시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
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문제

컴퓨터 프로그램에서는 논리식(logic expression)이 자주 나온다. 논리식의 요소들은 다음과 같다.

  • true 또는 false의 값을 가지고 있는 변수
  • 단항(unary), 이항(binary) 논리 연산자
  • 연산이 수행되는 순서에 영향을 주는 괄호

단항 연산자(Unary operators)는 하나의 변수에서 동작한다. 그에 반해 이항 연산자(Binary operators)는 두 개의 변수에서 동작한다. 일반적인 단항 논리 연산자에는 NOT이 있고, 이항 연산자에는 AND, OR, XOR, NAND, NOR 등이 있다.

하나의 논리 연산자는 ‘진리표(truth table)‘로 정의 될 수 있다. (단항 연산자는 1차원, 이항 연산자에는 2차원) 예를 들어, 아래의 그림을 보자.

논리식의 두 가지 예 :

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

이 프로그램의 목적을 위해, 하나의 논리식의 정확한 구조가 문법적으로 정의되어 있다:

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

(여기에서 수직 막대 ‘|’는 ‘or’로 발음되고 문법적으로 정의되어 있다; 이는 실제 표현에서는 표시되지 않는다.) <lowercase_letter>와 <uppercase_letter>은 각각 소문자, 대문자를 의미한다.

일부 경우에서는 모든 변수의 값이 정해지지 않더라도 논리식의 값을 구하는 것이 가능하다. 위의 예시 (1)에서 y = false 이고 x, z의 값은 모른다고 가정해 보아라. 그러면 변수 x, z의 값에 관계없이 주어진 논리식은 false가 되는 것을 확인할 수 있다. 다른 한 편으로는, 예시 (1)에서 x = true, y = true, z는 모른다고 가정해보자. 이때는 z의 값을 알지 못하면 논리식의 값을 정할 수 없다.

입력

입력은 하나 또는 그보다 많은 test cases를 포함한다. 각각의 test case에서, 입력의 첫 번째 줄에서는 두 개의 양의 정수(100을 초과하지 않음)를 입력 받는다. 하나는 그 case에서 사용될 단항 연산자의 개수이고, 다른 하나는 이항 연산자의 개수이다.

첫 째 줄 다음에는(아래의 두 단락에 설명 되어있는 것처럼) 연산자의 이름과 진리표 형태의 연산자 정의를 입력 받기 위한 여러 줄(several lines)이 따라올 것이다. 연산자의 이름은 20문자를 넘지 않도록 한다. 그리고 다른 연산자의 이름과 중복되지 않도록 한다.

첫째, 각각의 단항 연산자는 두 줄 안에 정의 되도록 한다. : 두 줄 중, 첫 번째 줄에는 연산자의 이름을 입력 받는다; 두 번 째 줄에는 true/false 항목을 두 번 입력 받는다. 이는 진리표를 정의하는 것이다.

단항 연산자의 정의를 마친 이후에, 각각의 이항 연산자는 세 줄 안에 정의 되도록 한다. : 세 줄 중 첫 번째 줄에는 연산자의 이름을 입력받는다; 두 번째, 세 번째 줄에는 true/false 항목을 각각 두 번 씩 입력 받는다. 이 두 줄은 이항 연산자의 진리표를 정의 하는 것이다.

연산자 표 이후에는 위의 문법을 만족하는 유효한 논리식을 입력 받는다. 변수들은 하나 이상의 공백으로 인접한 논리 연산자로부터 구분 될 수 있다. 괄호는 인접한 식의 원소들로부터 공백문자를 이용해 분리될 수도 있고 아닐 수도 있다. 당신은 하나의 식 안에서 한번 이상 나오는 변수가 없다고 가정할지도 모른다. 그 식은 최소 1에서 500을 넘지 않는 문자들로 구성되어 있다.

식 이후의 줄(lines, 0개 이상의 줄)에서는 변수의 값을 표로 입력 받는다. 각각의 줄은 두 가지 형태 중 하나를 갖는다. :

<variable> true
<variable> false

그 표에서 변수는 하나 이상 나타나지 않는다.

각 test case의 끝은 ‘*’로 표시된다. 두 개의 음의 정수를 입력하는 것으로 입력의 끝을 표시할 수 있다.

출력

각 test case의 출력을 한 줄씩 표시한다. case 번호를 표시한다. 그 이후에 문자 true, false, unknown 중 하나를 출력한다.; 위의 설명대로 주어진 식에 적합한 것이 출력됨.

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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(\ub2e8\ud56d \uc5f0\uc0b0\uc790\ub294 1\ucc28\uc6d0, \uc774\ud56d \uc5f0\uc0b0\uc790\uc5d0\ub294 2\ucc28\uc6d0) \uc608\ub97c \ub4e4\uc5b4, \uc544\ub798\uc758 \uadf8\ub9bc\uc744 \ubcf4\uc790.<\/p>\r\n\r\n<p><img src=\"https:\/\/onlinejudgeimages.s3-ap-northeast-1.amazonaws.com\/userupload\/oda\/20161103\/51c324fae368eb552f7e6068d156bc08.png\" \/><\/p>\r\n\r\n<p>\ub17c\ub9ac\uc2dd\uc758 \ub450 \uac00\uc9c0 \uc608 :<\/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 \ud504\ub85c\uadf8\ub7a8\uc758 \ubaa9\uc801\uc744 \uc704\ud574, \ud558\ub098\uc758 \ub17c\ub9ac\uc2dd\uc758 \uc815\ud655\ud55c \uad6c\uc870\uac00 \ubb38\ubc95\uc801\uc73c\ub85c \uc815\uc758\ub418\uc5b4 \uc788\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\uc5d0\uc11c \uc218\uc9c1 \ub9c9\ub300 &lsquo;|&rsquo;\ub294 &lsquo;or&rsquo;\ub85c \ubc1c\uc74c\ub418\uace0 \ubb38\ubc95\uc801\uc73c\ub85c \uc815\uc758\ub418\uc5b4 \uc788\ub2e4; \uc774\ub294 \uc2e4\uc81c \ud45c\ud604\uc5d0\uc11c\ub294 \ud45c\uc2dc\ub418\uc9c0 \uc54a\ub294\ub2e4.) &lt;lowercase_letter&gt;\uc640 &lt;uppercase_letter&gt;\uc740 \uac01\uac01 \uc18c\ubb38\uc790, \ub300\ubb38\uc790\ub97c \uc758\ubbf8\ud55c\ub2e4.<\/p>\r\n\r\n<p>\uc77c\ubd80 \uacbd\uc6b0\uc5d0\uc11c\ub294 \ubaa8\ub4e0 \ubcc0\uc218\uc758 \uac12\uc774 \uc815\ud574\uc9c0\uc9c0 \uc54a\ub354\ub77c\ub3c4 \ub17c\ub9ac\uc2dd\uc758 \uac12\uc744 \uad6c\ud558\ub294 \uac83\uc774 \uac00\ub2a5\ud558\ub2e4. \uc704\uc758 \uc608\uc2dc (1)\uc5d0\uc11c y = false \uc774\uace0 x, z\uc758 \uac12\uc740 \ubaa8\ub978\ub2e4\uace0 \uac00\uc815\ud574 \ubcf4\uc544\ub77c. \uadf8\ub7ec\uba74 \ubcc0\uc218 x, z\uc758 \uac12\uc5d0 \uad00\uacc4\uc5c6\uc774 \uc8fc\uc5b4\uc9c4 \ub17c\ub9ac\uc2dd\uc740&nbsp;false\uac00 \ub418\ub294 \uac83\uc744 \ud655\uc778\ud560 \uc218 \uc788\ub2e4. \ub2e4\ub978 \ud55c \ud3b8\uc73c\ub85c\ub294, \uc608\uc2dc (1)\uc5d0\uc11c x = true, y = true, z\ub294 \ubaa8\ub978\ub2e4\uace0 \uac00\uc815\ud574\ubcf4\uc790. \uc774\ub54c\ub294 z\uc758 \uac12\uc744 \uc54c\uc9c0 \ubabb\ud558\uba74 \ub17c\ub9ac\uc2dd\uc758 \uac12\uc744 \uc815\ud560 \uc218 \uc5c6\ub2e4.<\/p>\r\n","input":"<p>\uc785\ub825\uc740 \ud558\ub098 \ub610\ub294 \uadf8\ubcf4\ub2e4 \ub9ce\uc740 test cases\ub97c \ud3ec\ud568\ud55c\ub2e4. \uac01\uac01\uc758 test case\uc5d0\uc11c, \uc785\ub825\uc758 \uccab \ubc88\uc9f8 \uc904\uc5d0\uc11c\ub294 \ub450 \uac1c\uc758 \uc591\uc758 \uc815\uc218(100\uc744 \ucd08\uacfc\ud558\uc9c0 \uc54a\uc74c)\ub97c \uc785\ub825 \ubc1b\ub294\ub2e4. \ud558\ub098\ub294 \uadf8 case\uc5d0\uc11c \uc0ac\uc6a9\ub420 \ub2e8\ud56d \uc5f0\uc0b0\uc790\uc758 \uac1c\uc218\uc774\uace0, \ub2e4\ub978 \ud558\ub098\ub294 \uc774\ud56d \uc5f0\uc0b0\uc790\uc758 \uac1c\uc218\uc774\ub2e4.<\/p>\r\n\r\n<p>\uccab \uc9f8 \uc904 \ub2e4\uc74c\uc5d0\ub294(\uc544\ub798\uc758 \ub450 \ub2e8\ub77d\uc5d0 \uc124\uba85 \ub418\uc5b4\uc788\ub294 \uac83\ucc98\ub7fc)&nbsp;<strong>\uc5f0\uc0b0\uc790\uc758 \uc774\ub984<\/strong>\uacfc \uc9c4\ub9ac\ud45c \ud615\ud0dc\uc758 <strong>\uc5f0\uc0b0\uc790 \uc815\uc758<\/strong>\ub97c \uc785\ub825 \ubc1b\uae30 \uc704\ud55c \uc5ec\ub7ec \uc904(several lines)\uc774 \ub530\ub77c\uc62c&nbsp;\uac83\uc774\ub2e4. \uc5f0\uc0b0\uc790\uc758 \uc774\ub984\uc740 20\ubb38\uc790\ub97c \ub118\uc9c0 \uc54a\ub3c4\ub85d \ud55c\ub2e4. \uadf8\ub9ac\uace0 \ub2e4\ub978 \uc5f0\uc0b0\uc790\uc758 \uc774\ub984\uacfc \uc911\ubcf5\ub418\uc9c0 \uc54a\ub3c4\ub85d \ud55c\ub2e4.<\/p>\r\n\r\n<p>\uccab\uc9f8, \uac01\uac01\uc758 \ub2e8\ud56d \uc5f0\uc0b0\uc790\ub294&nbsp;\ub450 \uc904&nbsp;\uc548\uc5d0 \uc815\uc758 \ub418\ub3c4\ub85d \ud55c\ub2e4. :&nbsp;\ub450 \uc904 \uc911, \uccab \ubc88\uc9f8 \uc904\uc5d0\ub294 \uc5f0\uc0b0\uc790\uc758 \uc774\ub984\uc744 \uc785\ub825 \ubc1b\ub294\ub2e4;&nbsp;\ub450 \ubc88 \uc9f8 \uc904\uc5d0\ub294 true\/false \ud56d\ubaa9\uc744 \ub450 \ubc88 \uc785\ub825 \ubc1b\ub294\ub2e4. \uc774\ub294 \uc9c4\ub9ac\ud45c\ub97c \uc815\uc758\ud558\ub294 \uac83\uc774\ub2e4.<\/p>\r\n\r\n<p>\ub2e8\ud56d \uc5f0\uc0b0\uc790\uc758 \uc815\uc758\ub97c \ub9c8\uce5c \uc774\ud6c4\uc5d0, \uac01\uac01\uc758 \uc774\ud56d \uc5f0\uc0b0\uc790\ub294 \uc138 \uc904 \uc548\uc5d0 \uc815\uc758 \ub418\ub3c4\ub85d \ud55c\ub2e4. :&nbsp;\uc138 \uc904 \uc911 \uccab \ubc88\uc9f8 \uc904\uc5d0\ub294 \uc5f0\uc0b0\uc790\uc758 \uc774\ub984\uc744 \uc785\ub825\ubc1b\ub294\ub2e4;&nbsp;\ub450 \ubc88\uc9f8, \uc138 \ubc88\uc9f8 \uc904\uc5d0\ub294 true\/false \ud56d\ubaa9\uc744 \uac01\uac01 \ub450 \ubc88 \uc529 \uc785\ub825 \ubc1b\ub294\ub2e4. \uc774 \ub450 \uc904\uc740 \uc774\ud56d \uc5f0\uc0b0\uc790\uc758 \uc9c4\ub9ac\ud45c\ub97c \uc815\uc758 \ud558\ub294 \uac83\uc774\ub2e4.<\/p>\r\n\r\n<p>\uc5f0\uc0b0\uc790 \ud45c \uc774\ud6c4\uc5d0\ub294 \uc704\uc758 \ubb38\ubc95\uc744 \ub9cc\uc871\ud558\ub294 \uc720\ud6a8\ud55c \ub17c\ub9ac\uc2dd\uc744 \uc785\ub825 \ubc1b\ub294\ub2e4. \ubcc0\uc218\ub4e4\uc740 \ud558\ub098 \uc774\uc0c1\uc758&nbsp;\uacf5\ubc31\uc73c\ub85c \uc778\uc811\ud55c \ub17c\ub9ac \uc5f0\uc0b0\uc790\ub85c\ubd80\ud130 \uad6c\ubd84&nbsp;\ub420 \uc218 \uc788\ub2e4. \uad04\ud638\ub294 \uc778\uc811\ud55c \uc2dd\uc758 \uc6d0\uc18c\ub4e4\ub85c\ubd80\ud130 \uacf5\ubc31\ubb38\uc790\ub97c \uc774\uc6a9\ud574 \ubd84\ub9ac\ub420 \uc218\ub3c4 \uc788\uace0 \uc544\ub2d0 \uc218\ub3c4 \uc788\ub2e4. \ub2f9\uc2e0\uc740 \ud558\ub098\uc758 \uc2dd \uc548\uc5d0\uc11c \ud55c\ubc88 \uc774\uc0c1 \ub098\uc624\ub294 \ubcc0\uc218\uac00 \uc5c6\ub2e4\uace0 \uac00\uc815\ud560\uc9c0\ub3c4 \ubaa8\ub978\ub2e4. \uadf8 \uc2dd\uc740 \ucd5c\uc18c 1\uc5d0\uc11c 500\uc744 \ub118\uc9c0 \uc54a\ub294 \ubb38\uc790\ub4e4\ub85c \uad6c\uc131\ub418\uc5b4 \uc788\ub2e4.<\/p>\r\n\r\n<p>\uc2dd \uc774\ud6c4\uc758 \uc904(lines, 0\uac1c \uc774\uc0c1\uc758 \uc904)\uc5d0\uc11c\ub294 \ubcc0\uc218\uc758 \uac12\uc744 \ud45c\ub85c \uc785\ub825 \ubc1b\ub294\ub2e4. \uac01\uac01\uc758 \uc904\uc740 \ub450 \uac00\uc9c0 \ud615\ud0dc \uc911 \ud558\ub098\ub97c \uac16\ub294\ub2e4. :<\/p>\r\n\r\n<pre>\r\n&lt;variable&gt; true\r\n&lt;variable&gt; false<\/pre>\r\n\r\n<p>\uadf8 \ud45c\uc5d0\uc11c \ubcc0\uc218\ub294 \ud558\ub098 \uc774\uc0c1 \ub098\ud0c0\ub098\uc9c0 \uc54a\ub294\ub2e4.<\/p>\r\n\r\n<p>\uac01 test case\uc758 \ub05d\uc740 &lsquo;*&rsquo;\ub85c \ud45c\uc2dc\ub41c\ub2e4. \ub450 \uac1c\uc758 \uc74c\uc758 \uc815\uc218\ub97c \uc785\ub825\ud558\ub294 \uac83\uc73c\ub85c \uc785\ub825\uc758 \ub05d\uc744 \ud45c\uc2dc\ud560 \uc218 \uc788\ub2e4.<\/p>\r\n","output":"<p>\uac01 test case\uc758 \ucd9c\ub825\uc744 \ud55c \uc904\uc529 \ud45c\uc2dc\ud55c\ub2e4. case \ubc88\ud638\ub97c \ud45c\uc2dc\ud55c\ub2e4. \uadf8 \uc774\ud6c4\uc5d0 \ubb38\uc790 <strong>true, false, unknown<\/strong>&nbsp;\uc911 \ud558\ub098\ub97c \ucd9c\ub825\ud55c\ub2e4.; \uc704\uc758 \uc124\uba85\ub300\ub85c \uc8fc\uc5b4\uc9c4 \uc2dd\uc5d0 \uc801\ud569\ud55c \uac83\uc774 \ucd9c\ub825\ub428.<\/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"}]