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1.6 Behavioral Equivalence. Two very important concepts in the study and analysis of programs Equivalence between programs Congruence between statements Replacing statements and programs. •Consider the two programs: P1::[ out x:integer where x=0

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slide2
Two very important concepts in the study and analysis of programs
    • Equivalence between programs
    • Congruence between statements
    • Replacing statements and programs
slide3
•Consider the two programs:

P1::[ out x:integer where x=0

l0: x:=1 :l0’ ]

P2::[ out x:integer where x=0

local t:integer where t=0

l0: t:=1 :l0’

l1: x:=t :l1’ ]

slide4
Computation generated by P1
    • <{l0},0>,<{l0’,1}>,<{l0’},1>, …
  • Computation generated by P2
    • <{l0},0,0>,<{l1},0,1>,<{l1’},1,1>,<{l1’},1,1}>,…
  • Computations contain too much distinguishing information, irrelevant to the correctness of the program, like
    • Control variable
    • Local variables
slide5
•Observable variables: O a subset of state variables
    • Usually input or output variables
    • Control variables are never observable
  • Label renaming =>equivalent programs
slide6
We define the observable state corresponding to s, denoted by s|O, to be the restriction of s to just the observable variables O.
  • Thus, s|O is an interpretation of O that coincides with s on all the variables in O.
slide7
Given a computation
  • σ :s0, s1, …
  • We define the observable behavior corresponding to َ σ to be the sequence
  • َ σo : s0 |O, s1 |O, …
slide8
Computation generated by P1
    • <{l0},0>,<{l0’,1}>,<{l0’},1>, …
  • Computation generated by P2
    • <{l0},0,0>,<{l1},0,1>,<{l1’},1,1>,<{l1’},1,1}>,…
  • For P1 and P2, and O={x}, observable behaviors:
  • σ1O : <0>, <1>, <1>, …
  • σ2O : <0>, <0>, <1>, <1>, …
reduced behavior
Reduced behavior
  • The reduced behavior σr
    • relative to O,
    • corresponding to a computation σ,

is the sequence obtained from σby thefollowing transformations:

  • Replace each state si by its observable part si|O
  • Omit from the sequence each observable state that is identical to its predecessor but not identical to all of its successors.
    • Not to delete the infinite suffix.
slide10
Applying these transformations to the computations σ1 and σ2
  • or just the second transformation to σ2O
  • σ1r : <0>, <1>, <1>, …
  • σ2r : <0>, <1>, <1>, …
equivalence of transition systems
Equivalence of transition systems
  • For a basic transition system P, we denote by R(P) the set of all reduced behaviors generated by P.
  • Let P1 and P2 be two basic transition systems
  • and O subsetof Π1 intersectΠ2 be a set of variables (observable variables for both systems).
  • The systems P1 and P2 are defined to be equivalent (relative to O), denoted by P1~P2,

if R(P1)=R(P2).

slide12
Which is equivalent to which?
  • Q1::[out x: integer where x=0; x:=2]
  • Q2::[out x: integer where x=0; x:=1; x:=x+1]
  • Q3::[out x: integer where x=0; [local t: integer; t:=1; x:=t+1]]
  • Observable set?
slide13
Congruence between statements
    • To explain the meaning of a statement S by another more familiar statement S’, that is congruent to S (perform the same task as S), but may be more efficient.
congruence of statements
Congruence of statements
  • Consider the two statements:
  • T1::[x:=1;x:=2]
  • T2::[x:=1;x:=x+1]

Viewing them as the bodies of programs, they are equivalent:

  • P1::[out x: integer where x=0;T1]
  • P2::[out x: integer where x=0;T2]
slide15
Our expectation about equivalent statements is that they are completely interchangeable:
    • the behavior of a program containing T1 will not change when we replace an occurrence of T1 with T2.
slide16
Consider Q1 and Q2:
  • Q1:: [out x: integer where x=0;[T1 || x:=0]]
  • Q2:: [out x: integer where x=0;[T2 || x:=0]]
  • Are they equivalent?

Obtain the set of reduced behaviors of Q1 and Q2.

slide17
Let P[S] be a program context, which is a program in which statement variable S appears as one of the statements.
  • For example:

Q[S]:: [out x: integer where x=0;[S|| x:=0]]

  • Let programs P[S1] and P[S2] be the programs obtained by replacing statement variable S with the concrete statements S1 and S2, respectively.
  • Statements S1 and S2 are defined to be congruent,denoted by S1~S2, if P[S1]~P[S2] for every program context P[S].
examples
examples
  • Commutativity
    • Selection and cooperation constructions are commutative.
      • [S1 or S2] ~ [S2 or S1]
      • [S1 || S2] ~ [S2 || S1]
  • Associativity
    • Concatenation, selection, and cooperation constructions are all associative.
      • [S1;[S2;S3]] ~ [[S1;S2];S3]~[S1;S2;S3]
      • For or and ||
slide19
S~ [S; skip]

What about:

  • S1 :: [await x]
  • S2 :: [skip; m: await x] ?

Consider:

P[S]:: [out x: boolean where x=F

l0: [S or [await !x]]; l1: x:=T :l1’]

slide21
await c ~ while !c do skip
  • Implementing await by busy waiting
  • Problem 1.3
implementation versus emulation
Implementation versus emulation
  • Replacement of two programs may be desirable, for example in the case that one is expressed in terms of high-level constructs that are not directly available on a considered machine.
  • There are two possible relations;
    • Emulation
    • implementation
slide23
P2 emulates P1 if they are equivalent, i.e., if their sets of reduced behaviors are equal (a symmetric relation).
  • P2 implements P1 if the set of reduced behaviors of P2 is a subset of the set of reduced behaviors of P1.
example
Example:

P1::[ out x, y: integer where x=0, y=0

loop forever do

[x:=x+1 or y:=y+1]]

P2::[ out x, y: integer where x=0, y=0

loop forever do

[x:=x+1 ; y:=y+1]]

slide25
Emulation and implementation relations between statements:
    • The statement S2 emulates statement S1 if P[S2] emulates P[S1] for every program context P[S].
    • S2 emulates S1 iff S2 is congruent to S1.
    • The statement S2 implements statement S1 if P[S2] implements P[S1] for every program context P[S].
slide26
What are the relations?
    • While !c do skip ?? await c
    • x:=x+1 ?? [[x:=x+1] or [y:=y+1]]
    • S2= await x ?? S1=[await x] or [await y]
    • S3=await (x or y) ?? S1=[await x] or [await y]
slide27
An example to compare S1 and S2 and S3:

[local x,y : boolean where x=F, y=T

out z: integer where z=0

S; z:=1]

1 7 grouped statements
1.7 Grouped Statements
  • In our text language, an atomic step (corresponding to a single transition taken in a computation),

consists of the execution of at most one statement of the program.

slide29
We definea class of statements as elementary statements.
  • These statements can be grouped together.
  • The elementary statements:
    • Skip, assignment, and await statements
    • If S, S1, …, Sk are elementary statements, then so are:
      • When c do S
      • If c then S1 else S2
      • [S1 or … or Sk]
      • [S1; …; Sk]
    • Any statement containing: cooperation or a while statement is not elementary.
slide30
If S is an elementary statement, then <S> is a grouped statement.
  • Example: <y:=y-1; await y=0; y:=1>
  • Execution of this grouped statement calls for the uninterrupted and successful execution of the three statements participating in the group in succession.
  • This grouped statement is congruent to the statement

await y=1

  • Thisinterpretation implies that execution of a grouped statement cannot be started unless its successful termination is guaranteed.
the transition associated with a grouped statement
The transition associated with a grouped statement
  • Product of transitions
    • Let t1 and t2 be two transitions.
    • Product of t1 and t2 , denoted by t1o t2 , is
      • s”∈t1 o t2 iff there exists an s’

such that s’ ∈t1(s) and s” ∈t2(s’)

      • Thus, the (t1o t2)-successors of s can be obtained by the application of t1 to s, followed by the application of t2 to the resulting states.
slide32
Assume that
  • t1: C1 /\ (y’ = e1) and t2: C2 /\ (y’ = e2)

Where we assume that t1 andt2 have the same set of modifiable variables.

  • t1 0 t2 =

t1 o t2 : C1 /\ C2[e1/y] /\ (y’=e2[e1/y])

slide33
Example:
  • t1 : (x>y) /\ (x’ = x-y) /\ (y’ = y)
  • t2 : (x<y) /\ (x’ = x) /\ (y’ = y-x)
  • What is the transition relation for the product?