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Automatic Verification Book: Chapter 6

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What is verification? Traditionally, verification means proof of correctness automatic: model checking deductive: theorem proving Practical view: automated systematic debugging VERY good at finding errors!

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How can we check the model? The model is a graph. The specification should refer to the graph representation. Apply graph theory algorithms.

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What properties can we check? Invariants: a property that need to hold in each state. Deadlock detection: can we reach a state where the program is blocked? Dead code: does the program have parts that are never executed.

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How to perform the checking? Apply a search strategy (Depth first search, Breadth first search). Check states/transitions during the search. If property does not hold, report counter example!

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If it is so good, is this all we need? Model checking works only for finite state systems. Would not work with Unconstrained integers. Unbounded message queues. General data structures: queues trees stacks Parametric algorithms and systems.

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The state space explosion Need to represent the state space of a program in the computer memory. Each state can be as big as the entire memory! Many states: Each integer variable has 2^32 possibilities. Two such variables have 2^64 possibilities. In concurrent protocols, the number of states usually grows exponentially with the number of processes.

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If it is so constrained, is it of any use? Many protocols are finite state. Many programs or procedure are finite state in nature. Can use abstraction techniques. Sometimes it is possible to decompose a program, and prove part of it by model checking and part by theorem proving. Many techniques to reduce the state space explosion (BDDs, Partial Order Reduction).

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Depth First Search Program DFS For each s such that Init(s) dfs(s) end DFS Procedure dfs(s) for each s’ such that R(s,s’) do If new(s’) then dfs(s’) end dfs.

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Start from an initial state q3 q4 q2 q1 q5 q1 Stack: Hash table:

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Continue with a successor q3 q4 q2 q1 q5 q1 q2 q1 q2 Stack: Hash table:

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One successor of q2. q3 q4 q2 q1 q5 q1 q2 q4 q1 q2 q4 Stack: Hash table:

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Backtrack to q2 (no new successors for q4). q3 q4 q2 q1 q5 q1 q2 q4 q1 q2 Stack: Hash table:

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Backtracked to q1 q3 q4 q2 q1 q5 q1 q2 q4 q1 Stack: Hash table:

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Second successor to q1. q3 q4 q2 q1 q5 q1 q2 q4 q3 q1 q3 Stack: Hash table:

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Backtrack again to q1. q3 q4 q2 q1 q5 q1 q2 q4 q3 q1 Stack: Hash table:

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How can we check properties with DFS? Invariants: check that all reachable states satisfy the invariant property. If not, show a path from an initial state to a bad state. Deadlocks: check whether a state where no process can continue is reached. Dead code: as you progress with the DFS, mark all the transitions that are executed at least once.

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[] ¬ (PC0=CR0/\PC1=CR1) is an invariant! Turn=0 L0,L1 Turn=0 L0,NC1 Turn=0 NC0,L1 Turn=0 CR0,NC1 Turn=0 NC0,NC1 Turn=0 CR0,L1 Turn=1 L0,CR1 Turn=1 NC0,CR1 Turn=1 L0,NC1 Turn=1 NC0,NC1 Turn=1 NC0,L1 Turn=1 L0,L1

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Want to do more! Want to check more properties. Want to have a uniform algorithm to deal with all kinds of properties. This is done by writing specification is temporal logics. Temporal logic specification can be translated into graphs (finite automata).

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[](Turn=0 --> <>Turn=1) Turn=0 L0,L1 Turn=0 L0,NC1 Turn=0 NC0,L1 Turn=0 CR0,NC1 Turn=0 NC0,NC1 Turn=0 CR0,L1 Turn=1 L0,CR1 Turn=1 NC0,CR1 Turn=1 L0,NC1 Turn=1 NC0,NC1 Turn=1 NC0,L1 Turn=1 L0,L1

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Turn=0 L0,L1 Turn=0 L0,NC1 Turn=0 NC0,L1 Turn=0 CR0,NC1 Turn=0 NC0,NC1 Turn=0 CR0,L1 Turn=1 L0,CR1 Turn=1 NC0,CR1 Turn=1 L0,NC1 Turn=1 NC0,NC1 Turn=1 NC0,L1 Turn=1 L0,L1 init

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Turn=0 L0,L1 Turn=1 L0,L1 init Add an additional initial node. Propositions are attached to incoming nodes. All nodes are accepting. Turn=1 L0,L1 Turn=0 L0,L1

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Correctness condition We want to find a correctness condition for a model to satisfy a specification. Language of a model: L(Model) Language of a specification: L(Spec). We need: L(Model) L(Spec).

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Correctness All sequences Sequences satisfying Spec Program executions

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How to prove correctness? Show that L(Model) L(Spec). Equivalently: ______ Show that L(Model) L(Spec) = Ø. Also: can obtain L(Spec) by translating from LTL!

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What do we need to know? How to intersect two automata? How to complement an automaton? How to translate from LTL to an automaton?

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Intersecting two automata A1= and A2= Each state is a pair (x,y): a state x from S1 and a state y from S1. Initial states: x is from I1 and y is from I2. Accepting states: x is from F1. ((x,y) a (x’,y’)) is a transition if (x,a,x’) is in 1, and (y,a,y’) is in 2.

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Example A B CT0 T1 A A B,C S0 S1 States: (S0,T0), (S0,T1), (S1,T0), (S1,T1). Accepting: (S0,T0), (S0,T1). Initial: (S0,T0).

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A B CT0 T1 A A B,C S0 S1 S0,T0 S0,T1 S1,T1 S1,T0 B B A C A C

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How to check for emptiness? S0,T0 S0,T1 S1,T1 S1,T0 B B A C A C

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Emptiness... Need to check if there exists an accepting run (passes through an accepting state infinitely often).

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Finding accepting runs If there is an accepting run, then at least one accepting state repeats on it forever. This state appears on a cycle. So, find a reachable accepting state on a cycle.

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Equivalently... A strongly connected component: a set of nodes where each node is reachable by a path from each other node. Find a reachable strongly connected component with an accepting node.

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How to complement? Complementation is hard! Can ask for the negated property (the sequences that should never occur). Can translate from LTL formula to automaton A, and complement A. But: can translate ¬ into an automaton directly!

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From LTL to automata “always eventually p”: []<>p “always p until q”: [](pUq) Exponential blow-up Formulas are usually small p q p\/q p true

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Model Checking under Fairness Express the fairness as a property φ. To prove a property ψ under fairness, model check φ ψ. Fair (φ) Bad (¬ψ)Program Counter example

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[](Turn=0 --> <>Turn=1) Turn=0 L0,L1 Turn=0 L0,NC1 Turn=0 NC0,L1 Turn=0 CR0,NC1 Turn=0 NC0,NC1 Turn=0 CR0,L1 Turn=1 L0,CR1 Turn=1 NC0,CR1 Turn=1 L0,NC1 Turn=1 NC0,NC1 Turn=1 NC0,L1 Turn=1 L0,L1

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Model Checking under Fairness Specialize model checking. For weak process fairness: search for a reachable strongly connected component, where for each process P either it contains on occurrence of a transition from P, or it contains a state where P is disabled.

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