Pls. repharse this notations and concept.Thank you.
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 "/>
Extracted 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
Extracted 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, ∞}). 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,="">