### Table 1: The multi-valued attributes for the Geographical scale

1995

"... In PAGE 6: ... Hierarchical scales represent, of course, hierarchical information; either in the single-inheritance case of a tree hierarchy, or in the multi-inheritance case. The geographical scale of server site addresses in Table1 and Table 2 is hierarchically scaled by level. 3.... ..."

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### Table 1. Example context with three multi-valued attributes

2001

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### Table 2: Results for the multi-valued feature system.

2000

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### Table 1. Example context with three multi-valued attributes

2001

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### Table 3: Definition of Addition for Multi-Valued Data Domain

1994

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### Table 3: Definition of Addition for Multi-Valued Data Domain

### Table 1. Results of multi-valued bi-decomposition for POLO benchmarks [13]

"... In PAGE 5: ... We tested our program on a 933Mhz Pentium III PC under Windows 2000 using multi-valued machine-leaning and data- mining benchmarks available from Portland Logic Optimization Group (POLO) [13]. Experimental results are listed in Table1 . Column Bmark gives the benchmark name.... ..."

### Table 1: Correct multi-valued simpli cation rules Rule Single-valued

"... In PAGE 2: ...s simplify(...,symbolic). Most standard simpli cation \rules quot;, which are in fact not true in the single-valued case, are true in the multi-valued sense. A partial list of such rules in given in Table1 , generally with a counter-example for the single-valued case. At this stage of the research, we do not have a complete list of these rules, which seems to be an under-researched area of mathematics.... In PAGE 3: ... EXAMPLES 5.1 Square roots: p1 zp1 + z ? = p1 z2 It is clearly true, from the fth rule in Table1 , that Sqrt(1 z) Sqrt(1 + z) = Sqrt(1 z2), so that p1 zp1 + z 2 Sqrt(1 z2). Hence all that remains to do is check the single-valued correctness.... In PAGE 4: ...Logarithms: log 1 z ? = log x The multivalued form of this is already the third rule of Table1 , so we know that log 1 z 2 Log(z) = f log(z) + 2n i j n 2 Zg, and all that remains is to check the single- valued correctness. We proceed as follows.... In PAGE 4: ... We will consider one example, rst over R2 then over C2 = R4. Table1 states Arctan(x) + Arctan(y) = Arctan x + y 1 xy : (5) To what extent is this valid as a single-valued equation, i.e.... In PAGE 5: ... an accuracy better than =4, and therefore test whether the two are equal to an accuracy better than =2, will do. If we were considering the very similar proposed simpli - cation x arctan(x) + x arctan(y) ? =xarctan x+y 1 xy , and we knew that xarctan(x) + xarctan(y) 2 xArctan x + y 1 xy (9) from the rules in Table1 , the procedure above for R1 would not quite have worked. The indeterminacy in the right-hand side of equation (9) is fxn j n 2 Zg, which is zero at the sample point x = y = 0.... In PAGE 6: ...2 Future Algorithmic Development The algorithm we have presented uses the following sub- algorithms, all of which could bene t from improvement. Simpli cation of multi-valued elementary functions, us- ing rules as in Table1 . This table could do with be- ing completed, and it has to be recognised that these 11We can make the absolute value as small as we like by choosing n such that n is su ciently close to an integer... In PAGE 7: ...example of Arcsin(x) Arcsin(x) after Table1 | and hence standard computer algebra systems may need substantial modi cation to handle this multi-valued simpli cation. Cylindrical algebraic decomposition [6].... ..."

### Table 1. Multi-valued multi-output (combinational) relation in tabular form

in Evolvable hardware or learning hardware? induction of state machines from temporal logic constraints

1999

"... In PAGE 5: ... Table1 . Rows correspond to objects (examples, samples) a,b,c and d and columns to input variables (attributes) x 1 and x 2 , and output variables y 1 and y 2 .... In PAGE 5: ... Extension of this language can be done by introducing variables that depend on discrete time. For instance, the example a (row a ) from Table1 can be rewritten to our language as follows: x 1 [0,2](t) amp; x 2 [1](t) = gt; y 2 [2](t) , because both input variables x 1 (t) , x 2 (t) and the output variable y 2 (t) are defined in the same moment of time. By allow ing previous or next ticks of time, for instance: x 1 [0,2](t-2) amp; x 2 [1](t-3) = gt; y 2 [2](t+3) , we can specify arbitrary regular grammars, regular expressions, sequential netlists, or state machines with multi-valued inputs and outputs.... ..."

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