Rephrase q3
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@ -19,8 +19,8 @@
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@ -124,8 +124,8 @@
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@ -168,8 +168,8 @@
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@ -261,8 +261,8 @@
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@ -302,8 +302,8 @@
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@ -351,8 +351,8 @@
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@ -398,9 +398,9 @@
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"source": [
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"First it could be helpful to find the set of all valid games to compare against. This is a little tricky as first you need to know the rules of the game and encode them correctly. The rules that I have discerned are as follows:\n",
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"First, it could be helpful to find the set of all valid games to compare against. This is a little tricky as first you need to know the rules of the game and encode them correctly as a language. The rules that I have discerned are as follows:\n",
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"1. Each valid game has 3 a's or 3 b's, denoting a victory for one side\n",
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"2. Once it has either victories on either side, the game does not continue, and the string is complete\n",
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"2. Once either side has won, the game does not continue, and the string is complete\n",
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"\n",
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"$$L = \\{w \\in \\{a,b\\}^* | n_a(w) = 3 \\text{ or } n_b(w) = 3 \\text{ and } w \\text{ does not end with the opposite letter}\\}$$"
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]
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@ -412,8 +412,8 @@
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@ -455,7 +455,7 @@
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"source": [
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"(a) $L_1 = \\{w \\in \\{a,b\\}^* | n_a(w) = 3 \\text{ or } n_b(w) = 3 \\}$\n",
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"\n",
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"This one is not representable with code, as the set is produces is infinite. This language places no restrictions on the length of the string, so \"aaabbbbbbbbbbbbbbbbbbbbbbbbbbb\" is a valid string in this language. So here is a few examples of valid strings in this language that are not valid in the original language L but are in this language:\n",
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"This language is not representable with code, as the set is produces is infinite. This language places no restrictions on the length of the string, so \"aaabbbbbbbbbbbbbbbbbbbbbbbbbbb\" is a valid string in this language. So here is a few examples of valid strings in this language that are not valid in the original:\n",
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"\n",
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"1. \"aaabbbbbbbbbbbbbbbbbbbbbbbbbbb\"\n",
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"2. \"bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbaaa\"\n",
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@ -471,7 +471,7 @@
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"source": [
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"(b) $L_2 = \\{w \\in \\{a,b\\}^* | |w| \\leq 5, \\text{ and } n_a(w) = 3 \\text{ or } n_b(w) = 3 \\}$\n",
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"\n",
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"This language has the problem is not encoding the rule that the game stops once a player has 3 a's or 3 b's. This can be seen in these strings that are valid in this language but not in the original language L:"
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"This language differs as it is not encoding the rule that the game stops once a player has 3 a's or 3 b's. This can be seen in these strings that are valid in this language but not in the original language L:"
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{
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@ -481,8 +481,8 @@
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@ -514,7 +514,7 @@
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"(c) $L_3 = \\{w \\in \\{a,b\\}^* | (n_a(w) = 3 \\text{ and } n_b(w) < 3) \\text{ or } (n_b(w) = 3 \\text{ and } n_a(w) < 3) \\}$\n",
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"\n",
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"This language has a similar problem to the previous one, in that it does not encode the rule that the game stops once a player has 3 a's or 3 b's. The difference in this language is that the limit of string length 5 is not explicitly stated, however it is encoded to be that way by saying that if there are 3 a's then there must be less than 3 b's, and vice versa. This means that the effective maximum length of the string is 5, and thus the language is equivalent to the language from the previous question.\n",
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"This language is similar to the previous one, in that it does not encode the rule that the game stops once a player has 3 a's or 3 b's. The difference in this language is that the limit of string length 5 is not explicitly stated, however it is encoded to be that way by saying that if there are 3 a's then there must be less than 3 b's, and vice versa. This means that the effective maximum length of the string is 5, and thus the language is equivalent to the language from the previous question.\n",
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"$$\n",
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"x, y \\in \\mathbb{Z}^* \\\\\n",
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"x = 3 \\\\\n",
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@ -531,8 +531,8 @@
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