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6.5.2008 Formal Methods of Systems Specification Logical Specification of Hard- and Software Prof. Dr. Holger Schlingloff Institut für Informatik der Humboldt Universität and Fraunhofer Institut für Rechnerarchitektur und Softwaretechnik

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Slide 2 H. Schlingloff, Logical Specification 6.5.2008 Boolean Normal Forms DNF, CNF, NAND-, NOR-normal form (p|q)=(p ¬q); ¬p =(p|p); (p q)= (p| ¬ q) used for gate arrays Algebraic normal form XOR of conjunction of (positive) propositions later: tree normal forms (ordering of propositions)

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Slide 3 H. Schlingloff, Logical Specification 6.5.2008 Boolean Modelling of Reactive Systems (Parallel) transition systems, shared variables programs shared variables program (V,D,T,s 0 ) - V=(v 1,…,v n ) is a set (sequence) of program variables - D=(D 1,…,D n ) is a tuple of corresponding finite domains D i ={d i1,…,d im } - T D D is a transition relation, and - s 0 = (d 11,…,d n1 ) is the initial state Propositional representation of programs T=((request=true) (state=ready) (state‘=busy)) Representation of non-boolean domains?

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Slide 4 H. Schlingloff, Logical Specification 6.5.2008 Binary Encoding of Domains Any variable on a finite domain D can be replaced by log(D) binary variables similar to encoding of data types by compilers e.g. var v: {0..15} can be replaced by var v1,v2,v3,v4: boolean (0=0000, 1= 0001, 2=0010, 3=0011,..., 15=1111) State space still in the order of original domain! e.g. three int8-variables can have 2 24 =10 8 states e.g. array of length 10 with 10-bit values 10 30 states Representation of large sets of states?

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Slide 5 H. Schlingloff, Logical Specification 6.5.2008 Representation of Sets

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Slide 6 H. Schlingloff, Logical Specification 6.5.2008 Ordered Tree Form Normal form for propositional formulas Uses only the connective Ite Linear ordering on the set of propositions e.g., most significant bit first Shannon expansion

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Slide 7 H. Schlingloff, Logical Specification 6.5.2008 Truth table and tree form formula Reduction: Replace Ite (v,ψ,ψ) by ψ

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Slide 8 H. Schlingloff, Logical Specification 6.5.2008 Abbreviations Introduce abbreviations maximally abbreviated

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Slide 9 H. Schlingloff, Logical Specification 6.5.2008 Binary Decision Trees (BDTs) Binary decision tree Elimination of isomorphic subtrees (abbreviations)

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Slide 10 H. Schlingloff, Logical Specification 6.5.2008 Binary Decision Diagrams (BDDs) Elimination of redundant nodes (redundant subformulas) Ite (v,ψ,ψ) by ψ

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Slide 11 H. Schlingloff, Logical Specification 6.5.2008 A Toy Example How many states are reachable? How to check whether a given state is reachable?

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Slide 12 H. Schlingloff, Logical Specification 6.5.2008 Coding in nuSMV

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Slide 13 H. Schlingloff, Logical Specification 6.5.2008 Coding in SMV (cont.) SMV quickly finds a solution (rrddlluurrddlluurrddlluurrdd)

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Slide 14 H. Schlingloff, Logical Specification 6.5.2008 Another Toy Example gibts vielleicht noch besser (color)

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Slide 15 H. Schlingloff, Logical Specification 6.5.2008 Verification Model of Shift Register

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Slide 16 H. Schlingloff, Logical Specification 6.5.2008 Non-toy Examples Software verification: Correctness of aerospace and train computers, automobile controllers, nontrivial search problems,... Hardware verification: ALUs, PLAs, memory controllers, complete chip design,... For safety-critical systems formal validation is mandatory, for widely deployed systems highly recommended

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Slide 17 H. Schlingloff, Logical Specification 6.5.2008 Calculation of BDDs

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Slide 18 H. Schlingloff, Logical Specification 6.5.2008 The Influence of Variable Ordering Heuristics: keep dependent variables close together!

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Slide 19 H. Schlingloff, Logical Specification 6.5.2008 Transitive Closure Each finite (transition) relation can be represented as a boolean formula / BDD The transitive closure of a relation R is defined recursively by Thus, transitive closure be calculated by an iteration on BDDs Logical operations ( , , ) can be directly performed on BDDs

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Slide 20 H. Schlingloff, Logical Specification 6.5.2008 Reachability State s is reachable iff s 0 R*s, where s 0 S 0 is an initial state and R is the transition relation Reachability is one of the most important properties in verification most safety properties can be reduced to it in a search algorithm, is the goal reachable? Can be arbitrarily hard for infinite state systems undecidable Can be efficiently calculated with BDDs

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Slide 21 H. Schlingloff, Logical Specification 6.5.2008 Intuitively, xR*y iff there is a sequence w 0 w 1... w n of nodes connecting x with y In a finite model, this sequence must be smaller than the number of states. In practice, usually a few dozen steps are sufficient

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