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SYMBOLIC MODEL CHECKING: 10 20 STATES AND BEYOND J.R. Burch E.M. Clarke K.L. McMillan D. L. Dill L. J. Hwang Presented by Rehana Begam

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Motivation Definitions Symbolic Model Checking Contribution Mu-Calculus Encoding Binary Decision Diagram Representation Model Checking Algorithm CTL Model Checking Empirical Results Summary Future Work OUTLINE

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Many different methods for automatically verifying finite state systems LTL CTL All rely on algorithms that explicitly represent a state space, using a list or table that grows in proportion to the number of states Number of states in the model grow exponentially with the number of concurrently executing components The size of the state table is the limiting factor in applying these algorithms to realistic systems MOTIVATION

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This “state explosion problem” can not be handled by the state enumeration methods Explicit state enumeration methods are limited to systems with at most 10 8 reachable states Can be eliminated by representing the state space symbolically instead of explicitly This technique verifies models with more than 10 20 states ! MOTIVATION

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Relational variable a predicate or a function Abstraction operator λ: used in lambda calculus f(x 1, x 2 ) is written as λ x1, x2 [f] Relational term f is a formula and y i are individual variables R is relational term and P is a relational variable with arity n Fixed point of function f An element x such that f(x) = x DEFINITIONS

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Least fixed point is the least element that is a fixed point. y is lfp of f in S iff (f(y) = y) ∧ (∀x S. (f(x) = x) ⇒ (y ⊆ x)) Greatest fixed point is the greatest element that is a fixed point. y is gfp of f in S iff (f(y) = y) ∧ (∀x S. (f(x) = x) ⇒ (x ⊆ y)) Fixed point operators μ and ν are the lfp and gfp operators used in mu-calculus Monotone function A function f is monotone iff for all P ⊆ S and Q ⊆ S, P ⊆ Q ⇒ f(P) ⊆ f(Q) DEFINITIONS

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Variable Interpretation Individual I P : for each individual variable y, I P (y) is a value in domain D Relational I R : for each n-ary relational variable P, I R (P) is an n-ary relation in domain D Substitution of Variables The substitution of a variable w for a variable v in a formula f, denoted f(v ← w) f ⇒ ∃ v [(v ⇔ w) ∧ f] DEFINITIONS

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In explicit state model checking, we represent the Kripke structure as a graph and implement the model checking algorithm as graph traversal. 2 main steps: Encode Model Domain: Describe sets of states as propositional logic formulae instead of enumeration: Mu-Calculus S = {1, 2, 3, 4, 5} = {x | 1 ≤ x ≤ 5} Compact Representation: Represent those logical formulae/boolean functions using efficient means of manipulating boolean functions: Binary Decision Diagrams SYMBOLIC MODEL CHECKING

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Provides a generalized symbolic model checking method by using a dialect of the Mu-Calculus as the primary specification language Describes a model checking algorithm for Mu- Calculus formulas that uses BDD to represent relations and formulas Shows how Mu-Calculus model checking algorithm can be used to derive efficient decision procedures for CTL, LTL model checking Discusses how it can be used to verify a simple synchronous pipeline circuit CONTRIBUTIONS

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Syntax: In this formula, R can be a Relational variable or a Relational term of the following two forms: Second one represents the least fixed point of R where R be formally monotone with P MU-CALCULUS

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Example: MU-CALCULUS

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Formal Definition: given a finite signature each symbol in is either an Individual variable or a Relational variable with some positive arity. recursively define two syntactic categories: formulas and relational terms. Formula: MU-CALCULUS

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Relational term: ∀, ∧, ⇒, and ⇔ are treated as abbreviations in the usual manner ¬R is an abbreviation for R ∨ R’ is an abbreviation for MU-CALCULUS

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Model M = (D, I R, I D ), where D is the domain Semantic function MU-CALCULUS

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MU-CALCULUS

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Widely used in various tools for the design and analysis of digital circuits Canonical form representation for Boolean formulas Similar to binary decision tree Allows many practical systems with extremely large state spaces to be verified-which are impossible to handle with explicit state enumeration methods BINARY DECISION DIAGRAM

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DAG Occurrence of variables is ordered from root to a leaf. Example: Formula: (a ∧ b) ∨ (c ∧ d) Ordering: a < b < c < d (a ←1, b ← 0, c ← 1, d ← 1) leads to a leaf node labeled 1 BINARY DECISION DIAGRAM

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For the Mu-Calculus that uses BDDs as its internal representation BDDATOM(f) returns BDD iff f = 1 Last case substitutes x i by dummy d i FixedPoint() is the standard technique MODEL CHECKING ALGORITHM

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CTL formula f is true of Kripke structure M= (A, S, L, N, S O ) ⇔ Mu-Calculus formula f' is true of a structure M’ = (S, I R, I D ) If CTL formula f is an abbreviation for the Mu- Calculus relational term R, then f is true at state s iff R(s) is true If f has no temporal operators, then it represents the relational term R CTL MODEL CHECKING

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EX f = λ S [ ∃ t [ f(t) ∧ N(s, t) ] ] EG f = f ∧ EX EG f = νQ [ f ∧ EX Q ] = νQ [ λ S [ f(s) ∧ ∃ t [ Q(t) ∧ N(s, t) ] ] E [ f ∪ g ] = g ∨ (f ∧ EX E[f ∪ g]) = μQ [g ∨ (f ∧ EX Q]] = μQ [λ S [g(s) ∨ (f(s) ∧ ∃ t [Q(t) ∧ N(s, t)]] CTL MODEL CHECKING

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Performs three-address logical and arithmetic operations on a register 3 Pipeline stages: Operand read from the register file ALU (Arithmetic Logic Unit) operation Write back to register EMPIRICAL RESULTS

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Pipeline with 12 bits has approximately 1.5 x 1O 29 reachable states The number of nodes in BDD is asymptotically linear in the number of bits, not exponential The verification time is polynomial in the number of bits EMPIRICAL RESULTS

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Suitable encoding of the model domain and compact representation for relations, the complexity of various graph-based verification algorithms is reduced Regular structure of the data path logic captured by the BDD representation results in a linear space complexity in the number of circuit components rather than exponential SUMMARY

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Characterization of the models for which the BDD Mu-Calculus checker is efficient Applicability of developed technique in common graph algorithms whose results can be expressed as relations, such as minimum spanning trees, graph isomorphism etc. FUTURE WORKS

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