(Denotational Semantics)

1. (Denotational Semantics) ICS535-091

2. References • “Concepts in Programming Languages” by J. Mitchel [textbook] Chapter 4 • “Programming Languages: Principles and Paradigms” by Allan Tucker and R. Noonan, Chapter 3 • “Concepts of Programming Languages” by R. Sebesta, 6th Edition, Chapter 3. ICS535-091

3. Denotational Semantics • The most widely known method for describing the meaning of programs. • Based on recursive function theory. • Originally developed by Scott and Strachey (1970) ICS535-091

4. Denotational Semantics (continued) • The process of building a denotational specification for a language 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 difficulty with this method lies in creating the objects and the mapping functions. • The method is named denotational because the mathematical object denote the meaning of their corresponding language entity . ICS535-091

5. Example – Binary Number • The syntax of a binary number is: <bin_num> → 0 | 1 | <bin_num> 0 | <bin_num> 1 • To describe the meaning of a binary number using denotational semantics we associate the actual meaning with each rule that has a single terminal symbol in its RHS. • The syntactic entities in this case are ‘0’ and ‘1’. • The objects are the decimal equivalent. ICS535-091

6. Example – Binary Number • Let the domain of semantic values of the objects be N, the set of nonnegative decimal integer values. • The function Mbin maps the syntactic entities of the previous grammar to the objects in N. • The function Mbin ,for the above grammar, is defined as follows: Mbin(‘0’) = 0 Mbin(‘1’) = 1 Mbin(<bin_num> ‘0’) = 2 * Mbin(<bin_num>) Mbin(<bin_num> ‘1’) = 2 * Mbin(<bin_num>) + 1 ICS535-091

7. Example - Decimal Numbers <dec_num>  0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | <dec_num> ( 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9) • The denotational semantics for these syntax rules are: Mdec('0')=0, Mdec('1') =1, …, Mdec('9') = 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 ICS535-091

8. Denotational Semantics: Program Constructs • Let the state of a program be represented as a set of ordered pairs as follows: s = {<i1, v1>, <i2, v2>, …, <in, vn>} • Each i is a variable and the associated v is its current value. • Any of the v’s can have the special value undef. • Let VARMAP be a function that, when given a variable name and a state, returns the current value of the variable VARMAP(ij, s) = vj • The state changes are used to define the meanings of programs and program constructs. • Some constructs, such as expressions, are mapped to values, not states. ICS535-091

9. Denotational Semantics: Expressions • We assume here that we deal with only simple expressions: • Only + and * operators. • An expression can have at most one operator. • The only operands are scalar variables and integer literals. • No parenthesis. • The value of an expression is integer. ICS535-091

10. Denotational Semantics: Expressions • The BNF description of these expressions: <expr>  <dec_num> | <var> | <binary_exp> <binary_exp>  <left_exp> <operator> <right_exp> <left_exp>  <dec_num> | <var> <right_exp>  <dec_num> | <var> <operator>  + | * • The only error we consider in expressions is that a variable has an undefined value. • Let Z be the set of integers, and let error be the error value. • Then Z U {error} is the set of values to which an expression can evaluate. ICS535-091

11. Denotational Semantics: Expressions • The DS of expressions are (dot notation refer to child nodes of a node) Me(<expr>, s) = case <expr> of <dec_num> = Mdec(<dec_num>, s) <var> = if VARMAP(<var>, s) == undef then error else VARMAP(<var>, s) <binary_expr> = if (Me(<binary_expr>.<left_expr>, s) == error OR Me(<binary_expr>.<right_expr>, s) == error) then error else if (<binary_expr>.<operator> == ‘+’ then Me(<binary_expr>.<left_expr>, s) + Me(<binary_expr>.<right_expr>, s) else Me(<binary_expr>.<left_expr>, s) * Me(<binary_expr>.<right_expr>, s) ICS535-091

12. Assignment Statements • An assignment statement is an expression evaluation plus the setting of the left-side variable to the expression’s value. Maps state sets to state sets Ma(x := <expr>, s) = if Me(<expr>, s) == error then error else s’ = {<i1’,v1’>, <i2’,v2’>,..., <in’,vn’>}, where for j = 1, 2, ..., n, vj’ = VARMAP(ij, s) if ij <> x = Me(<expr>, s) if ij == x ICS535-091

13. Logical Pretest Loops • Assume we have two mapping functions, Msl and Mb • Msl Maps statement list to states. • Mb Maps boolean expression to boolean value. • The DS of a simple loop are: Ml(while B do L, s) = if Mb(B, s) == undef then error else if Mb(B, s) == false then s else if Msl(L, s) == error then error else Ml(while B do L, Msl(L, s)) ICS535-091

14. Loop Meaning • The meaning of the loop is the value of the program variables after the statements in the loop have been executed the prescribed number of times, assuming there have been no errors • In essence, the loop has been converted from iteration to recursion, where the recursive control is mathematically defined by other recursive state mapping functions ICS535-091