Formal Semantics of Programming Languages

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Topic 2: Operational Semantics

IMP: A Simple Imperative Language

numbers N Positive and negative numbers n, m N

truth values T={true, false} locations Loc

X, Y Loc arithmetic Aexp

a Aexp boolean expressions Bexp

b Bexp commands Com

c Com

(3+5 ) 3 + 5

3 + 5 5+ 3

Abstract Syntax for IMP Aexp

a ::= n | X | a0 + a1 | a0 – a1 | a0 a1

Bexp b ::= true | false | a0 = a1 | a0 a1 | b | b0 b1

| b0 b1

Com c ::= skip | X := a | c0 ; c1 | if b then c0 else c1

| while b do c

2+34-5

(2+(34))-5 ((2+3)4))-5

Example Program

Y := 1;

while (X=1) do

Y := Y * X;

X := X - 1

But what about semantics

Expression Evaluation States

Mapping locations to values - The set of states

: Loc N (X)= X=value of X in = [ X 5, Y 7]

The value of X is 5 The value of Y is 7 The value of Z is undefined

For a Exp, , n N, <a, > n

a is evaluated in to n

Evaluating (a0 + a1) at

Evaluate a0 to get a number n0 at

Evaluate a1 to get a number n1 at Add n0 and n1

Expression Evaluation Rules Numbers

<n, > n Locations

<X, > (X) Sums

Subtractions

Products

10,10

1,1,0,0nnnwhere

naa

nana

10,10

1,1,0,0nnnwhere

naa

nana

10,10

1,1,0,0nnnwhere

naa

nana

Axiom

s

Derivations

A rule instance Instantiating meta variables with

corresponding values

632

3322

0,

0,,0,

1232

4332

0,

0,,0,

Derivation (Tree)

Axioms in the leafs Rule instances at internal nodes

0Init 0 , 55 0 , 77 0 , 99 0 ,

55)Init( 0 , 1697 0 ,

219)(75)Init( 0 ,

Computing a derivation

We write <a, > n when there exists a derivation tree whose root is <a, > n

Can be computed in a top-down manner At every node try all derivations “in parallel”

0Init 0 , 55 0 , 77 0 , 99 0 ,

?5)Init( 0 , ?97 0 ,

?9)(75)Init( 0 ,

5 16

21

Recap

Operational Semantics The rules can be implemented easily Define interpreter

Structural Operational Semantics Syntax directed

Natural semantics

Equivalence of IMP expressions

a0 a1 iff

nanaNn ,,. 10

Boolean Expression Evaluation Rules <true, > true <false, > false

mnifaa

mana

true

,10

,1,,0

mnifaa

mana

false

,10

,1,,0

mnifaa

mana

true,10

,1,,0

mnnotifaa

mana

true

,10

,1,,0

Boolean Expression Evaluation Rules(cont)

otherwisetand

whentwhere 10

,10

,1,0,0 1

false

truetrue

tt

tbb

tbtb

false

true

,

,

b

b

true

false

,

,

b

b

otherwisetand

whentwhere 10

,10

,1,0,0 1

true

falsefalse

tt

tbb

tbtb

Equivalence of Boolean expressions

b0 b1 iff

tbtbTt ,,. 10

Extensions

Shortcut evaluation of Boolean expressions “Parallell” evaluation of Boolean expressions Other data types

The execution of commands <c, > ’

c terminates on in a final state ’

Initial state 0

0(X)=0 for all X

Handling assignments <X:=5, > ’

A notation:

XY

XYmYXm

if(Y)

if{)](/[

<X:=5, > [5/X]

Rules for commands

<skip, >

Sequencing:

Conditionals:

]/[: ,

,

XmaX

na

'

'

,10

,0,

ccb

cb

elsethenif

true

Atom

ic com

mands

'

'

,10

,1,

ccb

cb

elsethenif

false

'

'''1''1

,10

,,

cc

cc

;

Rules for commands (while)

,

,

cb

b

dowhile

false

'

'''

,

,,

cb

cbb

dowhile

dowhile '' c, true

Example Program

Y := 1;

while (X=1) do

Y := Y * X;

X := X - 1

Equivalence of commands

c0 c1 iff

',',.', 10 cc

Proposition 2.8

while b do c if then (c; while b do c) else skip

Small Step Operational Semantics

The natural semantics define evaluation in large steps Abstracts “computation time”

It is possible to define a small step operational semantics <a, > 1 <a’, ’>

“one” step of executing a in a state yields a’ in a state ’

Small Step Semantics for Additions

,101,10

,01,0

'

'

aaaa

aa

,11,1

,11,1

'

'

anan

aa

mnpwherepmn

,1,

Summary

Operational semantics enables to naturally express program behavior

Can handle Non determinism Concurrency Procedures Object oriented Pointers and dynamically allocated structures

But remains very closed to the implementation Two programs which compute the same

functions are not necessarily equivalent

Exercise 2

(1) Evaluate a (X4) + 3 in a state s.t. (X)=0.

(2) Write down rules which express the “parallel” evaluation of b0 and b1 in b0 V b1 so that evaluates to true if either b0 evaluates to true, and b1 is unevaluated, or b1 evaluates to true, and b0 is unevaluated.