto(s_A,\sharp,0) & (s_R,\sharp)\mapsto(s_R,\sharp,0)\\
+\hline \end{array} </math></center>
+<center><math>\begin{array} {c|c|c|c|c|}  & s_0 & s_1 & s_2\\
+\hline \tau _{\mathcal{A}}(1) & s_0 & s_1 & s_2\\
+\hline \tau _{\mathcal{A}}(a) & s_1 & s_2 & s_2\\
+\hline \tau _{\mathcal{A}}(b) & s_0 & s_0 & s_0\\
+\hline \tau _{\mathcal{A}}(a^{2}) & s_2 & s_2 & s_2\\
+\hline \tau _{\mathcal{A}}(ab) & s_0 & s_0 & s_2\\
+\hline \tau _{\mathcal{A}}(ba) & s_1 & s_1 & s_1\\
+\hline \tau _{\mathcal{A}}(b^{2}) & s_0 & s_0 & s_0\\
+\hline \tau _{\mathcal{A}}(aba) & s_1 & s_1 & s_2\\
+\hline ... & ... & ... & ...\\
+\hline \end{array} </math></center>
+<center><math>\begin{array} {c|c|c|c|c|} f & s_0 & s_1 & s_2 & s_3\\
+\hline a & s_1 & s_2 & s_0 & s_2\\
+\hline b & s_3 & s_2 & s_2 & s_2\\
+\hline \end{array} </math></center>
+{{algorytm|Minimalizuj2 - algorytm minimalizacji automatu
+wykorzystujący stabilizujący się ciąg relacji|algorytm minimalizacji automatu wykorzystujący stabilizujący się ciąg relacji|
+  Wejście: <math>\mathcal{A}=(S, A, f, s_0, T)</math> - automat taki, że <math>L=L(\mathcal{A})</math>.
+  Wyjście: automat minimalny <math>\mathcal{A}'=(S',A',f', s_0',
+T')</math> dla <math>\mathcal{A}</math>.
+  <math>\overline{\rho}_1\leftarrow\approx_{\mathcal{A}}</math>;
+  <math>i \leftarrow 1</math>;
+  '''repeat'''
+    <math>\slash \slash</math> oblicz <math>\overline{\rho}_i</math>: <math>s_1
+\overline{\rho}_i s_2 \Leftrightarrow (s_1 \overline{\rho}_{i-1}
+s_2) \wedge (\forall a \in A\ f(s_1, a) \overline{\rho}_{i-1}
+f(s_2,a))</math>;
+    <math>i \leftarrow i+1</math>;
+    '''empty'''<math>(\overline{\rho}_i)</math>
+    '''for''' '''each''' <math>(s_1,s_2)\in S\times S</math> '''do'''
+      flag<math>\leftarrow</math>'''true''';
+      '''for''' '''each''' <math>a\in A</math>
+        '''if''' '''not''' <math>f(s_1, a) \overline{\rho}_{i-1} f(s_2,a)</math> '''then'''
+          flag<math>\leftarrow</math>'''false''';
+        '''end''' '''if'''
+      '''end''' '''for'''
+      '''if''' flag<nowiki>=</nowiki>'''true''' '''and''' <math>s_1 \overline{\rho}_{i-1} s_2</math> '''then'''
+        <math>\overline{\rho}_{i} \leftarrow \overline{\rho}_{i} \cup \{(s_1,s_2)\}</math>;
+      '''end''' '''if'''
+    '''end''' '''for'''
+  '''until''' <math>\overline{\rho}_i = \overline{\rho}_{i-1}</math>
+  <math>S' \leftarrow S \slash \overline{\rho}_i</math>;
+  '''for''' '''each''' <math>[s]_{\overline{\rho}_i} \in S \slash \overline{\rho}_i</math> '''do'''
+    '''for''' '''each''' <math>a \in A</math> '''do'''
+      <math>f'([s]_{\overline{\rho}_i},a) \leftarrow
+[f(s,a)]_{\overline{\rho}_i}</math>;
+    '''end''' '''for'''
+  '''end''' '''for'''
+  <math>s_0' \leftarrow [s_0]_{\overline{\rho}_i}</math>;
+  <math>T' \leftarrow \{[t]_{\overline{\rho}_i}:\ t \in T\}</math>;
+  '''return''' <math>\mathcal{A}'=(S', A, f', s_0', T')</math>;
+}}
+{| border=1
+|+ <span style="font-variant:small-caps">Uzupelnij tytul</span>
+|-
+|
+||
+<math>s_{0} </math>  ||
+<math>s_{1} </math>  ||
+<math>s_{2} </math>
+|-
+|
+<math>\tau _{\mathcal{A}}(1) </math>  ||
+<math>s_{0} </math>  ||
+<math>s_{1} </math>  ||
+<math>s_{2} </math>
+|-
+|
+<math>\tau _{\mathcal{A}}(a) </math>  ||
+<math>s_{1} </math>  ||
+<math>s_{2} </math>  ||
+<math>s_{2} </math>
+|-
+|
+<math>\tau _{\mathcal{A}}(b) </math>  ||
+<math>s_{0} </math>  ||
+<math>s_{0} </math>  ||
+<math>s_{0} </math>
+|-
+|
+<math>\tau _{\mathcal{A}}(a^{2}) </math>  ||
+<math>s_{2} </math>  ||
+<math>s_{2} </math>  ||
+<math>s_{2} </math>
+|-
+|
+<math>\tau _{\mathcal{A}}(ab) </math>  ||
+<math>s_{0} </math>  ||
+<math>s_{0} </math>  ||
+<math>s_{2} </math>
+|-
+|
+<math>\tau _{\mathcal{A}}(ba) </math>  ||
+<math>s_{1} </math>  ||
+<math>s_{1} </math>  ||
+<math>s_{1} </math>
+|-
+|
+<math>\tau _{\mathcal{A}}(b^{2}) </math>  ||
+<math>s_{0} </math>  ||
+<math>s_{0} </math>  ||
+<math>s_{0} </math>
+|-
+|
+<math>\tau _{\mathcal{A}}(aba) </math>  ||
+<math>s_{1} </math>  ||
+<math>s_{1} </math>  ||
+<math>s_{2} </math>
+|-
+|
+... ||
+... ||
+... ||
+...
+|-
+|
+|}
+<math>
+\begin{array}{lll}
+\text{b) } \lim_{x\rightarrow 2^+} (x-2)e^{\frac{1}{x-2}}&=&\lim_{x\rightarrow
+^+} \frac{e^{\frac{1}{x-2}}}{(x-2)^{-1}}\begin{array} {c}\left[\frac{\infty}{\infty}\right]\\=\\H\end{array}
+\lim_{x\rightarrow 2^+}
+\frac{-(x-2)^{-2}e^{\frac{1}{x-2}}}{-(x-2)^{-2}}=\\
+&=&\lim_{x\rightarrow 2^+} e^{\frac{1}{x-2}}=+\infty;
+\end{array}
+</math><br>
+  alalalalaa
+ alala
 {| border="1" cellspacing="0"
 ! !! Złożoność czasowa !! Złożoność pamięciowa
@@ Linia 50: / Linia 305: @@
 {| border="1"
-! <math>\textnormal{p}</math>!! <math>\textnormal{q}</math>!! <math>\textnormal{p} \wedge \textnormal{q}</math>!! <math>\neg( p \wedge q)</math>!! <math>\neg p</math>!! <math>\neg q</math>!! <math>\neg p \vee \neg q</math>
+! <math>\text{p}</math>!! <math>\text{q}</math>!! <math>\text{p} \wedge \text{q}</math>!! <math>\neg( p \wedge q)</math>!! <math>\neg p</math>!! <math>\neg q</math>!! <math>\neg p \vee \neg q</math>
 |-
 | &nbsp;0&nbsp;|| &nbsp;0&nbsp;|| 0|| 1|| 1|| 1|| 1
@@ Linia 63: / Linia 318: @@
 {| border="1"
-! <math>\textnormal{p}</math>!! <math>\textnormal{q}</math>!! <math>\textnormal{r}</math>!!<math>(\textnormal{p} \wedge \textnormal{q})</math>!! <math>( p \wedge r)</math>!! <math>( q \wedge \neg r)</math>!! <math>(p \wedge r) \vee (q \wedge \neg r)</math>!! <math>(p \wedge q) \Rightarrow ((p \wedge r) \vee (q \wedge \neg r))</math>
+! <math>\text{p}</math>!! <math>\text{q}</math>!! <math>\text{r}</math>!!<math>(\text{p} \wedge \text{q})</math>!! <math>( p \wedge r)</math>!! <math>( q \wedge \neg r)</math>!! <math>(p \wedge r) \vee (q \wedge \neg r)</math>!! <math>(p \wedge q) \Rightarrow ((p \wedge r) \vee (q \wedge \neg r))</math>
 |-
 | 0|| 0|| 0|| 0|| 0|| 0|| 0|| 1
@@ Linia 92: / Linia 347: @@
 | 2|| 0|| 0|| 1|| 0|| &nbsp;|| <math>\neg (p \Rightarrow q)</math>
 |-
-| 3|| 0|| 0|| 1|| 1|| &nbsp;|| <math>\textnormal{p}</math>
+| 3|| 0|| 0|| 1|| 1|| &nbsp;|| <math>\text{p}</math>
 |-
 | 4|| 0|| 1|| 0|| 0|| &nbsp;|| <math>\neg (q \Rightarrow p)</math>
 |-
-| 5|| 0|| 1|| 0|| 1|| &nbsp;|| <math>\textnormal{q}</math>
+| 5|| 0|| 1|| 0|| 1|| &nbsp;|| <math>\text{q}</math>
 |-
 | 6|| 0|| 1|| 1|| 0|| &nbsp;|| <math>XOR</math>
@@ Linia 121: / Linia 376: @@
 {| border="1"
-! <math>\textnormal{p}</math>!! <math>\textnormal{q}</math>!! <math>\textnormal{r}</math>!!<math>(p \leftrightarrow q)</math>!! <math>(p \leftrightarrow q) \leftrightarrow r</math>!! <math>(q \leftrightarrow r)</math>!! <math>p \leftrightarrow (q \leftrightarrow r)</math>
+! <math>\text{p}</math>!! <math>\text{q}</math>!! <math>\text{r}</math>!!<math>(p \leftrightarrow q)</math>!! <math>(p \leftrightarrow q) \leftrightarrow r</math>!! <math>(q \leftrightarrow r)</math>!! <math>p \leftrightarrow (q \leftrightarrow r)</math>
 |-
 | 0|| 0|| 0|| 1|| 0|| 1|| 0
@@ Linia 142: / Linia 397: @@
 {| border="1"
-! <math>\textnormal{p}</math>!! <math>\textnormal{q}</math>!! <math>\textnormal{r}</math>!!<math>\circ (p, q, r)</math>
+! <math>\text{p}</math>!! <math>\text{q}</math>!! <math>\text{r}</math>!!<math>\circ (p, q, r)</math>
 |-
 | 0|| 0|| 0|| 0
@@ Linia 163: / Linia 418: @@
 {| border="1"
-! <math>\textnormal{p}</math>!! <math>\textnormal{q}</math>!! <math>\textnormal{r}</math>!!<math>f_{p \rightarrow _(q \rightarrow r)}</math>
+! <math>\text{p}</math>!! <math>\text{q}</math>!! <math>\text{r}</math>!!<math>f_{p \rightarrow _(q \rightarrow r)}</math>
 |-
 | 0|| 0|| 0|| 1
@@ Linia 200: / Linia 455: @@
-[[grafika:Todd.jpeg|thumb|right||John Arthur Todd (1908-1994)<br>[[Biografia Todd|Zobacz biografię]]]]
-[[grafika:Knuth1.jpeg|thumb|right||Donald Ervin Knuth (1938-)<br>[[Biografia Knuth|Zobacz biografię]]]]
-[[grafika:nagroda.jpeg|thumb|right||Nagroda Turinga<br>[[Nagroda Turinga|Zobacz Nagroda Turinga]]]]
-[[grafika:Goedel3.jpeg|thumb|right||Kurt Goedel (1906-1978)<br>[[Biografia Goedel|Zobacz biografię]]]]
-[[grafika:Chomsky1.jpg|thumb|right||Noam Avram Chomsky (1928-)<br>[[Biografia Chomsky|Zobacz biografię]]]]
-[[grafika:Kleene.gif|thumb|right||Stephen Cole Kleene (1909-1994)<br>[[Biografia Kleene|Zobacz biografię]]]]
-[[grafika:Thue.jpg|thumb|right||Axel Thue (1863-1922)<br>[[Biografia Thue|Zobacz biografię]]]]
-[[grafika:Church.jpg|thumb|right||Alonso Church (1903-1995)<br>[[Biografia Church|Zobacz biografię]]]]
-[[grafika:Greibach.jpg|thumb|right||Sheila Greibach (1939-)<br>[[Biografia Greibach|Zobacz biografię]]]]
-[[grafika:Post.jpg|thumb|right||Emil Leon Post (1897-1954)<br>[[Biografia Post|Zobacz biografię]]]]
 -----------------------------------------------------------
-[[grafika:Czebyszew.jpg|thumb|right||Pafnutij Lwowicz Czebyszew (1821-1894)<br>[[Biografia Czebyszew|Zobacz biografię]]]]
-[[grafika:Kołmogorow.jpg|thumb|right||Andriej Nikołajewicz Kołmogorow (1903-1987)<br>[[Biografia Kołmogorow|Zobacz biografię]]]]
-[[grafika:Markow.jpg|thumb|right||Andriej Markow (1856-1922)<br>[[Biografia Markow|Zobacz biografię]]]]
-[[grafika:Bernoulli.jpg|thumb|right||Jakob Bernoulli (1654-1705)<br>[[Biografia Bernoulli|Zobacz biografię]]]]
+Nagroda Goedla<br>[[Nagroda Goedla|Zobacz Nagroda Goedla]]]]
-[[grafika:Borel.jpg|thumb|right||Félix Édouard Justin Émile Borel (1871-1956)<br>[[Biografia Borel|Zobacz biografię]]]]
+Nagroda Turinga<br>[[Nagroda Turinga|Zobacz Nagroda Turinga]]
-[[grafika:Gauss.jpg|thumb|right||Carl Friedrich Gauss (1777-1855)<br>[[Biografia Gauss|Zobacz biografię]]]]
+Nagroda Knutha<br>[[Nagroda Knutha|Zobacz Nagroda Knutha]]
-[[grafika:Lebesgue.jpg|thumb|right||Henri Léon Lebesgue (1875-1941)<br>[[Biografia Lebesgue|Zobacz biografię]]]]
-[[grafika:Poisson.jpg|thumb|right||Siméon Denis Poisson (1781-1840)<br>[[Biografia Poisson|Zobacz biografię]]]]
+<center><math>g(C)=\begin{cases} C\cup \{f(C')\} C \end{cases}
+</math></center>
-Nagroda Goedla<br>[[Nagroda Goedla|Zobacz Nagroda Goedla]]]]
+<math>g(C)=\left\{\begin{align} C\cup \{f(C')\}\\C\end{align} \right</math>
-Nagroda Turinga<br>[[Nagroda Turinga|Zobacz Nagroda Turinga]]
+<math>c\forall d\; c\in C \land d\in C \land c\sqsubseteq d\implies c\sqsubseteq' d,
+(C,\sqsubseteq) \preccurlyeq (C',\sqsubseteq') \iff C\subset C' \land \left\{\begin{align} \forall c \forall d\; &(c\in C\land d\in C) \implies (c\sqsubseteq d \iff c\sqsubseteq' d) \text{ oraz }\\
+\forall c \forall d\; &(c\in C\land d\in C'\setminus C) \implies c\sqsubseteq' d \end{align} \right</math>
-Nagroda Knutha<br>[[Nagroda Knutha|Zobacz Nagroda Knutha]]
+<math>h(0, a) = f(a)</math> dla każdego <math>a \in A \\h(n', a) = g(h(n, a), n, a)</math> dla każdego <math>a \in A</math> i <math>n \in \mathbb{N}</math>
-<center><math>g(C)=\begincases C\cup \{f(C')\} C \endcases
-</math></center>
-<math> \displaystyle g(C)=\left\{\aligned C\cup \{f(C')\}\\C\endaligned \right</math>
+<math>e(0, a) = f(a)</math> dla każdego <math>a \in A \\ e(g(n, a), n, a)</math> dla każdego <math>a \in A</math> i <math>n \in m</math>

$p$	$q$	$r$	$(p \land q)$	$(p \land r)$	$(q \land \neg r)$	$(p \land r) \lor (q \land \neg r)$	$(p \land q) \Rightarrow ((p \land r) \lor (q \land \neg r))$
0	0	0	0	0	0	0	1
0	0	1	0	0	0	0	1
0	1	0	0	0	1	1	1
0	1	1	0	0	0	0	1
1	0	0	0	0	0	0	1
1	0	1	0	1	0	1	1
1	1	0	1	0	1	1	1
1	1	1	1	1	0	1	1

$p$	$q$	$r$	$(p \land q)$	$(p \land r)$	$(q \land \neg r)$	$(p \land r) \lor (q \land \neg r)$	$(p \land q) \Rightarrow ((p \land r) \lor (q \land \neg r))$
0	0	0	0	0	0	0	1
0	0	1	0	0	0	0	1
0	1	0	0	0	1	1	1
0	1	1	0	0	0	0	1
1	0	0	0	0	0	0	1
1	0	1	0	1	0	1	1
1	1	0	1	0	1	1	1
1	1	1	1	1	0	1	1

$p$	$q$	$r$	$(p \land q)$	$(p \land r)$	$(q \land \neg r)$	$(p \land r) \lor (q \land \neg r)$	$(p \land q) \Rightarrow ((p \land r) \lor (q \land \neg r))$
0	0	0	0	0	0	0	1
0	0	1	0	0	0	0	1
0	1	0	0	0	1	1	1
0	1	1	0	0	0	0	1
1	0	0	0	0	0	0	1
1	0	1	0	1	0	1	1
1	1	0	1	0	1	1	1
1	1	1	1	1	0	1	1