Some theoretical aspects of verification (with a focus on bisimilarity)

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Program <strong>verification</strong> Late 1960s: specialized logics and pro<strong>of</strong> procedures for <strong>verification</strong> <strong>of</strong> computer programs and systems Among the pioneers: Floyd, Hoare, Dijkstra, Gries, Lamport, Owicki, Manna, Pnueli Petr Jančar (TU Ostrava) <strong>Some</strong> <strong>aspects</strong> <strong>of</strong> <strong>verification</strong> Kolloquium Jena, 14 May 2007 27 / 89
Verification <strong>of</strong> a computer program (division <strong>of</strong> numbers) { x1, x2 are integers satisfying C1: x1 ≥ 0, x2 > 0 } Program P { C2: x1 = z1x2 + z2 ∧ 0 ≤ z2 < x2 } We want to verify: {C1}P{C2} ... (specification <strong>of</strong> P) Generated <strong>verification</strong> conditions: {C1} y1 := 0; y2 := x1 {INV} {INV ∧ y2 ≥ x2} y1 := y1 + 1; y2 := y2 − x2 {INV} {INV ∧ ¬(y2 ≥ x2)} z1 := y1; z2 := y2 {C2} Petr Jančar (TU Ostrava) <strong>Some</strong> <strong>aspects</strong> <strong>of</strong> <strong>verification</strong> Kolloquium Jena, 14 May 2007 28 / 89
Page 1 and 2: Some theor
Page 3 and 4: Outline of the tal
Page 5 and 6: Propositional logic Augustus de Mor
Page 7 and 8: First order predicate calculus Alfr
Page 9 and 10: Axiomatization of
Page 11 and 12: Presburger arithmetic Example sente
Page 13 and 14: (Nondeterministic) finite automata;
Page 21 and 22: Finite automata - closure propertie
Page 23 and 24: Finite automata - closure propertie
Page 25 and 26: Finite automata - summary Nondeterm
Page 27 and 28: Decidability of Pr
Page 29 and 30: Decidability of Pr
Page 31 and 32: Addition and multiplication Theory
Page 33 and 34: Absolute limits (for automated <str
Page 35 and 36: Automated theorem proving M. Davis
Page 37: Some available the
Page 41 and 42: Verification of a
Page 43 and 44: Program-and-proof
Page 45 and 46: Denotational semantics, while-comma
Page 47 and 48: Fixpoints of monot
Page 53 and 54: Fixpoints for continuous functional
Page 55 and 56: Kripke structure (transition system
Page 57 and 58: First order logic of</stron
Page 59 and 60: Decidability of va
Page 61 and 62: Monadic second order logic
Page 63 and 64: Automata on infinite words (Nondete
Page 65 and 66: Linear Temporal Logic (LTL) In prac
Page 67 and 68: Vending machines V1 V3 def = 5k.5k.
Page 69 and 70: Branching time LTL does not make di
Page 71 and 72: Model checking CTL The ‘most comp
Page 81 and 82: More on tree unfoldings Unfolding <
Page 83 and 84: µ-calculus Kozen, Pratt (1980s) Br
Page 85 and 86: Specification; process algebras Ope
Page 87 and 88: Synchronisation Merge ✓ E a → E
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Railway Components Road Rail down r
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The Complete Railway System ✞ Cro
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Bisimulation equivalence Milner, Pa
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Minsky counter machines A Minsky co
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Reduction of halti
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Reduction of halti
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Reduction of halti
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Reduction of halti
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Reduction - cont. • • . • •
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Bisimilarity over ‘context-free
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Bisimulation game 2 Players: Attack
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Type -1 systems; rules R a → w X
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Our results Normed Processes Unnorm
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Our results Normed Processes Unnorm
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Inf-PCP (a version of</stro
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Implementation of
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Implementation of
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Checking rules CI7I15I3I8I1 ⊥ C
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Formal methods (formal veri
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Conferences Some s
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Some theoretical aspects of verification (with a focus on bisimilarity)

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