시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
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문제

논리식은 컴퓨터 프로그램에 자주 사용된다. 논리식은 다음으로 구성된다.

  • true 또는 false의 값을 가지는 변수
  • 단항 및 이항 논리 연산자
  • 연산의 순서에 영향을 줄 수 있는 괄호

단항 연산자는 하나의 변수에 대해 동작하고, 이항 연산자는 두 개의 변수에 대해 동작한다. 자주 쓰이는 단항 연산자로 NOT이 있고, 자주 쓰이는 이항 연산자로 AND, OR, XOR, NAND, NOR 등이 있다.

논리 연산자는 '진리표'로써 정의할 수 있다. 이 진리표는 단항 연산자의 경우 1차원, 이항 연산자의 경우 2차원이다. 예를 들어, 다음 그림을 보자.

위쪽 여백과 왼쪽 여백 모두 첫 번째 피연산자가 false이고, 두 번째 피연산자가 true임에 주목하자.

논리식의 예시는 다음과 같다.

  1. (x AND (NOT(y NAND z)))
  2. (x OR ((NOT y) XOR z))

이 문제에서 논리식의 정확한 구조는 다음 문법으로 정의된다.

<expression> = <variable> | ( <expression> <operator> <expression> ) | ( <operator> <expression>)
<variable> = <lowercase_letter>
<operator> = <uppercase_letter> | <operator> <uppercase_letter>

(여기서 수직 막대 '|'는 'or'라고 발음하며, 문법을 정의할 때 사용하는 글자이다. 실제 식에서는 나타나지 않는다.) <lowercase_letter>와 <uppercase_letter>는 알파벳 대소문자이다.

어떤 경우에서는 모든 변수의 값이 정해지지 않아도 논리식을 계산할 수 있다. 위의 첫 번째 예시에서, y는 false이고, x와 z의 값은 모른다고 해 보자. 그러면 나머지 변수의 값이 무엇이든 해당 논리식은 항상 false로 계산됨을 알 수 있다. 반면, 같은 예시에서, x = true, y = true이고, z의 값은 모른다고 해보자. 그러면 z의 값을 알지 않고서는 논리식의 값을 결정할 수 없다.

입력

입력은 하나 이상의 테스트 케이스로 이루어져 있다. 각 테스트 케이스마다 첫 줄에 100 이하의 음이 아닌 정수가 2개 주어지며, 각각 그 케이스에서 사용할 단항 연산자와 이항 연산자의 종류이다.

그 다음에는 아래 두 문단에서 설명하는 형식으로 각 연산자의 이름과 진리표 형태의 연산자 정의가 여러 줄에 걸쳐 주어진다. 모든 연산자의 이름은 20글자를 넘지 않으며, 서로 다르다.

우선 단항 연산자를 각각 2줄씩에 걸쳐 정의한다. 첫째 줄에는 연산자의 이름이 주어지고, 둘째 줄에는 두 개의 참/거짓 값이 주어지며, 이것이 그 연산자의 진리표가 된다. 피연산자의 순서는 위 그림에 주어진 것과 같이 false, true이다.

단항 연산자 다음에는 이항 연산자를 각각 3줄씩에 걸쳐 정의한다. 첫째 줄에는 연산자의 이름이 주어지고, 다음 두 줄에는 두 개씩의 참/거짓 값이 주어지며, 이것이 그 연산자의 진리표가 된다. 왼쪽 피연산자는 위 그림의 왼쪽 여백에, 오른쪽 피연산자는 위쪽 여백에 해당한다.

연산자를 모두 정의한 다음에는 위의 문법을 만족하는 유효한 논리식을 한 줄에 입력받는다. 변수와 연산자는 하나 이상의 띄어쓰기로 구분되어 있으며, 괄호로 묶인 식과 연산자 사이에는 띄어쓰기가 있을 수도 없을 수도 있다. 논리식에는 같은 변수가 두 번 이상 등장하지 않음이 보장된다. 논리식의 길이는 1 이상 500 이하이다.

논리식 다음에는 0개 이상의 변수의 값이 각각 한 줄로 주어진다. 각 줄은 아래 둘 중 하나의 형태이다.

<variable> true
<variable> false

같은 변수는 두 번 이상 주어지지 않는다.

각 테스트 케이스의 끝에는 별표(*) 하나가 주어지며, 입력의 끝에는 음의 정수 두 개가 한 줄로 주어진다.

출력

각 테스트 케이스마다 한 줄씩 케이스 번호를 표시하고, 그 뒤에 truefalseunknown 중 주어진 논리식에 알맞은 것을 문제에서 설명한 대로 출력한다.

출력 포맷은 예제 출력을 통해 확인할 수 있다.

예제 입력 1

1 2
NOT
true false
AND
false false
false true
TWEEK
true false
true false
(x AND (NOT(y TWEEK z)))
x true
y true
*
1 1
MOCK
true true
NAND
true true
true false
(x NAND (MOCK (y NAND z)))
x false
y false
*
0 2
XOR
false true
true false
FAKE
true true
false false
((p XOR q) FAKE r)
p true
q false
*
-1 -1

예제 출력 1

Case 1: unknown
Case 2: true
Case 3: false
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XOR, NAND, NOR \ub4f1\uc774 \uc788\ub2e4.<\/p>\r\n\r\n<p>\ub17c\ub9ac \uc5f0\uc0b0\uc790\ub294 &#39;\uc9c4\ub9ac\ud45c&#39;\ub85c\uc368 \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \uc774 \uc9c4\ub9ac\ud45c\ub294 \ub2e8\ud56d \uc5f0\uc0b0\uc790\uc758 \uacbd\uc6b0 1\ucc28\uc6d0, \uc774\ud56d \uc5f0\uc0b0\uc790\uc758 \uacbd\uc6b0 2\ucc28\uc6d0\uc774\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \ub2e4\uc74c \uadf8\ub9bc\uc744 \ubcf4\uc790.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/0b4a9a58-7f6a-47ca-9568-522930c5b45b\/-\/preview\/\" \/><\/p>\r\n\r\n<p>\uc704\ucabd \uc5ec\ubc31\uacfc \uc67c\ucabd \uc5ec\ubc31 \ubaa8\ub450&nbsp;\uccab \ubc88\uc9f8 \ud53c\uc5f0\uc0b0\uc790\uac00 false\uc774\uace0, \ub450 \ubc88\uc9f8 \ud53c\uc5f0\uc0b0\uc790\uac00 true\uc784\uc5d0 \uc8fc\ubaa9\ud558\uc790.<\/p>\r\n\r\n<p>\ub17c\ub9ac\uc2dd\uc758 \uc608\uc2dc\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\r\n\r\n<ol>\r\n\t<li>(x AND (NOT(y NAND z)))<\/li>\r\n\t<li>(x OR ((NOT y) XOR z))<\/li>\r\n<\/ol>\r\n\r\n<p>\uc774 \ubb38\uc81c\uc5d0\uc11c \ub17c\ub9ac\uc2dd\uc758 \uc815\ud655\ud55c \uad6c\uc870\ub294 \ub2e4\uc74c \ubb38\ubc95\uc73c\ub85c \uc815\uc758\ub41c\ub2e4.<\/p>\r\n\r\n<pre>\r\n&lt;expression&gt; = &lt;variable&gt; | ( &lt;expression&gt; &lt;operator&gt; &lt;expression&gt; ) | ( &lt;operator&gt; &lt;expression&gt;)\r\n&lt;variable&gt; = &lt;lowercase_letter&gt;\r\n&lt;operator&gt; = &lt;uppercase_letter&gt; | &lt;operator&gt; &lt;uppercase_letter&gt;<\/pre>\r\n\r\n<p>(\uc5ec\uae30\uc11c \uc218\uc9c1 \ub9c9\ub300 &#39;|&#39;\ub294 &#39;or&#39;\ub77c\uace0 \ubc1c\uc74c\ud558\uba70, \ubb38\ubc95\uc744 \uc815\uc758\ud560 \ub54c \uc0ac\uc6a9\ud558\ub294 \uae00\uc790\uc774\ub2e4. \uc2e4\uc81c \uc2dd\uc5d0\uc11c\ub294 \ub098\ud0c0\ub098\uc9c0 \uc54a\ub294\ub2e4.) &lt;lowercase_letter&gt;\uc640 &lt;uppercase_letter&gt;\ub294 \uc54c\ud30c\ubcb3 \ub300\uc18c\ubb38\uc790\uc774\ub2e4.<\/p>\r\n\r\n<p>\uc5b4\ub5a4 \uacbd\uc6b0\uc5d0\uc11c\ub294 \ubaa8\ub4e0 \ubcc0\uc218\uc758<em> <\/em>\uac12\uc774 \uc815\ud574\uc9c0\uc9c0 \uc54a\uc544\ub3c4 \ub17c\ub9ac\uc2dd\uc744 \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4. \uc704\uc758 \uccab \ubc88\uc9f8 \uc608\uc2dc\uc5d0\uc11c, y\ub294 false\uc774\uace0, x\uc640 z\uc758 \uac12\uc740 \ubaa8\ub978\ub2e4\uace0 \ud574 \ubcf4\uc790. \uadf8\ub7ec\uba74 \ub098\uba38\uc9c0 \ubcc0\uc218\uc758 \uac12\uc774 \ubb34\uc5c7\uc774\ub4e0 \ud574\ub2f9 \ub17c\ub9ac\uc2dd\uc740 \ud56d\uc0c1 false\ub85c \uacc4\uc0b0\ub428\uc744 \uc54c \uc218 \uc788\ub2e4. \ubc18\uba74, \uac19\uc740 \uc608\uc2dc\uc5d0\uc11c, x = true, y = true\uc774\uace0, z\uc758 \uac12\uc740 \ubaa8\ub978\ub2e4\uace0 \ud574\ubcf4\uc790. \uadf8\ub7ec\uba74 z\uc758 \uac12\uc744 \uc54c\uc9c0 \uc54a\uace0\uc11c\ub294 \ub17c\ub9ac\uc2dd\uc758 \uac12\uc744 \uacb0\uc815\ud560 \uc218 \uc5c6\ub2e4.<\/p>\r\n","input":"<p>\uc785\ub825\uc740 \ud558\ub098 \uc774\uc0c1\uc758 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4. \uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub9c8\ub2e4 \uccab \uc904\uc5d0 100 \uc774\ud558\uc758 \uc74c\uc774 \uc544\ub2cc \uc815\uc218\uac00 2\uac1c \uc8fc\uc5b4\uc9c0\uba70, \uac01\uac01 \uadf8 \ucf00\uc774\uc2a4\uc5d0\uc11c \uc0ac\uc6a9\ud560 \ub2e8\ud56d \uc5f0\uc0b0\uc790\uc640 \uc774\ud56d \uc5f0\uc0b0\uc790\uc758 \uc885\ub958\uc774\ub2e4.<\/p>\r\n\r\n<p>\uadf8 \ub2e4\uc74c\uc5d0\ub294 \uc544\ub798 \ub450 \ubb38\ub2e8\uc5d0\uc11c \uc124\uba85\ud558\ub294 \ud615\uc2dd\uc73c\ub85c \uac01&nbsp;<strong>\uc5f0\uc0b0\uc790\uc758 \uc774\ub984<\/strong>\uacfc \uc9c4\ub9ac\ud45c \ud615\ud0dc\uc758&nbsp;<strong>\uc5f0\uc0b0\uc790 \uc815\uc758<\/strong>\uac00 \uc5ec\ub7ec \uc904\uc5d0 \uac78\uccd0 \uc8fc\uc5b4\uc9c4\ub2e4. \ubaa8\ub4e0 \uc5f0\uc0b0\uc790\uc758 \uc774\ub984\uc740 20\uae00\uc790\ub97c \ub118\uc9c0 \uc54a\uc73c\uba70, \uc11c\ub85c \ub2e4\ub974\ub2e4.<\/p>\r\n\r\n<p>\uc6b0\uc120 \ub2e8\ud56d \uc5f0\uc0b0\uc790\ub97c \uac01\uac01 2\uc904\uc529\uc5d0 \uac78\uccd0 \uc815\uc758\ud55c\ub2e4. \uccab\uc9f8 \uc904\uc5d0\ub294 \uc5f0\uc0b0\uc790\uc758 \uc774\ub984\uc774 \uc8fc\uc5b4\uc9c0\uace0, \ub458\uc9f8 \uc904\uc5d0\ub294 \ub450 \uac1c\uc758 \ucc38\/\uac70\uc9d3 \uac12\uc774 \uc8fc\uc5b4\uc9c0\uba70, \uc774\uac83\uc774 \uadf8 \uc5f0\uc0b0\uc790\uc758 \uc9c4\ub9ac\ud45c\uac00 \ub41c\ub2e4. \ud53c\uc5f0\uc0b0\uc790\uc758 \uc21c\uc11c\ub294 \uc704 \uadf8\ub9bc\uc5d0 \uc8fc\uc5b4\uc9c4 \uac83\uacfc \uac19\uc774 false, true\uc774\ub2e4.<\/p>\r\n\r\n<p>\ub2e8\ud56d \uc5f0\uc0b0\uc790 \ub2e4\uc74c\uc5d0\ub294 \uc774\ud56d \uc5f0\uc0b0\uc790\ub97c \uac01\uac01 3\uc904\uc529\uc5d0 \uac78\uccd0 \uc815\uc758\ud55c\ub2e4. \uccab\uc9f8 \uc904\uc5d0\ub294 \uc5f0\uc0b0\uc790\uc758 \uc774\ub984\uc774 \uc8fc\uc5b4\uc9c0\uace0, \ub2e4\uc74c \ub450 \uc904\uc5d0\ub294 \ub450 \uac1c\uc529\uc758 \ucc38\/\uac70\uc9d3 \uac12\uc774 \uc8fc\uc5b4\uc9c0\uba70, \uc774\uac83\uc774 \uadf8 \uc5f0\uc0b0\uc790\uc758 \uc9c4\ub9ac\ud45c\uac00 \ub41c\ub2e4. \uc67c\ucabd \ud53c\uc5f0\uc0b0\uc790\ub294 \uc704 \uadf8\ub9bc\uc758 \uc67c\ucabd \uc5ec\ubc31\uc5d0, \uc624\ub978\ucabd \ud53c\uc5f0\uc0b0\uc790\ub294 \uc704\ucabd \uc5ec\ubc31\uc5d0 \ud574\ub2f9\ud55c\ub2e4.<\/p>\r\n\r\n<p>\uc5f0\uc0b0\uc790\ub97c \ubaa8\ub450 \uc815\uc758\ud55c \ub2e4\uc74c\uc5d0\ub294 \uc704\uc758 \ubb38\ubc95\uc744 \ub9cc\uc871\ud558\ub294 \uc720\ud6a8\ud55c \ub17c\ub9ac\uc2dd\uc744 \ud55c \uc904\uc5d0 \uc785\ub825\ubc1b\ub294\ub2e4. \ubcc0\uc218\uc640 \uc5f0\uc0b0\uc790\ub294 \ud558\ub098 \uc774\uc0c1\uc758 \ub744\uc5b4\uc4f0\uae30\ub85c \uad6c\ubd84\ub418\uc5b4 \uc788\uc73c\uba70, \uad04\ud638\ub85c \ubb36\uc778 \uc2dd\uacfc \uc5f0\uc0b0\uc790 \uc0ac\uc774\uc5d0\ub294 \ub744\uc5b4\uc4f0\uae30\uac00 \uc788\uc744 \uc218\ub3c4 \uc5c6\uc744 \uc218\ub3c4 \uc788\ub2e4. \ub17c\ub9ac\uc2dd\uc5d0\ub294 \uac19\uc740 \ubcc0\uc218\uac00 \ub450 \ubc88 \uc774\uc0c1 \ub4f1\uc7a5\ud558\uc9c0 \uc54a\uc74c\uc774 \ubcf4\uc7a5\ub41c\ub2e4. \ub17c\ub9ac\uc2dd\uc758 \uae38\uc774\ub294 1 \uc774\uc0c1 500 \uc774\ud558\uc774\ub2e4.<\/p>\r\n\r\n<p>\ub17c\ub9ac\uc2dd \ub2e4\uc74c\uc5d0\ub294 0\uac1c \uc774\uc0c1\uc758 \ubcc0\uc218\uc758 \uac12\uc774 \uac01\uac01 \ud55c \uc904\ub85c \uc8fc\uc5b4\uc9c4\ub2e4. \uac01 \uc904\uc740 \uc544\ub798 \ub458 \uc911 \ud558\ub098\uc758 \ud615\ud0dc\uc774\ub2e4.<\/p>\r\n\r\n<pre>\r\n&lt;variable&gt; true\r\n&lt;variable&gt; false\r\n<\/pre>\r\n\r\n<p>\uac19\uc740 \ubcc0\uc218\ub294 \ub450 \ubc88 \uc774\uc0c1 \uc8fc\uc5b4\uc9c0\uc9c0 \uc54a\ub294\ub2e4.<\/p>\r\n\r\n<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \ub05d\uc5d0\ub294 \ubcc4\ud45c(*) \ud558\ub098\uac00 \uc8fc\uc5b4\uc9c0\uba70, \uc785\ub825\uc758 \ub05d\uc5d0\ub294 \uc74c\uc758 \uc815\uc218 \ub450 \uac1c\uac00 \ud55c \uc904\ub85c \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub9c8\ub2e4 \ud55c \uc904\uc529 \ucf00\uc774\uc2a4 \ubc88\ud638\ub97c \ud45c\uc2dc\ud558\uace0, \uadf8 \ub4a4\uc5d0&nbsp;<strong>true<\/strong>,&nbsp;<strong>false<\/strong>,&nbsp;<strong>unknown<\/strong>&nbsp;\uc911 \uc8fc\uc5b4\uc9c4 \ub17c\ub9ac\uc2dd\uc5d0 \uc54c\ub9de\uc740 \uac83\uc744 \ubb38\uc81c\uc5d0\uc11c \uc124\uba85\ud55c \ub300\ub85c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>\ucd9c\ub825 \ud3ec\ub9f7\uc740 \uc608\uc81c \ucd9c\ub825\uc744 \ud1b5\ud574 \ud655\uc778\ud560 \uc218 \uc788\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"1901","problem_lang":"1","title":"Evaluating Logic Expressions","description":"<p><img alt=\"\" src=\"https:\/\/onlinejudgeimages.s3-ap-northeast-1.amazonaws.com\/problem\/1901\/1.gif\" style=\"float:right; height:122px; width:247px\" \/>Logic expressions occur frequently in computer programs. The elements of logic expressions are:<\/p>\r\n\r\n<ul>\r\n\t<li>variables, which may have the values true or false<\/li>\r\n\t<li>unary and binary logic operators<\/li>\r\n\t<li>parentheses which may affect the order in which the operations are carried out<\/li>\r\n<\/ul>\r\n\r\n<p>Unary operators operate on one variable, whereas binary operators operate on two variables. A common unary logic operator is NOT; some common binary logic operators are AND, OR, XOR, NAND and NOR.<\/p>\r\n\r\n<p>A logic operator can be defined by a &#39;truth table&#39; (one-dimensional for unary operators, two-dimensional for binary operators). For examples, see the diagram.<\/p>\r\n\r\n<p><img alt=\"\" src=\"https:\/\/onlinejudgeimages.s3-ap-northeast-1.amazonaws.com\/problem\/1901\/2.jpg\" \/><\/p>\r\n\r\n<p>Note that false comes before true in the header across the top (for unary or the right operand of binary operators), as well as on the left margin (for the left operand of binary operators).<\/p>\r\n\r\n<p>Two examples of logic expressions are:<\/p>\r\n\r\n<ol>\r\n\t<li>(x AND (NOT(y NAND z)))<\/li>\r\n\t<li>(x OR ((NOT y) XOR z))<\/li>\r\n<\/ol>\r\n\r\n<p>For the purposes of this problem, the precise structure of a logic expression is defined by the grammar:<\/p>\r\n\r\n<pre>\r\n&lt;expression&gt; = &lt;variable&gt; | ( &lt;expression&gt; &lt;operator&gt; &lt;expression&gt; ) | ( &lt;operator&gt; &lt;expression&gt;)\r\n&lt;variable&gt; = &lt;lowercase_letter&gt;\r\n&lt;operator&gt; = &lt;uppercase_letter&gt; | &lt;operator&gt; &lt;uppercase_letter&gt;<\/pre>\r\n\r\n<p>(Here the vertical bar &#39;|&#39; is pronounced &#39;or&#39; and is used to define the grammar; it does not actually show up in the expression.) &lt;lowercase_letter&gt; and &lt;uppercase_letter&gt; have their usual meanings.<\/p>\r\n\r\n<p>In some cases, it is possible to evaluate a logic expression even when not all of the variables have been assigned values. Consider Example (1) above, and suppose that y = false, but the values of x and z are not known. It can be seen that the given expression evaluates to false regardless of the values of the unassigned variables. On the other hand, suppose that, in the same Example (1), x = true, y = true, and z is unknown. it is not possible to determine the value of the expression without knowing the value of z.<\/p>\r\n","input":"<p>The input will contain data for one or more test cases. For each test case, the first line of input will contain two non-negative integers (not exceeding 100), the number of unary operators and the number of binary operators to be considered for that case. The first line will be followed by several lines that will name each operator and define it in the form of a truth table, as described in the next two paragraphs. The names of the operators do not exceed 20 characters and are unique for each operator.<\/p>\r\n\r\n<p>First, each unary operator will be defined in two lines of input: the first of these two lines will contain the name of the operator; the second line will contain two true \/ false entries that define the table for the unary operator, without the implied column headers across the top.<\/p>\r\n\r\n<p>After the unary operators have been defined, each binary operator will be defined in three lines of input: the first of these three lines will contain the name of the operator; the second and third lines will each contain two true \/ false entries. These two lines define the table for the binary operator, without the implied column headers across the top and row headers on the left margin.<\/p>\r\n\r\n<p>The operator tables will be followed by a line containing a valid logic expression satisfying the above grammar. Variables will be separated from adjacent logic operators by one or more blank spaces. Parentheses may or may not be separated from adjacent elements of the expression by blank spaces. You may assume that no variable will occur more than once in an expression. The expression will consist of at least 1 but not more than 500 characters.<\/p>\r\n\r\n<p>The expression will be followed by zero or more lines that comprise a table of values. Each of these lines will have one of the two forms:<\/p>\r\n\r\n<pre>\r\n&lt;variable&gt; true\r\n&lt;variable&gt; false<\/pre>\r\n\r\n<p>No variable will appear more than once in the table.<\/p>\r\n\r\n<p>The end of each test case will be marked by a line containing a single asterisk. End of input will be marked by a line containing two negative integers.<\/p>\r\n","output":"<p>For each test case there will be one line of output. The case number will be printed in the format of the sample output.<\/p>\r\n\r\n<p>It will be followed by one of the words: true, false, or unknown; whichever is appropriate for the given expression, as explained above.<\/p>\r\n\r\n<p>The output format is illustrated in the sample output.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]