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The B Method by Péter Györök

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Contents Metadata The B language The Prover Demo

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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

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History, origin, versions Predecessor: Z-notation (also by Abrial) Newest incarnation: Event-B Tools: Atelier B, B4free, B-toolkit

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Primary application domain Software engineering – Specification – Design – Proof – Code generation Safety-critical systems Big companies that use it: Siemens, Alstom, Systerel…

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Success stories METEOR project – Paris Metro Line 14 – (Hungarian relevance?) Ariane 5 (rocket)

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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

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Language features Types: based on set theory Types 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

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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)

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Language features Some types of substitution – BEGIN…END – skip – := :() : ∈ – PRE – ASSERT – IF – CASE – LET – VAR – ; – || – WHILE

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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

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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

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Language features Example: adding assertions to help with a proof. MACHINE MA CONCRETE_VARIABLES var INVARIANT var ∈ INT ⋀ var 2 = 1 ASSERTIONS var = 1 ⋁ var = - 1... END This must be proven from the invariant. Then it can be used as a lemma in other proofs. Typing predicate

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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.

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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 = e 1 + e 2 where b ∈ BOOL and e 1 ∈ ℤ, e 2 ∈ ℤ is a legal goal which is unprovable Well-definedness must be proved too e.g. 8/c is well-defined if c ≠ 0

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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

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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

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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

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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)

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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, …]

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Commands ah: Add Hypothesis – If the goal was h 1, …, h n ⇒ G, ah(P) replaces it with h 1, …, h n ⇒ P h 1, …, h n, P ⇒ G ct: proof by contradiction – Replaces a goal h 1, …, h n ⇒ G with h 1, …, h n, ¬ G ⇒ bfalse

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Commands dc: Do Cases – If the goal is G, use dc(P) to split it into ¬ P ⇒ G P ⇒ G se: Suggest for Exist – If the goal is ∃ (w 1, …, w n ).P(w 1, …, w n ) se(v 1, …, v n ) turns it into P(v 1, …, v n )

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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

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Demo The task: decide if a given number is prime

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Creating a project

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Adding a component Let’s add something to the empty project…

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

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

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Insert Theorem Here What we want to enter there: MACHINE prim OPERATIONS p ← is_prim ( n ) = PRE n ∈ [3.. MAXINT] THEN p := bool (∀ i. ( i ∈ [ 2.. n-1 ] ⇒ ( n mod i ) ≠ 0 ) ) END

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Insert Theorem Here What it will look like in B: Atelier B hates single- letter identifiers so we reduplicate everything

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Adding an implementation IMPLEMENTATION 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

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Generate PO’s Click „Po”, then „F0” to try to prove… Interactive Proof time!

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Interactive Prover Double-click one

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Interactive Prover Now we can enter commands.

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Completing the proof Here are the commands to complete the proof: dc(jj = kk-1) pr ah(jj: kk..nn-1) pp(100) pr dc(ll$7777 = TRUE) dd ah(kk$7777 = 2) pr pp pr dd ah(ll$7777 = FALSE) pp dd pr se(kk$7777) pr

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Completing the proof Green means success!

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THE END

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