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Weak Arithmetic Completeness of Object-Oriented First-Order Assertion Networks

This paper presents the completeness of object-oriented first-order assertion networks, demonstrating how to express and prove statements in a proof system using both rules and axioms. The paper also discusses the expressiveness of the system and provides a proof sketch of the results. Furthermore, it introduces the concept of uniform instrumentation with auxiliary variables and demonstrates its use in reasoning about reachable states. The conclusion highlights the decidable nature of the approach and suggests potential improvements for special cases.

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Weak Arithmetic Completeness of Object-Oriented First-Order Assertion Networks

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  1. 28-01-2013 Weak Arithmetic Completeness of Object-Oriented First-Order Assertion Networks Stijn de Gouw, Frank S. de Boer, Richard Bubel, Wolfgang Ahrendt 1

  2. Completeness: Hoare Logic Example Hoare triple: {x=0} x := x+3 {x>0} Completeness if S |= {p} Stmt {q} then {p} Stmt {q} is provable • Proof system (rules + axioms) for statements • Proof system (rules + axioms) for assertions • Expressiveness: strongest postcondition (in the example: x=3) 2

  3. Existing Results Harel: completeness for arithmetical structures (incl. finite ADTs) Assertion language: first-order, addition and multiplication Tucker & Zucker: completeness for arbitrary structures Assertion language: (weak) second-order Apt: decidable assertions suffice, but only with auxiliary variables 2

  4. Our result z.val := 2 3

  5. Arrays as Objects Semantics: many-sorted structure S = (N, D1, …, Dn, I) where I(op) is a function/relation and op is a function/relation symbol 4

  6. Proof sketch of our result • Uniforminstrumentationwithauxiliary variables • For eachcomputation step, record ifandhow the state changes Example: field assignmente.x := e’. Add array variables • pc[i] = j if line j was executed in i-thcomputation step • x’’[i]=trueif in the i-th step, the field x of some object was changed • x’[i]= <o,v> if in the i-th step, the value v was assignedto field x of object o j: pc[|pc|] := j; x’[|pc] := <e, e’>; e.x := e’; x’’[|pc|] := true; |pc| := |pc| + 1 Instrumentation allows ‘recovering’ computation in anassertion, andconsequentlycandefine ‘reachablestates’ 5

  7. Conclusion • Express heap properties with auxiliary variables, only Presburger needed (decidable) • Uniform instrumentation, but ‘heavy’:can do better in special cases (example) • KeY Java theorem prover available, reasoning of object creation at abstraction level of prog language: http://keyaoc.hats-project.eu 6

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