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This is precisely the Nonogram-type description a1, a2, ...,a, for a line (row or column). Note that it has length 2r +1 and can also be written as oʻ1ª*o+1®°0+ ...1ªo*. We denote the set of all Nonogram-type descriptions by Daonogran C D. In the sequel we will concentrate on this type of description. Let s be a finite string over I. If zero or more occurrences of x are replaced with elements from E, the resulting string is called a specification of s. A specification to a string over E (i.e., no longer containing any "x" symbols) is called a fir. If a string s has a fix that adheres to a given description d, s is called firable with respect to d. By definition, the boolean function Fir(s,d)

This is precisely the Nonogram-type description a1, a2, ...,a, for a line (row or column). Note that it has length 2r +1 and can also be written as oʻ1ª*o+1®°0+ ...1ªo*. We denote the set of all Nonogram-type descriptions by Daonogran C D. In the sequel we will concentrate on this type of description. Let s be a finite string over I. If zero or more occurrences of x are replaced with elements from E, the resulting string is called a specification of s. A specification to a string over E (i.e., no longer containing any "x" symbols) is called a fir. If a string s has a fix that adheres to a given description d, s is called firable with respect to d. By definition, the boolean function Fir(s,d)

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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Pls. repharse this notations and concept.Thank you.

This is precisely the Nonogram-type description a1, a2, . .., a, for a line (row or column). Note that it has length 2r + 1 and can also be written as 0*1ª o+1®°0+ ...1ªro*. We denote the set of all Nonogram-type descriptions by Dnonogram C D. In the sequel we will concentrate on this type of description. Let s be a finite string over r. If zero or more occurrences of x are replaced with elements from E, the resulting string is called a specification of s. A specification to a string over E (i.e., no longer containing any “x" symbols) is called a fir. If a string s has a fix that adheres to a given description d, s is called fixable with respect to d. By definition, the boolean function Fiæ(s, d) is true if and only if s is fixable with respect to d. In a somewhat different context, we also use the term fixing a pixel to indicate that a pixel has only one possible value, and can therefore be assigned that value. An m x n Nonogram description N consists of m > 0 row descriptions r1, r2, ...,"m € Dnonogram and n > 0 column descriptions c, C2, . .. , Cn E Dnonogram- A partial filling is an m x n matrix over I. The set of all partial fillings is de- noted by Imxn; its elements can also be considered as strings of length m x n. If a partial filling contains no occurrences of x, it is called a full fix. A full fix F€ Emxn adheres to the Nonogram description N if the ith row of F adheres to r; (for all i = 1,2, ..., m) and the jth column of F adheres to c; (for all j = 1,2, ..., n). We generalize the concepts of specification and fixable that defined for single lines in the natural way to mxn Nonograms. |were
Transcribed Image Text:This is precisely the Nonogram-type description a1, a2, . .., a, for a line (row or column). Note that it has length 2r + 1 and can also be written as 0*1ª o+1®°0+ ...1ªro*. We denote the set of all Nonogram-type descriptions by Dnonogram C D. In the sequel we will concentrate on this type of description. Let s be a finite string over r. If zero or more occurrences of x are replaced with elements from E, the resulting string is called a specification of s. A specification to a string over E (i.e., no longer containing any “x" symbols) is called a fir. If a string s has a fix that adheres to a given description d, s is called fixable with respect to d. By definition, the boolean function Fiæ(s, d) is true if and only if s is fixable with respect to d. In a somewhat different context, we also use the term fixing a pixel to indicate that a pixel has only one possible value, and can therefore be assigned that value. An m x n Nonogram description N consists of m > 0 row descriptions r1, r2, ...,"m € Dnonogram and n > 0 column descriptions c, C2, . .. , Cn E Dnonogram- A partial filling is an m x n matrix over I. The set of all partial fillings is de- noted by Imxn; its elements can also be considered as strings of length m x n. If a partial filling contains no occurrences of x, it is called a full fix. A full fix F€ Emxn adheres to the Nonogram description N if the ith row of F adheres to r; (for all i = 1,2, ..., m) and the jth column of F adheres to c; (for all j = 1,2, ..., n). We generalize the concepts of specification and fixable that defined for single lines in the natural way to mxn Nonograms. |were
We now define notation for a single line (i.e., row or column) of a Nonogram. After that, we combine these into rectangular puzzles. Let E be a finite alphabet. Its elements are referred to as pixel values. In this paper we focus on the case E = {0, 1}, but most concepts apply to sets consisting of more than two elements as well. The symbols 0 and 1 represent the white (0) and black (1) pixels in the puzzle. In addition, we introduce a special symbol, x ¢ E, indicating that a pixel is not decided yet. Put I' = EU {x}. For l > 0, let E' (resp. I*) denote the set of all strings over E (resp. T) of length l. For describing a Nonogram, we introduce more general concepts of row and column descriptions, such that Nonograms are in fact a special case. Most of the concepts in this paper can be applied to all logic problems that follow the more general definitions. A description d of length k > 0 is an ordered series (d, d2, ..., dr) with d; = o;{a,, b;}, where o; e £ and a;, b, € {0, 1, 2, ...} with a; < b; (j = 1,2, ...,k). The curly braces are used here in order to stick to the conventions from regular expressions; so, in o;{a;, b;} they do not refer to a set, but to an ordered pair. Any such d, will correspond with between a, and b; characters o,, as defined below. Without loss of generality we will assume that consecutive characters o; differ, so o; # Oj+1 for j = 1,2, ..., k – 1. Let Dr denote the (infinite) set of all descriptions of length k, and put D = U,Dk, where Do consists of the empty description e. A single d; = o;{a;,b;} is called a segment description. We will sometimes write o* as a shortcut for o{0, } (for o e E) and o+ as a shortcut for o{1,00}, where ∞ is suitably large number. We use oª as a shortcut for o{a, a} (a € {0, 1, 2, ...}), and we sometimes omit parentheses and commas; also oº is omitted. A finite string s over E adheres to a description d (as defined above) if s = of o..o, where a; < c; < b; for j = 1,.,k. As an example, consider the following description for E = {0, 1}: d = (0{0, 0}, 1{a1, a1}, 0{1, 0}, 1{a2, a2}, 0{1, 0}, ..., 1{a,, a, }, 0{0, ∞}).
Transcribed Image Text:We now define notation for a single line (i.e., row or column) of a Nonogram. After that, we combine these into rectangular puzzles. Let E be a finite alphabet. Its elements are referred to as pixel values. In this paper we focus on the case E = {0, 1}, but most concepts apply to sets consisting of more than two elements as well. The symbols 0 and 1 represent the white (0) and black (1) pixels in the puzzle. In addition, we introduce a special symbol, x ¢ E, indicating that a pixel is not decided yet. Put I' = EU {x}. For l > 0, let E' (resp. I*) denote the set of all strings over E (resp. T) of length l. For describing a Nonogram, we introduce more general concepts of row and column descriptions, such that Nonograms are in fact a special case. Most of the concepts in this paper can be applied to all logic problems that follow the more general definitions. A description d of length k > 0 is an ordered series (d, d2, ..., dr) with d; = o;{a,, b;}, where o; e £ and a;, b, € {0, 1, 2, ...} with a; < b; (j = 1,2, ...,k). The curly braces are used here in order to stick to the conventions from regular expressions; so, in o;{a;, b;} they do not refer to a set, but to an ordered pair. Any such d, will correspond with between a, and b; characters o,, as defined below. Without loss of generality we will assume that consecutive characters o; differ, so o; # Oj+1 for j = 1,2, ..., k – 1. Let Dr denote the (infinite) set of all descriptions of length k, and put D = U,Dk, where Do consists of the empty description e. A single d; = o;{a;,b;} is called a segment description. We will sometimes write o* as a shortcut for o{0, } (for o e E) and o+ as a shortcut for o{1,00}, where ∞ is suitably large number. We use oª as a shortcut for o{a, a} (a € {0, 1, 2, ...}), and we sometimes omit parentheses and commas; also oº is omitted. A finite string s over E adheres to a description d (as defined above) if s = of o..o, where a; < c; < b; for j = 1,.,k. As an example, consider the following description for E = {0, 1}: d = (0{0, 0}, 1{a1, a1}, 0{1, 0}, 1{a2, a2}, 0{1, 0}, ..., 1{a,, a, }, 0{0, ∞}).
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