1. CSC 4504 : Langages formels et applications
(La méthode Event-B)
J Paul Gibson, A207
Proof
Thanks to Jean-Raymond Abrial
2009: J Paul Gibson
T&MSP-CSC 4504 : Langages formels et applications

2. T&MSP-CSC 4504 : Langages formels et applications
Language of Predicates
2009: J Paul Gibson
T&MSP-CSC 4504 : Langages formels et applications

3. T&MSP-CSC 4504 : Langages formels et applications
Language of Predicates: Classical Results
2009: J Paul Gibson
T&MSP-CSC 4504 : Langages formels et applications

4. T&MSP-CSC 4504 : Langages formels et applications
Language of Predicates: Classical Results
2009: J Paul Gibson
T&MSP-CSC 4504 : Langages formels et applications

5. T&MSP-CSC 4504 : Langages formels et applications
Language of Predicates: Classical Results
2009: J Paul Gibson
T&MSP-CSC 4504 : Langages formels et applications

6. T&MSP-CSC 4504 : Langages formels et applications
Language of Predicates: Classical Results
2009: J Paul Gibson
T&MSP-CSC 4504 : Langages formels et applications

7. T&MSP-CSC 4504 : Langages formels et applications
Language of Predicates: Classical Results
2009: J Paul Gibson
T&MSP-CSC 4504 : Langages formels et applications

8. T&MSP-CSC 4504 : Langages formels et applications
Language of Predicates: Refining the language
2009: J Paul Gibson
T&MSP-CSC 4504 : Langages formels et applications

9. T&MSP-CSC 4504 : Langages formels et applications
Predicates & Expressions
A Predicate is a formal text that can be PROVED
An Expression DENOTES AN OBJECT.
A Predicate denotes NOTHING.
An Expression CANNOT BE PROVED
Predicates and Expressions are INCOMPATIBLE.
2009: J Paul Gibson
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10. T&MSP-CSC 4504 : Langages formels et applications
VARIABLES, PROPOSITIONS AND PREDICATES
2009: J Paul Gibson
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11. T&MSP-CSC 4504 : Langages formels et applications
WHAT CAN WE DO WITH A PREDICATE ?
2009: J Paul Gibson
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12. T&MSP-CSC 4504 : Langages formels et applications
SUBSTITUTION
2009: J Paul Gibson
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13. T&MSP-CSC 4504 : Langages formels et applications
UNIVERSAL QUANTIFICATION
2009: J Paul Gibson
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14. T&MSP-CSC 4504 : Langages formels et applications
Well-formedness
Each occurrence of an identifier in a formula (that is a predicate or an expression)
can be either free or bound.
Intuitively, a free occurrence of an identifier
refers to a declaration of that identifier in a scope outside of the formula,
while a bound occurrence corresponds to a local declaration introduced by a quantifier
in the formula itself.
For a formula to be considered well-formed, we ask that, beyond being syntactically
correct, it also satisfies the two following conditions:
1. Any identifier that occurs in the formula, should have only free occurrences
or bound occurrences, but not both.
2. Any identifier that occurs bound in the formula, should be bound in exactly
one place (i.e., by only one quantifier).
2009: J Paul Gibson
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15. T&MSP-CSC 4504 : Langages formels et applications
Well-formedness: checking automatically
There are pages of rules for checking this on the abstract syntax of Event-B expressions.
For example:
2009: J Paul Gibson
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16. T&MSP-CSC 4504 : Langages formels et applications
Type Checking
Type checking consists of checking, statically, that a formula is meaningful in
a certain context.
For that, we associate a type with each expression that occurs in a formula.
This type is the set of all values that the expression can take.
Then, we check that the formula abides by some type checking rules.
Those rules enforce that the operators used can be meaningful.
Unfortunately, type checking, as it is a static check, cannot by itself prove that a formula is meaningful.
For some operators, like integer division, we will also need to check
some additional dynamic constraints (e.g., that the denominator is not zero).
2009: J Paul Gibson
T&MSP-CSC 4504 : Langages formels et applications

17. T&MSP-CSC 4504 : Langages formels et applications
Type Checking
A type denotes the set of values that an expression can take.
Moreover, we want this set to be derived statically, based on the form of the expression and the context in which it appears.
As a consequence, a type can take one of the three following forms:
2009: J Paul Gibson
T&MSP-CSC 4504 : Langages formels et applications

18. T&MSP-CSC 4504 : Langages formels et applications
Type Checking
A type variable is a meta-variable that can denote any type.
We shall use lowercase Greek letters to denote type variables.
A typing environment represents the context in which a formula is to be type checked. A typing environment is a partial function from the set of all identifiers to the set of all possible types. For instance, the typing environment
2009: J Paul Gibson
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19. T&MSP-CSC 4504 : Langages formels et applications
Type Checking - Rules
There are pages of rules for checking this on the abstract syntax of Event-B expressions.
For example:
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20. T&MSP-CSC 4504 : Langages formels et applications
Type Checking – Rules (Example)
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21. T&MSP-CSC 4504 : Langages formels et applications
Dynamic Checking
Static checks are not enough to ensure that a formula is meaningful.
For instance, expression x÷y passes all the static checks described above, nevertheless it is meaningless if y is zero.
The aim of dynamic checking is to detect these kind of meaningless formulas.
This is done by generating (and then proving) some well-definedness lemma.
2009: J Paul Gibson
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22. T&MSP-CSC 4504 : Langages formels et applications
Dynamic Checking (Well-definedness WD)
WD lemmas for predicates
2009: J Paul Gibson
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23. T&MSP-CSC 4504 : Langages formels et applications
Dynamic Checking (Well-definedness WD)
WD lemmas for binary and unary expressions
2009: J Paul Gibson
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24. T&MSP-CSC 4504 : Langages formels et applications
Dynamic Checking (Well-definedness WD)
WD lemmas for other expressions
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25. T&MSP-CSC 4504 : Langages formels et applications
Inference Rules: for (automated) reasoning
“Ich wollte zunächst einmal einen Formalismus aufstellen, der dem wirklichen Schließen möglichst nahe kommt. So ergab sich ein „Kalkül des natürlichen Schließens”.
(“First I wished to construct a formalism that comes as close as possible to actual reasoning. Thus arose a "calculus of natural deduction".)
— Gentzen, Untersuchungen über das logische Schließen (Mathematische Zeitschrift 39, pp , 1935)
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26. T&MSP-CSC 4504 : Langages formels et applications
Inference Rules: for (automated) reasoning
Antécédent
nom
Conséquent
Tabular
Notation
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27. T&MSP-CSC 4504 : Langages formels et applications
Inference Rules: for (automated) reasoning
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28. T&MSP-CSC 4504 : Langages formels et applications
Inference Rules: for (automated) reasoning
Les règles d’inférence pour ^
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29. T&MSP-CSC 4504 : Langages formels et applications
Inference Rules: for (automated) reasoning
Les règles d’inférence pour ^
2009: J Paul Gibson
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30. T&MSP-CSC 4504 : Langages formels et applications
Inference Rules: for (automated) reasoning
Les règles d’inférence pour ¬
règles de contradiction ( “reductio ad absurdum”)
2009: J Paul Gibson
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31. T&MSP-CSC 4504 : Langages formels et applications
Inference Rules: for (automated) reasoning
Une preuve ( à la main)
2009: J Paul Gibson
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32. T&MSP-CSC 4504 : Langages formels et applications
Principe général d’un prouveur de prédicats
• On procède par induction sur la syntaxe du but P d’un séquent HYP |- P.
règles appliqué en arrière (backward).
• On ne monte en hypothèse (utilisation de DED) que des
– prédicats simples (pas de ^, =>, . . .) ou
– prédicats quantifiés universellement (∀) et normalisés
• on s’arrête avec un axiome ou sur HYP |- FAUX en cherchant une contradiction
dans les hypothèses.
• sinon on relance une preuve en cherchant de nouvelles instanciations pour les
variables des prédicats quantifiés (filtre + unification).
2009: J Paul Gibson
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33. T&MSP-CSC 4504 : Langages formels et applications
Règles d’un prouveur de prédicats (génériques)
opération générique
2009: J Paul Gibson
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34. T&MSP-CSC 4504 : Langages formels et applications
Règles d’un prouveur de prédicats (^)
générique
Instantiation with /\
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35. T&MSP-CSC 4504 : Langages formels et applications
Règles d’un prouveur de prédicats (=>)
générique
=>
Tactique gagnante: On utilise =>4 en dernier
2009: J Paul Gibson
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36. T&MSP-CSC 4504 : Langages formels et applications
Règles d’un prouveur de prédicats (not)
générique
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37. T&MSP-CSC 4504 : Langages formels et applications
Règles d’un prouveur de prédicats (les axioms)
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38. T&MSP-CSC 4504 : Langages formels et applications
Les prédicats quantifiés
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39. T&MSP-CSC 4504 : Langages formels et applications
Event-B is heavily influenced by Floyd-Hoare logic
Floyd­Hoare Logic is a method of reasoning mathematically about imperative programs.
It is the basis of most mechanized program verification systems
Tony Hoare introduced the notation {P} C {Q}, called a partial correctness specification for specifying what a program does, where:
C is a program (code) from the programming language whose programs are being specified
P and Q are conditions on the program variables used in C
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40. T&MSP-CSC 4504 : Langages formels et applications
Meaning of Hoare's Notation
{P} C {Q} is true if
whenever C is executed in a state satisfying P
and if the execution of C terminates
then the state in which C terminates satisfies Q
Example: {X = 1} X := X + 1 {X = 2}
P is the condition that the value of X is 1
Q is the condition that the value of X is 2
C is the assignment command X := X + 1 (i.e. `X becomes X + 1')
{X = 1} X := X + 1 {X = 2} is clearly true
{X = 1} X := X + 1 {X = 3} is clearly false
BE CAREUL with partial correctness:
{X = 1} WHILE true do skip {Y=3} is true
2009: J Paul Gibson
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41. T&MSP-CSC 4504 : Langages formels et applications
Total Correctness
Informally:
Total correctness = Termination + Partial correctness
Total correctness is the ultimate goal
It is usually easier to show partial correctness and termination separately
Termination is usually straightforward to show, but there are examples where it is not, e.g.: no one knows whether the program below terminates for all values of X
WHILE X > 1 DO
IF ODD(X) THEN X := (3 × X) + 1 ELSE X := X DIV 2
Where the expression X DIV 2 evaluates to the result of rounding down X/2 to a whole number
2009: J Paul Gibson
T&MSP-CSC 4504 : Langages formels et applications

42. T&MSP-CSC 4504 : Langages formels et applications
Specification can be Tricky
"The program must set Y to the maximum of X and Y"
[True] C [Y = max(X, Y)]
A suitable program (C)?:
IF X >= Y THEN Y := X ELSE SKIP
Another?
IF X >= Y THEN X := Y ELSE SKIP
Or even?
Y := X
WARNING: Later we will be able to prove that all these programs are "correct"
WHY?: The postcondition "Y = max(X, Y)" says "Y is the maximum of X and Y in the final state"
2009: J Paul Gibson
T&MSP-CSC 4504 : Langages formels et applications

43. T&MSP-CSC 4504 : Langages formels et applications
SKIP: possibly the simplest axiomatisation
Syntax: SKIP
Semantics: the state is unchanged
The SKIP Axiom :
|- {P} SKIP {P}
It is an axiom schema
P can be instantiated with arbitrary predicate calculus formulae (statements)
Instances of the SKIP axiom are:
|- {Y = 2} SKIP {Y = 2}
|- {True} SKIP {True}
|- {R = X + (Y × Q)} SKIP {R = X + (Y ×Q)}
2009: J Paul Gibson
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44. T&MSP-CSC 4504 : Langages formels et applications
Substitution Notation and assignment axiom: the most difficult axiomatisation
Define P [E/V ] to mean the result of replacing all occurrences of V in P by E
read P [E/V ] as `P with E for V '
for example: (X + 1 > X)[Y + Z/X] = ((Y + Z) + 1 > Y + Z)
Think of this notation as the `cancellation law':
V [E/V ] = E
which is analogous to the cancellation property of fractions:
v × (e/v) = e
The Assignment Axiom
|- {P [E/V ]} V := E {P}
Where V is any variable, E is any expression, P is any statement
and the notation P [E/V ] denotes the result of substituting the
term E for all occurrences of the variable V in the statement P .
Example:
|- {X + 1 = n + 1} X := X + 1 {X = n + 1}
|- can be proven
2009: J Paul Gibson
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45. T&MSP-CSC 4504 : Langages formels et applications
Precondition Strengthening is a typical development step
Recall that
|- S 1 , , |- Sn
|-S
means |- S can be deduced from |- S 1 , , |- Sn
Using this notation, the rule of precondition strengthening is:
|- P => P’ , |-{P’} C {Q}
|- {P} C {Q}
2009: J Paul Gibson
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46. T&MSP-CSC 4504 : Langages formels et applications
Postcondition Weakening
Just as the previous rule allows the precondition of a partial correctness specification to be strengthened, the following one allows us to weaken the postcondition:
|- {P} C {Q’}, |- Q’ => Q
|- {P} C {Q}
The rules precondition strengthening and postcondition weakening are sometimes called the rules of consequence
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47. T&MSP-CSC 4504 : Langages formels et applications
Existential Quantification
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Comparing the Quantification Rules
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Classical Results
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Classical Results
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Refining our Language: Equality (with classical results)
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Refining our Language: Set Theory
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53. T&MSP-CSC 4504 : Langages formels et applications
Basic Set Operator Memberships (Axioms)
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54. T&MSP-CSC 4504 : Langages formels et applications
Set Inclusion and Extensionality Axiom
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55. T&MSP-CSC 4504 : Langages formels et applications
Classical Results with Relation Operators
Relations (like r, q and p)
between Sets
(like S) containing elements (like a and b)
2009: J Paul Gibson
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Applying a Function
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Invariant Preservation
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Invariant Preservation
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Invariant Preservation
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Invariant Preservation: the rules
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Invariant Preservation: the rules
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Invariant Preservation: the rules
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Deadlock Freedom
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Event interpretation for refinement proofs
The execution of this event is enabled whenever there exist some values x and y such
that the guard P is true, then z is assigned x+y
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65. T&MSP-CSC 4504 : Langages formels et applications
Event interpretation – Example
This event is always enabled (there always exists a natural number x > 10)
The result of the event is that z is assigned an arbitrary natural number greater than 10.
The event is equivalet to
2009: J Paul Gibson
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66. T&MSP-CSC 4504 : Langages formels et applications
Refinement is used to transform an abstract machine into a
concrete machine which does the same computation, but possibly using a different data structure and/or different internal execution
can be refined (possibly) into
Also if in the abstract machine we have a nondeterministic event, then this could be refined into a deterministic one in the concrete machines :
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Names of context proof obligations:
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Names of machine proof obligations:
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Names of refinement proof obligations:
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Names of variant proof obligations:
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Names of Witness proof obligations:
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Names of Deadlock Freeness proof obligations:
At the moment, the deadlock freeness proof obligation generation is incomplete.
If you need it, you can generate it yourself as a theorem saying the the disjunction of the abstract guards imply the disjunction of the concrete guards.
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The Automatic Post-tactic: Rewrite rules
The following rewrite rules are applied automatically in a systematic fashion from left to right either in the goal or in the selected hypotheses.
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The Automatic Post-tactic: Rewrite rules
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The Automatic Post-tactic: Rewrite rules
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The Automatic Post-tactic: Rewrite rules
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The Automatic Post-tactic: Rewrite rules
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The Automatic Post-tactic: Rewrite rules
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The Automatic Post-tactic: Rewrite rules
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The Automatic Post-tactic: Rewrite rules
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Automatic inference rules
The following inference rules are applied automatically in a systematic fashion at the end of each proof step. They have the following possible effects:
• they discharge the goal,
• they simplify the goal and add a selected hypothesis,
• they simplify the goal by decomposing it into several simpler goals,
• they simplify a selected hypothesis,
• they simplify a selected hypothesis by decomposing it into several simpler selected hypotheses.
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Automatic inference rules
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Automatic inference rules
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Automatic inference rules
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Automatic inference rules
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Automatic inference rules
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Preferences for the Auto-prover
The auto-prover can be configured by means of a preference page, which can be obtained as follows:
press the ”Window” button on the top tooolbar. On the coming menu, press the ”Preferences” button. On the coming menu, press the ”Event-B” menue, then the ”Sequent Prover’, and finally the ”Auto-Tactic” button. This yields the following window:
2009: J Paul Gibson
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Interactive inference rules: through the red buttons in prover window
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Interactive inference rules: through the red buttons in prover window
… many more
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Interactive inference rules: through the red buttons in prover window
2009: J Paul Gibson
T&MSP-CSC 4504 : Langages formels et applications