Formal Definitions of Programming Languages: K
Semantics
Dorel Lucanu
Faculty of Computer Science
Alexandru Ioan Cuza University, Iaşi, Romania
[email protected]
FMSE 2014
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
1 / 24
Outline
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
2 / 24
Outline
K Project
Started in 2003 by Grigore Rosu at UIUC, motivated mainly by teaching
programming languages and noticing that the existing semantic frameworks
have limitations
Project thesis:
Rewriting gives an appropriate environment to formally define the
semantics of real-life programming languages and to test and analyze
programs written in those languages.
Joint work between Formal Systems Laboratory (FSL) from University of
Illinois at Urbana-Champaign (UIUC) and Formal Methods in Software
Engineering (FMSE) group from Al. I. Cuza University (UAIC)
web page: http://kframework.org/
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
3 / 24
Outline
K Ingredients
Computations
Sequences of tasks, including syntax
Capture the sequential fragment of programming languages
Syntax annotations specify order of evaluation
Configurations
Multisets (bags) of nested cells
High potential for concurrency and modularity
K rules
Specify only what needed, precisely identify what changes
More concise, modular, and concurrent than regular rewrite rule
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
4 / 24
Outline
Computations1/2
syntax Exp
::= Int | Bool | Id
| Exp "+" Exp
| Id "=" Exp
syntax Stmt ::= Exp ";"
syntax Stmts ::= Stmt
> Stmts Stmts
[left, strict]
[strict(2)]
[strict(2)]
[left, seqstrict]
Generated heating rules
E1 + E2 ⇒ E1 y + E2
E1 + E2 ⇒ E2 y E1 + X = E ⇒ E y X = E ; ⇒ E y ;
S1 S2 ⇒ S1 y S2
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
5 / 24
Outline
Computations 2/2
E1
E1
X
S1
+ E2 ⇒ E1 y + E2
+ E2 ⇒ E2 y E1 + = E ; ⇒ E y X = ;
S2 ⇒ S1 y S2
y = x + 2; x = x + y;
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
6 / 24
Outline
Computations 2/2
E1
E1
X
S1
+ E2 ⇒ E1 y + E2
+ E2 ⇒ E2 y E1 + = E ; ⇒ E y X = ;
S2 ⇒ S1 y S2
y = x + 2; x = x + y;
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
6 / 24
Outline
Computations 2/2
E1
E1
X
S1
+ E2 ⇒ E1 y + E2
+ E2 ⇒ E2 y E1 + = E ; ⇒ E y X = ;
S2 ⇒ S1 y S2
y = x + 2; y x = x + y;
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
6 / 24
Outline
Computations 2/2
E1
E1
X
S1
+ E2 ⇒ E1 y + E2
+ E2 ⇒ E2 y E1 + = E ; ⇒ E y X = ;
S2 ⇒ S1 y S2
y = x + 2; y x = x + y;
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
6 / 24
Outline
Computations 2/2
E1
E1
X
S1
+ E2 ⇒ E1 y + E2
+ E2 ⇒ E2 y E1 + = E ; ⇒ E y X = ;
S2 ⇒ S1 y S2
y = x + 2; y x = x + y;
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
6 / 24
Outline
Computations 2/2
E1
E1
X
S1
+ E2 ⇒ E1 y + E2
+ E2 ⇒ E2 y E1 + = E ; ⇒ E y X = ;
S2 ⇒ S1 y S2
x + 2 y y = ; y x = x + y;
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
6 / 24
Outline
Computations 2/2
E1
E1
X
S1
+ E2 ⇒ E1 y + E2
+ E2 ⇒ E2 y E1 + = E ; ⇒ E y X = ;
S2 ⇒ S1 y S2
x + 2 y y = ; y x = x + y;
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
6 / 24
Outline
Computations 2/2
E1
E1
X
S1
+ E2 ⇒ E1 y + E2
+ E2 ⇒ E2 y E1 + = E ; ⇒ E y X = ;
S2 ⇒ S1 y S2
x + 2 y y = ; y x = x + y;
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
6 / 24
Outline
Computations 2/2
E1
E1
X
S1
+ E2 ⇒ E1 y + E2
+ E2 ⇒ E2 y E1 + = E ; ⇒ E y X = ;
S2 ⇒ S1 y S2
x y + 2 y y = ; y x = x + y;
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
6 / 24
Outline
Computations 2/2
E1
E1
X
S1
+ E2 ⇒ E1 y + E2
+ E2 ⇒ E2 y E1 + = E ; ⇒ E y X = ;
S2 ⇒ S1 y S2
x y + 2 y y = ; y x = x + y;
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
6 / 24
Outline
K Configuration Specification, Graphically
configuration:
T
k
$PGM:K
env
.Map
store
.Map
ASCII:
configuration <T color="yellow">
<k color="green"> $PGM:K </k>
<env color="blue"> .Map </env>
<store color="red"> .Map </store>
</T>
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
7 / 24
Outline
A Cocrete K Configuration
T
k
x y + 2 y y = ; y x = x + y;
env
a 7→ 1 x 7→ 2 y 7→ 3 z 7→ 4
store
1 7→ 5 2 7→ 23 3 7→ 16 4 7→ −12
ASCII:
<T>
<k> x ∼> HOLE + 2 ∼> y = HOLE; ∼> x = x + y; </k>
<env> a |-> 1 x |-> 2 y |-> 3 z |-> 4 </env>
<store> 1 |-> 5 2 |-> 23 3 |-> 16 4 |-> -12 </store>
</T>
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
8 / 24
Outline
K Rules
T
k
x y + 2 y y = ; y x = x + y;
env
a 7→ 1 x 7→ 2 y 7→ 3 z 7→ 4
store
1 7→ 5 2 7→ 23 3 7→ 16 4 7→ −12
⇒
T
k
23 y + 2 y y = ; y x = x + y;
env
a 7→ 1 x 7→ 2 y 7→ 3 z 7→ 4
store
1 7→ 5 2 7→ 23 3 7→ 16 4 7→ −12
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
9 / 24
Outline
Abstraction 1/3
T
k
x y + 2 y y = ; y x = x + y;
env
a 7→ 1 x 7→ 2 y 7→ 3 z 7→ 4
store
1 7→ 5 2 7→ 23 3 7→ 16 4 7→ −12
⇒
T
k
23 y + 2 y y = ; y x = x + y;
env
a 7→ 1 x 7→ 2 y 7→ 3 z 7→ 4
store
1 7→ 5 2 7→ 23 3 7→ 16 4 7→ −12
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
10 / 24
Outline
Abstraction 2/3
T
k
x
23
env
x 7→ 2
store
2 7→ 23
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
11 / 24
Outline
Abstraction 3/3
rule
k
env
X :Id
X 7→ L
store
L 7→ I
I
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
12 / 24
Outline
Concretization
ASCII:
rule I1 + I2 => I1 +Int I2
Graphically:
rule
I1 + I2
I1 +Int I2
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
13 / 24
Outline
Concretization
k
I1 + I2
I1 +Int I2
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
13 / 24
Outline
Concretization
T
k
I1 + I2
T
I1 +Int I2
k
—
env
env
—
store
—
—
store
—
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
13 / 24
Outline
Concretization
T
k
I1 + I2 y κ
env
—
store
—
⇒
T
k
I1 +Int I2 y κ
env
—
rule
<T>
<k> I1:Int + I2:Int ∼> kappa:K </k>
<env> sigma:Map </env>
<store> rho:Map </store>
</T>
=>
<T>
<k> I1 +Int I2 ∼> kappa </k>
<env> sigma </env>
<store> rho </store>
</T>
store
—
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
13 / 24
Outline
Rules at Work
rule
k
state
X 7→ I
X :Id
I
T
k
x y + 2 y y = ; y x = x + y;
env
a 7→ 1 x 7→ 2 y 7→ 3 z 7→ 4
store
1 7→ 5 2 7→ 23 3 7→ 16 4 7→ −12
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
14 / 24
Outline
Rules at Work
rule
k
state
X 7→ I
X :Id
I
T
k
23 y + 2 y y = ; y x = x + y;
env
a 7→ 1 x 7→ 2 y 7→ 3 z 7→ 4
store
1 7→ 5 2 7→ 23 3 7→ 16 4 7→ −12
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
14 / 24
Outline
Rules at Work
rule
I y + E2
I + E2
T
k
23 y + 2 y y = ; y x = x + y;
env
a 7→ 1 x 7→ 2 y 7→ 3 z 7→ 4
store
1 7→ 5 2 7→ 23 3 7→ 16 4 7→ −12
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
14 / 24
Outline
Rules at Work
rule
I y + E2
I + E2
T
k
23 + 2 y y = ; y x = x + y;
env
a 7→ 1 x 7→ 2 y 7→ 3 z 7→ 4
store
1 7→ 5 2 7→ 23 3 7→ 16 4 7→ −12
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
14 / 24
Outline
Rules at Work
rule
I1 + I2
I1 +Int I2
T
k
23 + 2 y y = ; y x = x + y;
env
a 7→ 1 x 7→ 2 y 7→ 3 z 7→ 4
store
1 7→ 5 2 7→ 23 3 7→ 16 4 7→ −12
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
14 / 24
Outline
Rules at Work
rule
I1 + I2
I1 +Int I2
T
k
25 y y = ; y x = x + y;
env
a 7→ 1 x 7→ 2 y 7→ 3 z 7→ 4
store
1 7→ 5 2 7→ 23 3 7→ 16 4 7→ −12
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
14 / 24
Outline
Rules at Work
rule
I yX =;
X =I ;
T
k
25 y y = ; y x = x + y;
env
a 7→ 1 x 7→ 2 y 7→ 3 z 7→ 4
store
1 7→ 5 2 7→ 23 3 7→ 16 4 7→ −12
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
14 / 24
Outline
Rules at Work
rule
I yX =;
X =I ;
T
k
y = 25; y x = x + y;
env
a 7→ 1 x 7→ 2 y 7→ 3 z 7→ 4
store
1 7→ 5 2 7→ 23 3 7→ 16 4 7→ −12
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
14 / 24
Outline
K Syntax of CinK: Expressions
syntax Exp
::= Int | Bool | Id
| "(" Exp ")"
| Exp "/" Exp
| Exp "*" Exp
| Exp "%" Exp
> left:
Exp "+" Exp
| Exp "-" Exp
> left:
Exp "==" Exp
| Exp "<=" Exp
| Exp "<" Exp
> "!" Exp
> Exp "&&" Exp
> Id "=" Exp
D. Lucanu (FII - UAIC)
K Semantics
[bracket]
[left, strict]
[left, strict]
[left, strict]
[left, strict]
[left, strict]
[left, strict]
[left, strict]
[left, strict]
[strict]
[left, strict(1)]
[strict(2)]
FMSE 2014
15 / 24
Outline
K Syntax of CinK: Statements
syntax Stmt ::= Decl
| Exp ";" [strict]
| "{" "}"
| "{" Stmts "}"
| "while" "(" Exp ")" Stmt
| "if" "(" Exp ")" Stmt "else" Stmt [strict(1)]
>
"if" "(" Exp ")" Stmt
syntax Stmts ::= Stmt | Stmts Stmts [left]
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
16 / 24
Outline
K Semantics of CinK: Declarations
syntax Type ::= "int"
syntax Decl ::= Type Ids ";"
syntax Ids ::= List{Id,","}
syntax Pgm ::= Stmts
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
17 / 24
Outline
K Semantics of CinK: Configuration
configuration <T color="yellow">
<k color="green"> $PGM:K </k>
<env color="blue"> .Map </env>
<store color="red"> .Map </store>
</T>
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
18 / 24
Outline
K Semantics of CinK: Expressions 1/2
syntax KResult ::= Int | Bool
rule <k> X:Id => I ...</k>
<env>... X |-> L ...</env>
<store>... L |-> I ...</store>
rule
rule
rule
rule
rule
I1
I1
I1
I1
I1
/
*
%
+
-
I2
I2
I2
I2
I2
=>
=>
=>
=>
=>
D. Lucanu (FII - UAIC)
I1
I1
I1
I1
I1
/Int
*Int
%Int
+Int
-Int
I2
I2
I2
I2
I2
requires I2 =/=Int 0
K Semantics
FMSE 2014
19 / 24
Outline
K Semantics of CinK: Expressions 2/2
rule
rule
rule
rule
rule
rule
I1 == I2 => I1 ==Int I2
I1 <= I2 => I1 <=Int I2
I1 < I2 => I1 <Int I2
! B => notBool B
true && B => B
false && _ => false
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
20 / 24
Outline
K Semantics of CinK: Statements
rule {} => .
[structural]
rule { Ss } => Ss [structural]
rule I; => .K
rule <k> X = I:Int => I ...</k>
<env>... X |-> L ...</env>
<store>... L |-> (_ => I) ...</store>
rule
rule
rule
rule
rule
S1:Stmts S2:Stmts => S1 ~> S2 [structural]
if (E) S => if (E) S else {} [structural]
if (true) S else _ => S
if (false) _ else S => S
while (B) S => if (B) {S while (B) S} else {}
D. Lucanu (FII - UAIC)
K Semantics
[structur
FMSE 2014
21 / 24
Outline
K Semantics of CinK:
rule <k> int (X,Xs => Xs); ...</k>
<env> Rho:Map (.Map => X|-> !L:Int) </env>
<store>... .Map => !L:Int |-> 0 </store>
requires notBool (X in keys(Rho))
rule int .Ids; => .K
D. Lucanu (FII - UAIC)
[structural]
K Semantics
FMSE 2014
22 / 24
Outline
K Semantics vs Big-Step SOS
on the blackboard
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
23 / 24
Outline
K Semantics vs Small-Step SOS
on the blackboard
D. Lucanu (FII - UAIC)
K Semantics
FMSE 2014
24 / 24