1. Concepts of Programming Languages
Lecturer: Dr Emad Nabil
Lecture #9
Dynamic Semantic
Ch3
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2. Denotational semantic
Static Semantic
Dynamic Semantic
Operational semantic
Denotational semantic
Axiomatic semantic
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3. Denotational Semantics
Denotational semantics is the most rigorous (strict) and most widely known formal method for describing the meaning of programs.
Based on recursive function theory
The most abstract semantics description method
Originally developed by Scott and Strachey (1970)
Dana Scott
Christopher S. Strachey

4. Denotational Semantics
The meaning of language constructs are defined by only the values of the program's variables

5. Denotational Semantics
* In denotational semantics, mathematical objects are used to represent the meanings of language constructs. * Language entities are converted to these mathematical objects with recursive functions
are converted by Recursion
Language entities
mathematical objects

6. Denotational Semantics
The process of building a denotational specification for a language: -
Define a mathematical object for each language entity
Define a function that maps instances of the language entities onto instances of the corresponding mathematical objects
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7. Denotational Semantics: program state
The state of a program is the values of all its current variables
s = {<i1, v1>, <i2, v2>, …, <in, vn>}
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
Math. Map function
Var. name (lang. entity)
value
state
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8. Example: Binary Numbers
syntactic objects
Semantic function
Mbin
objects in N, nonnegative decimal numbers ex
110
The semantic function, named Mbin, maps the syntactic objects, as described in the previous grammar rules, to the objects in N,
N is the set of nonnegative decimal numbers.
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9. syntax-directed semantics.
Syntax Rules
The semantic function Mbin
syntax-directed semantics.

10. Example2: Decimal Numbers
The syntax rules:
<dec_num>  '0' |'1' |'2' |'3' |'4' |'5' |'6' | '7' | '8' | '9' | <dec_num> '0' | <dec_num> '1' | ……………. <dec_num> ‘9’
Semantic rules
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
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12. Expressions
Suppose that the only error we consider in expressions is a variable having an undefined value.
Map expressions onto Z  {error}: Z is the set in integers
We assume expressions are
decimal integer numbers,
variables,
or binary expressions
having
one arithmetic operator( * or + ) and
two operands,
each of which can be an expression

13. Syntax Rules for Expr. VARMAP(ij, s) = vj
Mdec(<dec_num>, s)= value
∆= to define mathematical functions
=> to connect an operand to its associated switch (case)
A=123;
A=B;
A=B+C;
Semantic Rules for Expr.
The return value is:Z ∪ error. Z is the set of int numbers
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14. s = {<i1, v1>, <i2, v2>, …, <in, vn>}
* An assignment statement is an expression evaluation plus the setting of the target variable to the expression’s value * the meaning function maps a state to a state.
Syntax rules
<assign> X = E
X A|B|C
Semantic rules
Returns the new state s’
The comparison here for names, not values
s = {<i1, v1>, <i2, v2>, …, <in, vn>}
Me(E,s)=value
VARMAP(ij, s) = vj
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15. Logical Pretest Loops
In essence, the loop has been converted from iteration to recursion, where the recursive control is mathematically defined by other recursive state mapping functions
- Recursion, when compared to iteration, is easier
to describe with mathematical rigor
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16. Logical Pretest Loops <while> while B do L
Semantic rules
Syntax rules
<while> while B do L
we assume that there are two other existing mapping functions: The meaning of the loop is simply the value of the program variables after the statements in the loop have been executed the prescribed number of times.
Msl: map statement lists and states to states (or error)
Mb: map Boolean expressions to Boolean values (or error)
The return value is :
s ∪ error.

17. Do-while in java public static void main(String args[]) { }
boolean b;
int x=5; do
x+=1;
}while (b); }
error: b is not initialized
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18. Do-Until Msl: map statement lists and states to states (or error)
Mb: map Boolean expressions to Boolean values (or error)
Maps state sets to state sets U {error}
Mr(Do L until B,S) ∆= if Mb(B, s) = error then error else if Msl(L, S) = error else s’=Msl(L, s) if Mb(B, s’) = True then S’ else Mr( Do L until B, S’)
The return value is :
s ∪ error.

19. Do-While in java
Msl: map statement lists and states to states (or error)
Mb: map Boolean expressions to Boolean values (or error)
Maps state sets to state sets U {error}
Mr(Do L until B,S) ∆= if Mb(B, s) = error then error else if Msl(L, S) = error else s’=Msl(L, s) if Mb(B, s’) = False then S’ else Mr(Do L until B, S’)
The return value is :
s ∪ error.

20. Java Boolean Expression
Example of B If(x==5||y==6)
Mb(B, s)∆= if VARMAP(i, s)=undef for some i in B then error else B', where B' is the result of evaluating B after setting each variable i in B to VARMAP(i, s)
Mb: map Boolean expressions to Boolean values (T or F) (or error)
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21. Mfor(for (<assign1> ; <boolean_expr>; <assign2>) {<stmts>}, S)  Assume the semantics mapping functions Ma, Mb, Me, are Msl given.
 Mfor(for (<assign1> ; <boolean_expr>; <assign2>) L, S)  ∆=
  If Ma (<assign1>, S) = error
    then error
    else 
S’ = Ma (<assign1>, S)
if Mb(<Boolean_expr>, S’) = error
         then error
        else  if Mb(<Boolean_expr>, S’) = false
           then S’
            else if Msl( L , S’) = error
               then error
               else 
S’’ = Msl( L, S’)
if Ma(<assign2>, S’’) = error
                    then error
                   else 
S’’’ = Ma(<assign2>, S’’)
MforHelp(for(<boolean_expr>; <assign2>) L , S’’’)
 MforHelp(for (<boolean_expr>; <assign2>) L, S)  ∆=
   if Mb(<Boolean_expr>, S) = false
          then S
   else 
S’ = Msl( L, S)
S’’ = Ma(<assign2>, S’)
MforHelp(for(<boolean_expr>; <assign2>) L , S’’)
For loop in Java
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22. Evaluation of Denotational Semantics
provide an excellent way to describe a language concisely “briefly ”.
Provides a rigorous “strict” way to think about programs
Can be an aid to language design
For example, statements for which the denotational semantic description is complex and difficult may indicate to the designer that such statements may also be difficult for language users to understand and that an alternative design may be in order.
Has been used in compiler generation systems. (work started but didn’t completed (Jones,1980; Milos et al., 1984; Bodwin et al., 1982)).
Because of its complexity, is little used by language users.
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23. operational semantics denotational semantics
In both the meaning of a program is the values of its variables (the state)
the state is changed by algorithmic code (ie. 3 address code)
the state is changed by mathematical objects
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24. حكمة اليوم
لا ينمو الجسد إلا بالطعام و الرياضة و لا ينمو العقل إلا بالمطالعة و التفكير.