c laus b rabrand s emantics (q1,’05) o ct 13, 2005 c laus b rabrand © 2005, university of aarhus...
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CLAUS BRABRAND SEMANTICS (Q1,’05) OCT 13, 2005
CLAUS BRABRAND
© 2005, University of Aarhus
[ brabrand@daimi.au.dk ] [ http://www.daimi.au.dk/~brabrand/ ]
SEMANTICS (Q1,’05)
WEEK 7: ”PROGRAM EQUIVALENCE AND IMPERATIVE FEATURES”
CLAUS BRABRAND © SEMANTICS (Q1,’05)[ 2 ]
OCT 13, 2005
Course Structure
Introduction [background]: Prerequisitional Math // 1 week
Part I [describe/explain/analyze]: Structural Operational Semantics // 3
weeks
Part II [compare/reason]: Concurrency and Communication (CCS) // 1 week
Part III [compare/prove/apply]: Equivalence: Bisimulation and Games // 1 week
Practice [link to real world]: Imperative Features + Sem in Practice // 1 week
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OCT 13, 2005
Week 7 - Outline
Issues from week 6
Course Evaluation Program Equivalence Imperative Blocks Sketch: Various Language Extensions “The Environment-Store Model” Other Semantic Formalisms“Semantics in Practice” [at 13:15] (Peter Gorm Larsen, IHA)
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OCT 13, 2005
“3x3 main issues” from week 6
Project: 1. sample solutions (which and when) ? [today] 2. how to specify an infinite trace ? [sample solutions] 3. grades (when) ? [Monday 17/10]
Exam: 4. prefer weekly hand-ins over project !!! [me+TAs too!!!] 5. size of exam: |exam| ~ |project| ? [slightly smaller] 6. more practical information ! [specific
questions?]
FYI (message from TAs): 7. “read” the project (and the exam) ! 8. “argue” vs. “prove” ! 9. Q/A session (on next Monday) !
Main Entry: 1read Pronunciation: 'rEdFunction: verbInflected Form(s): read /'red/; read·ing /'rE-di[ng]/
1a to receive or take in the sense of (as letters or symbols) especially by sight or touch
Meriam Webster(“read”)
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OCT 13, 2005
Example Exam Exercise:Prove by structural induction determinism for the SOS:
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OCT 13, 2005
Course Evaluation
“Tilfredshedsundersøgelse”: From an Evaluation-Theoretical perspective:
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OCT 13, 2005
Course Evaluation
Your e aluation is important and matters!: Gives you a chance to voice your opinion Helps improve next year’s course Helps improve my teaching (in general) Impacts (+/-) my personal employment opportunities at uni May influence larger didactic strategies for whole dept. / uni
Invariables (beyond my influence): Fixed project form (i.e., one (exam-like) project)
as opposed to weekly hand-ins (with a distributed workload)! Fixed exam form (i.e., 2-3 days “take-home”)
as opposed to 4-hr-written / oral exam Material situation
not easy to find adequate textbook (plan to write notes next year)
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OCT 13, 2005
Program Equivalence ()?
Program equivalence () ?: x1FV(E2) x2FV(E1)
...
How do we know they are “equivalent” ? …and what does that mean ?
C ; nil nil ; C C
if B then C else C’ if ~B then C’ else C
(C1 ; C2) ; C3 C1 ; (C2 ; C3)
repeat C until B C ; while ~B do C
x1 := E1 ; x2 := E2 x2 := E2 ; x1 := E1
nil nil ; nil
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OCT 13, 2005
Behavior and Behavioral Equivalence
Assume deterministic language L:
Def: Behavior: Partial function :
exec(C,) =
Def: Behavioral equivalence (C C’):
’ if <C,> * ’
undef otherwise e.g. nontermination,abnormal termination
exec : Com Store Store
Store: exec(C,) = exec(C’,)i.e. the two commands produce the same resulting store, ’, (but not necessarily in the same number of steps)
if both defined
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OCT 13, 2005
Congruence ()
Theorem: “” is a congruence [proof omitted]
i.e., we can substitute equivalent fragments in programs!
Example (Java):
C C’ => P[C] P[C’] , for all contexts P[]
class C { D void m() { S’ for (E1 ; E2 ; E3) S0
S” }}
safe transformation
who:compiler, homo-sapiens,
combination (refactoring tools), …
why:readability, optimization, simplification, …
class C { D void m() { S’ { E1 ; while (E2) { S0
E3 ; }} S” }}
class C { D void m() { S’ [ ] S’’ }}
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OCT 13, 2005
How to Prove Behavioral Equivalence?
How do we prove: (for given C, C’)? i.e.:
For derivation sequences of any length, n
C C’
,’: (<C,> * ’) (<C’,> * ’)
Store: exec(C,) = exec(C’,) if both defined
,’: (<C,> * ’) (<C’,> * ’)
,’: (<C,> * ’) (<C’,> * ’)
,’: (<C,> n ’) (<C’,> * ’)
,’: (<C,> * ’) (<C’,> n ’)
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OCT 13, 2005
Induction on the Length of Derivation Seq’s
Base case: P(k=0) Prove that the property, P, holds
for all derivation sequences of length 0 (zero)
Inductive step: P(k) P(k+1) Assume P(k):
that the property holds for derivation sequences of length k
Prove P(k+1): that it holds for derivation sequences of length k+1
Then: n: P(n) Property P holds for all derivation sequences (any length)
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OCT 13, 2005
…Or
How do we prove: (for given C, C’)? i.e.:
For some intermediate configuration,
C C’
,’: (<C,> * ’) (<C’,> * ’)
Store: exec(C,) = exec(C’,) if both defined
,’: (<C,> * ’) (<C’,> * ’)
,’: (<C,> * ’) (<C’,> * ’)
: (<C,> * ) (<C’,> * )
: (<C,> * ) (<C’,> * )
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OCT 13, 2005
Example (Proof Structure)
Example:
Prove “” (let be given w/o assumptions):
Assume [LHS]: Show [RHS]:
Case analysis on possible derivations for [LHS]…
if B then C else C’ if ~B then C’ else C
<if B then C else C’, > * <if ~B then C’ else C, > *
for some
<if B then C else C’, > *
<if ~B then C’ else C, > *
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OCT 13, 2005
Example (cont’d)
Case [B * tt]:
Then construct:
Analogous for [B * ff] Symmetric for the other direction “”
<if B then C else C’,> <C,’>C1
<B,> <tt,>B*
[IF1]
<if ~B then C’ else C,> <C,’>C1
<~B,> <ff,>B1
[IF2]
<B,> <tt,>B*
[NEG1]
proof
proof
C*
C*
’
’
proof ’
proof ’
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OCT 13, 2005
Blocks
Consider the language ABCD:
Example:
A ::= z | v | A0 + A1 | A0 - A1 | A0 A1
B ::= b | ~ B | B0 or B1 | A0 = A1
C ::= skip | x := A | if B then C else C’ | while B do C | begin D ; C end // local block
D ::= nil | var x := A | D0 ; D1 // local definitions
if (~ (x = y)) then begin var z := x ; x := y ; y := zend else skip
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OCT 13, 2005
Semantics of Definitions
Semantics of Definitions:
[NIL]D
<nil, > D
<var x := A, > D ’[x=z][VAR]D
<A, > A* <z, ’>
<D0 ; D1, > D <D0’ ; D1, ’>[SEQ1]D
<D0, > D <D0’, ’>
<D0 ; D1, > D <D1, ’>[SEQ2]D
<D0, > D ’
extend store
Note: [Plotkin] does this differently (through env-store model); read it yourselves…
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OCT 13, 2005
Semantics of Blocks
Semantics of Blocks:
[BLK1]C
<begin D ; C end, > C <begin(V,0) C end, ’><D, > D ’
[BLK2]C
<begin(V,0) C end, > C <begin(V,0) C’ end, ><C, > C <C’,’>
[BLK3]C
<begin(V,0) C end, > C (’ \ V) [0]
<C, > C ’
remember shadowed values : 0 = |V
remember set of locally defined variables : V = DV(D)
purge locally defined variables and restore old shadowed values
Def: |V := {v=(v)|vVDV()}
Def: \V := {v=(v)|vDV()\V}
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OCT 13, 2005
Dynamic vs. Static Scope Rules
Example: x := 2 ;begin var x := 7 ; nilend// here: x has the value...
[BLK3]C
<begin(V,0) C end, > C (’ \ V) [0]
<C, > C ’
purge locally defined variables and restore old shadowed values
“Static Scope Rules”x = 2
“Dynamic Scope Rules”x = 7
restoring old shadowed values not restoring …
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OCT 13, 2005
Inaccessible Val’s (Garbage Collection)
Example:
[BLK3]C
<begin(V,0) C end, > C (’ \ V) [0]
<C, > C ’
purge locally defined variables and restore old shadowed values
// x undefinedbegin var x := 7 ; nilend// here x is ...
“No Inaccessible Values”x isn’t in the store
(garbage collection)!
“Inaccessible Values”x is in the store
(but inaccessible)!
purging locally defined vars not purging …
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OCT 13, 2005
Language Extensions
Language extensions: Simple language without variables:
E E’
Adding variables (=> environments): |- E E’ : VAR VAL
Adding assignments (=> stores): <E,> <E’,’> : VAR VAL
Adding output (=> output “tape”): <E,> <E’,’> VAL*
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OCT 13, 2005
SOS Extensions (cont’d)
…more extensions: Adding stack operations (=> value stack)
<E,s> <E’,s’> s VAL*
Runtime-errors (=> error values): E errorK = Exp {errorK}
Exceptions (=> exception values): E exceptionK = Exp {exceptionK}
Adding jumps/gotos (=> labels and label map) L |- C C’ L : LABEL
COM
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OCT 13, 2005
SOS Extensions (cont’d)
…even more extensions: Adding functions (=> function environment):
F |- E E’ F : FNAME EXP
Adding statically scoped functions: F |- E E’ F : FNAME EXP
ENV
Adding procedures (=> procedure environment): P |- C C’ P : PNAME COM
Adding statically scoped procedures: P |- C C’ P : PNAME COM
ENV
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OCT 13, 2005
SOS Extensions (cont’d)
…still more extensions: “First class” functions (=> function values: “closures”)
|- E E’ : VAR CLOSURE
Adding call-by-refence / pointers / arrays / … (=> abstract locations / “the environment-store model”):
|- <E,> <E’,’> : VAR LOC : LOC VAL
Adding pointer arithmetic / “address-of” / …(=> phys. locations / addresses):
|- <E,> <E’,’> : VAR ADDR : ADDR VAL
…
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OCT 13, 2005
Expressible / Denotable / Storeable Values
Expressible Values: Values of expressions (i.e., E = EVAL)
Denotable Values: Values of identifiers/variables (i.e. : VAR DVAL)
Storeable Values: Values in the store (i.e. : LOC SVAL)
Printable Values: Values in the output (i.e. PVAL*)
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OCT 13, 2005
“The Environment-Store Model”
“The Environment-Store Model”: Introducing abstract locations:
Transitions: |- <E,> <E’,’>
x ℓ v
VAR LOC VAL
environment store
(x) ((x))x
: VAR LOC , : LOC VAL
env : doesn’t change w/ execstore: mutates with execution
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OCT 13, 2005
Examples (C-like)
Pointers (for the C-hackers: :)
Static Semantics:
Dynamic Semantics:
ptr p = 0xCAFEBABE; // p LocZ 0xff is a location constant
int x = *p; // *p Z (since p LocZ)
[DER]
|- * E :
|- E : LOC
[DER2]
|- <* E,> <* E’,’>v = (ℓ)
|- <* ℓ,> <v,>v = (ℓ)
[DER1]
|- <E,> <E’,’>
#define ptr (int*)
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OCT 13, 2005
Examples (cont’d)
Aliasing (similarly with call-by-reference):
Explicit allocation:
Explicit deallocation:
{ ptr p = malloc(1); // p LocZ
*p = 42; // side-effecting: ’ = [ℓ=42]} // ℓ, but is an “inaccessible reference”(!)
ptr p = ...;free(p);// (p)=ℓ, but ℓ ; aka. “dangling reference”!
ptr q = p; // location aliasing: (p) = ℓ = (q)*p = 42; // side-effecting: ’ = [ℓ=42]// now *q also has the value 42: ((q)) is 42
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OCT 13, 2005
Operational Semantics
Operational Semantics:
Labelled Transition System: 0 = <z=x;x=y;y=z, [x=1,y=2,z=3]> 1 = <x=y;y=z, [x=1,y=2,z=1]> 2 = <y=z, [x=2,y=2,z=1]> 3 = result = [x=2,y=1,z=1]
Variations in step-sizes (small-step, big-step, …)
The meaning of a construct is specified by the computation it induces when it is executed on a machine. In particular, it is of interest how the effect of a computation is produced.
-- [Nielson & Nielson, “Semantics with Applications”, ’93]
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OCT 13, 2005
Denotational Semantics
Denotational Semantics:
Describe everything as mathematical functions: [[ z=x;(x=y;y=z)]] =
[[ x=y;y=z ]] o [[ z=x ]] =[[ y=z ]] o [[ x=y ]] o [[ z=x ]] =s.s[y=s(z)] o s.s[x=s(y)] o s.s[z=s(x)] =s.s[x=s(y),y=s(x),z=s(x)]
Loops are expressed as fixed-points of rec’sive functors i.e., functions that takes functions as arguments
Meanings are modelled by mathematical objects that represent the effect of executing the constructs. Thus, only the effect is of interest, not how it is obtained.
-- [Nielson & Nielson, “Semantics with Applications”, ’93]
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OCT 13, 2005
Axiomatic Semantics
Axiomatic Semantics:
Partial correctness; Command C is partially correct wrt. a pre and a post-
condition if whenever the initial state fulfils the pre-condition and the program terminates, then the final state fulfils the post-condition.
{x=a,y=b} z=x;x=y;y=z {x=b,y=a}
Specific properties of the effect of executing the constructs are expressed as assertions. Thus, there may be aspects of the executions that are ignored.
-- [Nielson & Nielson, “Semantics with Applications”, ’93]
{ pre } C { post }
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OCT 13, 2005
Program Relationship (Example c)
Program worldModel world
ConcreteAbstract
~
P
P’
M
M’
1. P ~ P’ ?2. abstract
3. M ~ M’ ?
4. relate
5. M ~ M’ !6. concretize7. P ~ P’ !
CLAUS BRABRAND SEMANTICS (Q1,’05) OCT 13, 2005
Next week: Revision Period ; then Exam
Good Luck!
Any Questions?
CLAUS BRABRAND © SEMANTICS (Q1,’05)[ 41 ]
OCT 13, 2005
Note
Note on Structural Induction vs. Transitive 1-Step: You have only seen structural induction in the “shape” of C for 1-
step derivations where the induction hypothesis may be used to recompose insights according to 1-step SOS rules (not: * ’)
Solution: induction in the length of the derivation sequence
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