1.6 Behavioral Equivalence

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1.6 Behavioral Equivalence
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• Two very important concepts in the study and
  analysis of programs
• Equivalence between programs
• Congruence between statements
• Replacing statements and programs

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• Consider the two programs
• P1 out xinteger where x0
• l0 x1 l0
• P2 out xinteger where x0
• local tinteger where t0
• l0 t1 l0
• l1 xt l1

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• Computation generated by P1
• ltl0,0gt,ltl0,1gt,ltl0,1gt,
• Computation generated by P2
• ltl0,0,0gt,ltl1,0,1gt,ltl1,1,1gt,ltl1,1,1gt,
• Computations contain too much distinguishing
  information, irrelevant to the correctness of the
  program, like
• Control variable
• Local variables

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• Observable variables O a subset of state
  variables
• Usually input or output variables
• Control variables are never observable
• Label renaming gtequivalent programs

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• We define the observable state corresponding to
  s, denoted by sO, to be the restriction of s to
  just the observable variables O.
• Thus, sO is an interpretation of O that
  coincides with s on all the variables in O.

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• Given a computation
• s s0, s1,
• We define the observable behavior corresponding
  to ? s to be the sequence
• ? s o s0 O, s1 O,

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• Computation generated by P1
• ltl0,0gt,ltl0,1gt,ltl0,1gt,
• Computation generated by P2
• ltl0,0,0gt,ltl1,0,1gt,ltl1,1,1gt,ltl1,1,1gt,
• For P1 and P2, and Ox, observable behaviors
• s1O lt0gt, lt1gt, lt1gt,
• s2O lt0gt, lt0gt, lt1gt, lt1gt,

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Reduced behavior

• The reduced behavior sr
• relative to O,
• corresponding to a computation s,
• is the sequence obtained from s by the following
  transformations
• Replace each state si by its observable part siO
• 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.

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• Applying these transformations to the
  computations s1 and s2
• or just the second transformation to s2O
• s1r lt0gt, lt1gt, lt1gt,
• s2r lt0gt, lt1gt, lt1gt,

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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 ?1 intersect ?2 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 P1P2,
• if R(P1)R(P2).

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• Which is equivalent to which?
• Q1out x integer where x0 x2
• Q2out x integer where x0 x1 xx1
• Q3out x integer where x0 local t integer
  t1 xt1
• Observable set?

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• 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.

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Congruence of statements

• Consider the two statements
• T1x1x2
• T2x1xx1
• Viewing them as the bodies of programs, they are
  equivalent
• P1out x integer where x0T1
• P2out x integer where x0T2

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• 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.

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• Consider Q1 and Q2
• Q1 out x integer where x0T1 x0
• Q2 out x integer where x0T2 x0
• Are they equivalent?
• Obtain the set of reduced behaviors of Q1 and
  Q2.

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• Let PS be a program context, which is a program
  in which statement variable S appears as one of
  the statements.
• For example
• QS out x integer where x0S x0
• Let programs PS1 and PS2 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 S1S2, if PS1PS2 for every
  program context PS.

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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.
• S1S2S3 S1S2S3S1S2S3
• For or and

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• S S skip
• What about
• S1 await x
• S2 skip m await x ?
• Consider
• PS out x boolean where xF
• l0 S or await !x l1 xT l1

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• PS2 may deadlock, while PS1 may not.

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• await c while !c do skip
• Implementing await by busy waiting
• Problem 1.3

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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

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• 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.

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Example

• P1 out x, y integer where x0, y0
• loop forever do
• xx1 or
  yy1
• P2 out x, y integer where x0, y0
• loop forever do
• xx1
  yy1

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• Emulation and implementation relations between
  statements
• The statement S2 emulates statement S1 if PS2
  emulates PS1 for every program context PS.
• S2 emulates S1 iff S2 is congruent to S1.
• The statement S2 implements statement S1 if PS2
  implements PS1 for every program context PS.

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• What are the relations?
• While !c do skip ?? await c
• xx1 ?? xx1 or yy1
• S2 await x ?? S1await x or
  await y
• S3await (x or y) ?? S1await x or await y

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• An example to compare S1 and S2 and S3
• local x,y boolean where xF, yT
• out z integer where z0
• S z1

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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.

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• We define a 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.

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• If S is an elementary statement, then ltSgt is a
  grouped statement.
• Example ltyy-1 await y0 y1gt
• 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 y1
• This interpretation implies that execution of a
  grouped statement cannot be started unless its
  successful termination is guaranteed.

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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.

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• 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 /\ C2e1/y /\
  (ye2e1/y)

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• Example
• ?t1 (xgty) /\ (x x-y) /\ (y y)
• ?t2 (xlty) /\ (x x) /\ (y y-x)
• What is the transition relation for the product?

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• Transitions for ltSgt page 54.
• Example page 55