Fall 2002 1

Semantics

Juan Carlos GuzmánCS 3123 Programming Languages ConceptsSouthern Polytechnic State University

Fall 2002 2

Why Semantics?

We need to establish precisely the meaning of programs

English is not enough – it is imprecise!

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Examples

What does it mean to compute x+5 x++ x+y

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More Examples

What does it mean to compute fact(5) fact(-5)

where fact is defined asint fact(n) {

if (n==0)

return 1;

return n * fact(--n);

}

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Still more Examples

What does it mean to defineint fact(n) {

if (n==0)

return 1;

return n * fact(--n);

}

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Issues

Any construct can be given a semantics, if one is persistent enough!The more complex the semantics, the less intuitive your language is!Be simple and elegant

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Classes of Semantics

Operational Semantics Meaning is given by presenting an “interpreter”:

the meaning of a language is the behavior of such an interpreter

Axiomatic Semantics Meaning given by using Logic to establish

assertions (properties) about programs

Denotational Semantics Based on mathematics (Function Theory,

Calculus) Meaning given by modeling each construct as a

function

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Operational Semantics

Meaning by interpretationEasy to construct: just build the machineTo know the meaning of a program, run the program in the machineThe machine provided is simple, of easy implementationThe limitation of this approach is that the meaning is given in the realm of computation: only computable concepts can be understood

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Operational Semantics (II)

Typical machine: JVM von Neumann machine Turing Machine!

for (exp1; exp2; exp3)

body

exp1

loop: if exp2 = 0 goto out

body

exp3

goto loop

out: …

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Limitations of Operational Semantics

Because meaning is given by another machine, there is no reasoning about noncomputable properties: nontermination function equality

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Axiomatics Semantics

Meaning by using LogicAssertions everywhere!Each assertion establishes the state of the system at any given timeComputation changes the system from one state to anotherTherefore, computation can be thought of a transformer of states

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Axiomatic Semantics (II)

State Transformer:{ P } S { Q }S being the statement, or construct, P is its precondition, and Q its postconditionP being valid before S is executed implies that Q must be valid after SThe “name of the game” is finding the weakest precondition so that the postcondition holds true!

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Example of Axiomatic Semantics

{ x>10 } sum = 2*x+1 { sum>1 }

Why the weakest precondition? The expected result of the program happens

to be the postcondition of the program, i.e., the postcondition of its last statement

We can work all the way back to find the weakest precondition of the program that would satisfy the expected result!

Anything stronger that the weakest precondition will work

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More Examples

{ P1 } S1; S2 { P3 }

{ P1 } S1 { P2 } { P2 } S2 { P3 }

Sequence

{ P } if B then S1 else S2 { Q }

{ B P } S1 { Q } {B P } S2 { Q }

If

{ I } while B do S { I B }

{ I B } S { I }While

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A Correct Program

{ n >= 0 }count = n;

fact =1;

while count <> 0 do

fact = fact * count;

count = count –1;

end

{ fact = n! }

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A Correct Program

{ n 0 }count = n;

fact =1;

{count 0 n 0 fact * count! = n! }while count <> 0 do

{count > 0 n 0 fact * count! = n! } fact = fact * count;

count = count –1;

end

{count = 0 n 0 fact * count! = n! }{ i.e., fact = n! }

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Denotational Semantics

Meaning given by modeling each construct as a functionMaps all syntactic objects to corresponding semantic elements: The number 12 present in the

program means the mathematical concept 12

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Example

Let e ::= n (number)| e1+e2 (sum)

Eval [[n]] = Number [[n]]Eval [[e1+e2]] = (Eval [[e1]]) + (Eval [[e2]])

Number [[n]], not defined, maps the input n into its corresponding mathematical concept

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Other Semantics

Many other ways of expressing the semantics of a program have been invented: you can always express the semantics if you try hard enough!Simple semantics usually means simple implementation and a better mathematical foundation

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Attribute Grammars

Attribute grammars are extensions to Context-Free grammars that allow certain language rules to be expressedThey are used to further restrict the phrases in a language

building and use of symbol tables: enforce static binding (decl. & use of id’s)

compute properties of phrases Construction of the AST

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Attribute Grammars (II)

There is the tendency associate syntax with the CFG used in parsing, and to call all other properties pertaining to the phrases of the language as semantics, or, more properly, static semanticsYour professor would rather call them “abstract interpretations”, with AG being one convenient way of expressing these

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Attribute Grammars (III)

An attribute grammar is a context-free grammar, with

attributes attribute computation functions predicate functions

Attributes are variables that can hold values on specific nodes of a syntax treeAttribute computation functions indicate how the values to those attributes are computedPredicate functions state properties that must be satisfied by the tree

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Attributes

Attributes are variables, associated to tree nodes, that will be given a value Synthesized attribute: the value of the

attribute is dependant on the information contained in the subtree where it is located

Inherited attribute: the value of the attribute is determined by the environment where the attribute is located

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Grammar for Simple Assignment Statements

assign var = exprexpr var2 + var3

expr varvar A | B | C

Allowable types are int or realAttributes:

actual_type, for nonterminals expr and var expected_type, for nonterminal expr string, for var

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Attribute Grammar for Simple Assignment Statements

assign var = expr expr.expected_type := var.actual_type

expr var2 + var3

expr.actual_type := if (var2.actual_type==int) && (var3.actual_type==int)

then int else real expr.actual_type == expr.expected_type

expr var expr.actual_type := var.actual_type expr.actual_type == expr.expected_type

var A | B | C var.actual_type := lookup(var.string)

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Parse Tree for A=A+B

assign

expr

var var var

A = A + BThis tree does not reflect the checking of the Predicate Functions, in this case, just in the expr node

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Flow of Attributes for A=A+B

assign

expr

var var var

A = A + B

actual_type actual_type actual_type

actual_typeexpected_type

This tree does not reflect the checking of the Predicate Functions, in this case, just in the expr node

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Fully Attributed Parse Tree for A=A+B

assign

expr

var var var

A = A + B

actual_type = intstring = A

actual_type = intstring = A

actual_type = intstring = B

expected_type = intactual_type = int

This tree does not reflect the checking of the Predicate Functions, in this case, just in the expr node