The B Method

by Péter Györök

Contents

• Metadata• The B language• The Prover• Demo

People behind it

• Developed by Jean-Raymond Abrial– Other people: G. Laffite, F. Mejia, I. McNeal

• Currently big companies and various universities maintain it

• ClearSy, Oxford University (Programming Research Group)• Subsidised projects

History, origin, versions

• Predecessor: Z-notation (also by Abrial)• Newest incarnation: Event-B

• Tools: Atelier B, B4free, B-toolkit

Primary application domain

• Software engineering– Specification– Design– Proof– Code generation

• Safety-critical systems• Big companies that use it: Siemens, Alstom,

Systerel…

Success stories

• METEOR project – Paris Metro Line 14– (Hungarian relevance?)

• Ariane 5 (rocket)

System overview

• B notation based on group theory and first order logic• The method is heavily focused on system development

– Multiple versions of the system: abstract machine -> refiniements -> implementation

– The proofs are for the consistency between versions• Syntax is expressed using mathematical symbols or

their ASCII equivalents (e.g. ! for )∀• Lots of syntactic sugar for easily writing down

expressions

Language features

• Types: based on set theoryTypes are either basic (integer, bool, string, enum) or built using Cartesian product, power set or record– Types inferred by typing predicates (∈, ⊂, ⊆, =)– The type of something is „the biggest set that contains it”– The type of integer literals and expressions is ℤ– The type of a set literal or expression is p(set), e.g. ℤ ∈ p( )ℤ– The type of a function from X to Y is (X × Y)℘– Distinction of „concrete” types that can be used in implementation– Many advanced types such as array, sequence, relation, tree – each

with their own set of operators

Language features

• Expressions and predicates– Predicates use the syntax of first order logic– Expressions of various types use the types’ specific operators– Lambda expressions are allowed

• Substitutions– Allow a predicate to be transformed ( [x := E] P )– Resemble features of an imperative language– Also some „alien” features (precondition etc.)– Proof obligations are derived from substitutions– Can be nondeterministic (but the implementation must be

deterministic, cf. concrete types)

Language features• Some types of substitution

– BEGIN…END– skip– := :() :∈– PRE– ASSERT– IF– CASE– LET– VAR– ;– ||– WHILE

Language features

• Machine– The „thing” that we are reasoning about– Resembles classes from OOP– Can be abstract, refinement or implementation– Special constraints apply to implementations– Elements of a machine:

• Parameters and their constraints• Imports, sees, includes etc.• Sets (enum or „deferred”)• Abstract and concrete constants, variables

Language features

– Elements of a machine• Properties, invariants• Values (!)• Initialisation and operations – expressed as a

substitution• Operations can have multiple return values• Assertions – this makes it possible to use B as a

mathematical proof assistant

Language featuresExample: adding assertions to help with a proof.

MACHINEMA

CONCRETE_VARIABLESvar

INVARIANTvar ∈ INT ⋀var2 = 1

ASSERTIONSvar = 1 ⋁ var = - 1

...END

This must be proven from the invariant.Then it can be used as a lemma in other proofs.

Typing predicate

Language fetaures

• The B0 language– Restricted version of the B language– Used for implementation only– Substitutions are equivalent to instructions– Translated to C(++), Ada etc.

The Prover

• Atelier B uses both an automatic and interactive prover

• The basic concept is the proof obligation (PO):Goal + hypotheses

• The prover doesn’t type check – that’s part of the proof! e.g. b = e1 + e2 where b BOOL and ∈ e1 , ∈ ℤ e2 is a ∈ ℤlegal goal which is unprovable

• Well-definedness must be proved tooe.g. 8/c is well-defined if c ≠ 0

The Prover

• Proof obligations– The types of things match up– The refinements are consistent– The initialisation sets the invariants and the

operations keep them– The operations meet their pre/postconditions– Assertions are true

The Prover

• Rules: inductive, deductive and rewriting• Theory: a list of rules (higher index has

priority)• Tactic: a list of theories to search for an

applicable rule– Backward tactic divides the goal into subgoals– Forward tactic generates new hypotheses– A full tactic is the combination of the two

The Prover

• Procedure of applying the tactic:– Search the backward tactic for an applicable rule– If one is found, apply it and continue with the next

theory– Tilde (~) can be used as the „repeat” operator– The whole tactic is implicitly tilded– For every new hypothesis generated, run the

forward tactic with the same procedure

The Prover

• The theory is fully customizable, even with inconsistent rules!

• The prover might loop infinitely• Proof obligations are normalized

– Examples: n > m becomes m+1 <= n,a ⇔ b becomes (a ⇒ b) (∧ b ⇒ a),a ⊆ b becomes a ∈ (℘ b)

The Prover

• Commands can be given to the interactive prover

• The prover will try to prove what is needed to execute the command. If it fails, a new goal is created

• ae : Abstract expression– P[…, expr, …] after ae(expr, y) becomes

well-defined(expr) ∧ expr=y ⇒ P[…, y, …]

Commands

• ah: Add Hypothesis– If the goal was h1, …, hn ⇒ G,

ah(P) replaces it withh1, …, hn ⇒ Ph1, …, hn, P ⇒ G

• ct: proof by contradiction– Replaces a goal h1, …, hn ⇒ G with

h1, …, hn, ¬ G bfalse⇒

Commands

• dc: Do Cases– If the goal is G, use dc(P) to split it into

¬ P ⇒ GP ⇒ G

• se: Suggest for Exist– If the goal is (∃ w1, …, wn).P(w1, …, wn)

se(v1, …, vn) turns it intoP(v1, …, vn)

Commands• ap: Arithmetic Proof

– An automated mechanism for proving things about systems of linear equations and inequations

• pp: Predicate Prover– Another automated system

• pr: Prover Call– Yet another (these all solve different kinds of goals)

• ar: Apply Rule– Just applies a rule

• dd: Deduction– For a goal P ⇒ Q, raise P in the hypothesis stack then prove Q

• ba: Back• cg: display Current Goal• qu: Quit

Demo

• The task: decide if a given number is prime

Creating a project

Adding a component

• Let’s add something to the empty project…

Adding a component

• Since this is our first component, the only choice is „Machine”.

Editing

• Now that we have a machine, double click it on the „Components” list to edit

Insert Theorem Here

• What we want to enter there:MACHINE primOPERATIONS p ← is_prim ( n ) = PRE n ∈ [3 .. MAXINT] THEN p := bool (∀ i . ( i ∈ [ 2 .. n-1 ] ⇒ ( n mod i ) ≠ 0 ) ) ENDEND

Insert Theorem Here

• What it will look like in B:

Atelier B hates single-letter identifiers so we reduplicate everything

Adding an implementationIMPLEMENTATION

prim_i

REFINES

prim

OPERATIONS

pp <-- is_prim ( nn ) =

BEGIN

VAR ll , kk IN

ll := TRUE ;

kk := nn ;

WHILE ( 2 /= kk & ll = TRUE) DO

IF nn mod (kk-1) = 0 THEN

kk := kk-1;

ll := FALSE

ELSE

kk := kk-1

END

INVARIANT

ll : BOOL &

nn : NAT &

nn >= 3 &

kk : 2..nn &

(ll=TRUE => (! jj.(jj:kk..nn-1 => nn mod jj /=0))) &

(ll=FALSE=> ( kk: 2..nn-1 & nn mod kk = 0))

VARIANT

kk

END ;

pp :=ll

END

END

END

Generate PO’s

• Click „Po”, then „F0” to try to prove…

Interactive Proof time!

Interactive Prover

Double-click one

Interactive Prover

• Now we can enter commands.

Completing the proof

Here are the commands to complete the proof:dc(jj = kk-1)prah(jj: kk..nn-1)pp(100)pr

dc(ll$7777 = TRUE)ddah(kk$7777 = 2)prppprddah(ll$7777 = FALSE)ppddprse(kk$7777)pr

Completing the proof

• Green means success!

THE END