Mobile Calculi

Prof. Diletta Romana Cacciagrano

Operational Semantics based on reduction

Reduction semantics

Reduction semantics

(red-cong)

:

Alpha-conversion

Structural congruence

Structural congruence

Operational Semantics based on labels

Labeled semantics (Late operational

semantics)

Labeled semantics (Late operational) semantics+alpha conv

Labeled semantics (Late operational semantics+alpha conv

+struct cong)

α

αCONG

Labeled semantics (Early operational

semantics)

Labeled semantics (Early operational semantics+alpha conv)

Labeled semantics (Early operational semantics+alpha conv

+struct cong)

α

αCONG

Labeled semantics

Labeled semantics

Labeled semantics

Early and late LTSs

Reduction and Labeled semantics

Operational Equivalencesbased on labels

Bisimulation

Bisimulation on Pi-calculus

Strong late bisimulation

L

Strong late bisimulation

L

Late Instantiation

Strong early bisimulation(finer than late)

Early Instantiation

Input

L

Congruence

Congruence w.r.t. parallel(proof for early. Similarly for late)

Theorem:

Congruence w.r.t. parallel

(proof for early. Similarly for late)

Congruence w.r.t. parallel (proof for early. Similarly for late)

Substitution preservation

Strong bisimilarity is not a congruence(proof for early. Similarly for late))

Open bisimulation(Bisimulation for Pi)

Name instantation is moved inside the definition of bisimulation.

The open bisimilarity, written , is the largest open bisimulation.

Open bisimilarity (Full bisimilarity)

Operational Equivalencesbased on reduction:Testing Preorders

Testing machinery

A set of processes to be test.

A set of tests or observers. These are obtained by extending the syntax of processes to generate processes which can perform a particular action (omega) reporting success.

A way to exercise a process on a given test: it is done by letting the process and the test to run in parallel and by looking at the computations which the embedded process can perform. These computations can be successful or failing, depending on whether or not they allow the execution of omega.

A general criterion (semantics) for interpreting the results of these exercises.

Testing machinery

Observer (Tests)

Experiments

Maximal computations

May Testing

Must Testing

Fair Testing

Testing preorders

Testing and Bisimulation equivalences

Bisimulation equivalences are usually rather strict: they depend on the whole branching structure of processes which, in some cases, are not relevant.

Weak bisimulation incorporates a particular notion of fairness: it abstracts from the tau-loops (i.e infinite sequences of tau-moves): the “normal” behavior can be resumed each time after a finite sequence of tau-moves.

Must testing semantics is based on the interpretation of tau-loops as divergences, making them quasi-observable as a chaotic or under-specified behavior. For this, it has been defined fair-testing semantics.

The standard testing equivalences are coarser than weak bisimulation in the case of divergence-free processes, and they are incomparable in general.