Dr. Muhammed Al-Mulhem 1ICS ICS 535 Design and Implementation of Programming Languages Part 1 Fundamentals (Chapter 4) Denotational Semantics ICS 535 Design and Implementation of Programming Languages Part 1 Fundamentals (Chapter 4) Denotational Semantics

Dr. Muhammed Al-Mulhem 2ICS Outline Compilers: A review Compilers: A review Syntax specification Syntax specification BNF (EBNF) BNF (EBNF) Semantics specification Semantics specification Static semantics Static semantics Attribute grammar Attribute grammar Dynamic semantics Dynamic semantics Operational semantics Operational semantics Axiomatic semantics Axiomatic semantics Denotational semantics Denotational semantics

Dr. Muhammed Al-Mulhem 3ICS References 1. “Concepts in Programming Languages” by J. Mitchel [textbook] Chapter 4 2. “Programming Languages: Principles and Paradigms” by Allan Tucker and R. Noonan, Chapter 3 3. “Concepts of Programming Languages” by R. Sebesta, 6 th Edition, Chapter 3.

Dr. Muhammed Al-Mulhem 4ICS Denotational Semantics The most widely known method for describing the meaning of programs. The most widely known method for describing the meaning of programs. Based on recursive function theory. Based on recursive function theory. Originally developed by Scott and Strachey (1970) Originally developed by Scott and Strachey (1970)

Dr. Muhammed Al-Mulhem 5ICS Denotational Semantics (continued) The process of building a denotational specification for a language define for each language entity both The process of building a denotational specification for a language define for each language entity both a mathematical object and a mathematical object and a function that maps instances of that entity onto instances of the mathematical object. 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 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. The method is named denotational because the mathematical object denote the meaning of their corresponding language entity.

Dr. Muhammed Al-Mulhem 6ICS Example – Binary Number The syntax of a binary number is: The syntax of a binary number is: → 0 → 0 | 1 | 0 | 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. 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 syntactic entities in this case are ‘0’ and ‘1’. The objects are the decimal equivalent. The objects are the decimal equivalent.

Dr. Muhammed Al-Mulhem 7ICS Example – Binary Number Let the domain of semantic values of the objects be N, the set of nonnegative decimal integer values. Let the domain of semantic values of the objects be N, the set of nonnegative decimal integer values. The function M bin maps the syntactic entities of the previous grammar to the objects in N. The function M bin maps the syntactic entities of the previous grammar to the objects in N. The function M bin,for the above grammar, is defined as follows: The function M bin,for the above grammar, is defined as follows: M bin (‘0’) = 0 M bin (‘1’) = 1 M bin ( ‘0’) = 2 * M bin ( ) M bin ( ‘1’) = 2 * M bin ( ) + 1

Dr. Muhammed Al-Mulhem 8ICS Example - Decimal Numbers  0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ( 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9)  0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ( 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9) The denotational semantics for these syntax rules are: The denotational semantics for these syntax rules are: M dec ('0')=0, M dec ('1') =1, …, M dec ('9') = 9 M dec ( '0') = 10 * M dec ( ) M dec ( '1’) = 10 * M dec ( ) + 1 … M dec ( '9') = 10 * M dec ( ) + 9

Dr. Muhammed Al-Mulhem 9ICS Denotational Semantics: Program Constructs Let the state of a program be represented as a set of ordered pairs as follows: Let the state of a program be represented as a set of ordered pairs as follows: s = {,, …, } s = {,, …, } Each i is a variable and the associated v is its current value. Each i is a variable and the associated v is its current value. Any of the v’s can have the special value undef. 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 Let VARMAP be a function that, when given a variable name and a state, returns the current value of the variable VARMAP(i j, s) = v j VARMAP(i j, s) = v j The state changes are used to define the meanings of programs and program constructs. 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. Some constructs, such as expressions, are mapped to values, not states.

Dr. Muhammed Al-Mulhem 10ICS Denotational Semantics: Expressions We assume here that we deal with only simple expressions: We assume here that we deal with only simple expressions: Only + and * operators. Only + and * operators. An expression can have at most one operator. An expression can have at most one operator. The only operands are scalar variables and integer literals. The only operands are scalar variables and integer literals. No parenthesis. No parenthesis. The value of an expression is integer. The value of an expression is integer.

Dr. Muhammed Al-Mulhem 11ICS Denotational Semantics: Expressions The BNF description of these expressions: The BNF description of these expressions:  | |  | |    |  |  + | *  + | * The only error we consider in expressions is that a variable has an undefined value. 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. 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. Then Z U {error} is the set of values to which an expression can evaluate.

Dr. Muhammed Al-Mulhem 12ICS Denotational Semantics: Expressions The DS of expressions are (dot notation refer to child nodes of a node) The DS of expressions are (dot notation refer to child nodes of a node) M e (, s) = case of = M dec (, s) = M dec (, s) = if VARMAP(, s) == undef = if VARMAP(, s) == undef then error then error else VARMAP(, s) else VARMAP(, s) = if (M e (., s) == undef OR = if (M e (., s) == undef OR M e (., s) = undef) M e (., s) = undef) then error then error else if (. == ‘+’ then else if (. == ‘+’ then M e (., s) + M e (., s) + M e (., s) M e (., s) else M e (., s) * else M e (., s) * M e (., s) M e (., s)

Dr. Muhammed Al-Mulhem 13ICS 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 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 M a (x := E, s) = if M e (E, s) == error then error else s’ = {,,..., }, where for j = 1, 2,..., n, where for j = 1, 2,..., n, v j ’ = VARMAP(i j, s) if i j <> x v j ’ = VARMAP(i j, s) if i j <> x = M e (E, s) if i j == x = M e (E, s) if i j == x

Dr. Muhammed Al-Mulhem 14ICS Logical Pretest Loops Logical Pretest Loops Assume we have two mapping functions, M sl and M b Assume we have two mapping functions, M sl and M b M sl Maps statement list to states. M sl Maps statement list to states. M b Maps boolean expression to boolean value. M b Maps boolean expression to boolean value. The DS of a simple loop are: The DS of a simple loop are: M l (while B do L, s) = if M b (B, s) == undef M l (while B do L, s) = if M b (B, s) == undef then error else if M b (B, s) == false then s else if M sl (L, s) == error then error else M l (while B do L, M sl (L, s))

Dr. Muhammed Al-Mulhem 15ICS 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 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 In essence, the loop has been converted from iteration to recursion, where the recursive control is mathematically defined by other recursive state mapping functions