Eiffel Refinement Calculus for OOSystems R'Paige and J'Ostroff

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Eiffel Refinement Calculus for
OO-Systems(R.Paige and J.Ostroff)
ERC

• Presented by Jared Schirm

2
Overview

• Motivation
• Refinement
• Eiffel
• Eiffel Refinement
• Outlook and Conclusion

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Motivation

• Seamless Development with OO-Techniques
• A core set of modelling constructs
• .. applied throughout entire development process
• OO Refinement Calculus
• for provably correct code
• achieved in modular manner
• with Eiffels built-in specification mechanism
• .. all in one

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Seamless Development

• OO can model everything
• late requirements
• SW architectures
• detailed specification and design
• implementation
• testing
• .. so a refinement calculus should directly lead
  to all OO-language constructs (including the
  standard imperative/procedural)

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Refinement

• first order logic for specification
• for imperative programs
• from specification to implementation
• constructive approach
• via proof obligations
• complementary to static checking

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Refinement An Approach

• math
• Equation x² 1
• Solutions x 1 , x -1
• refinement
• Specification x x² 1
• Programs x 1 , x -1

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Contract Notation

• specification w pre, post
• frame w
• the bunch of variables, which may be modified
• precondition pre
• a single state predicate on the prestate space
• postcondition post
• a double state predicate on the space of pre-
  poststate
• Example a,y,z y?0, xy² ? ?yltza

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Special Contracts(with abbreviation w post
for w true, post )

• abort w false, true
• choose w true
• skip true
• magic w false

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The Refinement Process

• spec ? mixed1 ? .. ? mixedn ? code
• more clear more executable
• step by step
• x,y true, x9 ? y²x spec
• ? x true, x9 y true, y²x spec
• ? x9 y true, y²x mixed
• ? x9 ysqrt(x) code
• ? x9 y3 code

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Basic Refinement Laws

• strengthen postcondition
• iff post ? post then w pre, post ? w
  pre, post
• weaken precondition
• iff pre ? pre then w pre, post ? w
  pre, post
• assign
• iff pre ? post w\E then w,x pre, post ?
  wE
• feasibility
• iff pre ? (?wT ? post) then wpre, post is
  feasible

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Eiffel The Language

• is purely OO
• goes the next natural steps in this technique
• multiple inheritance with name conflict
  management
• expanded classes
• addition to OO paradigm
• separate queries and commands
• pre- and postconditions for routines,
• class- and loop invariants
• has the desired specification through contracts ..

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Eiffel The Class
class CITIZEN feature ANY name, sex
STRING spouse CITIZEN children
SETCITIZEN single BOOLEAN ensure Result
(spouse Void) divorce modifies single,
spouse require ?single ensure single ? (old
spouse).single invariant single_or_married
single ? spouse.spouse Current number_of_paren
ts parents.count ? 2 symmetry ? c ? children
? ? p ? c.parents ? p Current end
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ERC - Notaion

• ?s, s pre- and poststate
• r.spec ?r.pre ? r.post ? t ? ?t ? t ? ?
• time
• e ?S, D? ?S ? D ? time ? ?(e)
• ?(w) ?(s-w)
• spec ? impl (? ?s,s ? impl ? spec)

??????
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ERC Core Rule Setfor imperative program
constructs
skip
e1 ?e1 ? e2 ?e2 ? ... ? en ?en ? time
e1exp
defined(?exp ) ? e1 ?exp ? e2 ?e2 ? ... ? time
if b then P else Q
( ?b ? P ) ? ( ?b ? Q ) ? time
P Q
? s1 ? P s s1 ? Q ?s s1
(local e T P)
(? e,?e T ? P)
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Reuse of Z and Morgan Rules

• infeasible specs cant be expressed in Z
• Z-spec P is typedecl ? pre ? post ,
• P.pre is (? s ? P) ? (P.pre, P)
• Z style
• P1 ?z P2 (? ?s,s ? P1.pre ? P2.pre ? P2 ? P1)
• ? ERC, Morgan style
• P1 ? P2 (? ?s,s ? (P2.pre ? P2) ? (P1.pre ? P1))
  
• ?
• ?z ? ?

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Entity Groups for References

• r.? reference entities reachable from r
• attrs of class, args and local vars
• e entity group the set of entities referring
  to same object as e
• r.? bunch of entity groups
• ee2,e3 e e,e2,e3
• reference equality
• (e1 e2) ? (e1 e2)

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Rule for create statement

• create e
• modifies e, t
• ensure ee ? remove(e) ? default(e)
• remove(e) ? ei ? ?e ? ei ? e ? ei ?ei e
• default(e) each e.a is set to default vaule

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Rule for assigned feature call

• e1 e2.q(e3)
• modifies e1, t
• require e2 ? Void
• q.preae2.a, Currente2, x2 e3
• ensure q.posta?e2.a, Current?e2, x e3,
  Resulte1
• e1 e2.q(e3)

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Eiffel Refinement Process

• write Eiffel classes with contracts
• start refinement at the root class
• refine only one class at a time
• .. routine by routine
• from the dependees use the contracts
• take some dependee and refine, refine, ..

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Outlook and Conclusion

• What is the benefit?
• a refinement calculus for producing immediately
  executable Eiffel programs
• What is missing?
• support for expanded types, exception handlers
  and agents
• Tool support for the calculus