1. Denotational Semantics
Based on recursive function theory
Define for each language entity both a mathematical object and a function that maps instances of that entity onto instances of the mathematical object.
The method is named denotational because the mathematical objects denote the meaning of their corresponding syntactic entities.
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2. Denotational Semantics (continued)
Two simple examples:
Binary numbers
<bin_num> -> 0
| 1
| <bin_num> 0
| <bin_num> 1
<bin_num>
<bin_num>
<bin_num> 1
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3. Denotational Semantics (continued)
The objects are simple decimal numbers. The
meaning of a binary number will be its decimal
equivalent.
Let the domain of semantic values of the objects be N, the set of nonnegative decimal integer values. It is these objects that we wish to associate with binary numbers.
The semantic function, Mbin maps the syntactic objects to the objects in N.
Mbin(‘0’) = 0
Mbin(‘1’) = 1
Mbin(<bin_num>’0’) = 2 * Mbin(<bin_num>)
Mbin(<bin_num>’1’) = 2 * Mbin(<bin_num>) + 1
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4. Denotational Semantics (continued)
6 <bin_num>
3 <bin_num>
1<bin_num> 1
Mbin(<bin_num>’0’) = 2 * Mbin(<bin_num>) + 0
= 2* Mbin(‘11’) + 0 = 6
Mbin(<bin_num>’1’) = 2 * Mbin(<bin_num>) + 1
= 2* Mbin(‘1’) + 1 = 3
Mbin(‘1’) = 1
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5. Denotational Semantics (continued)
Another example
<dec_num> - > 0|1|2|3|4|5|6|7|8|9
|<dec_num>(0|1|2|3|4|5|6|7|8|9)
Mdec(‘0’)=0, Mdec(‘1’) = 1, ..., Mdec(‘0’) = 9
Mdec(<dec_num>’0’) = 10 * Mdec(<dec_num>)
Mdec(<dec_num>’1’) = 10 * Mdec(<dec_num>) + 1 …
Mdec(<dec_num>’9’) = 10 * Mdec(<dec_num>) + 9
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