An explicit state model checker SPIN An explicit state model checker

Explict State Model Checker Represents the system as an finite state machine Visits each reachable state (state space) explicitly Checks some property Property is satisfied Counterexample

DFS DFS visit of the state space procedure DFS(s) visited = visited  {s}; for each successor s’ of s if s’  visited then DFS(s’); end if end for end procedure

DFS How do we: Represent the transition relation Store the visited set Needs fast access (hash table) State space explosion Check properties

Promela Process Algebra An algebraic approach to the study of concurrent processes. Its tools are algebraical languages for the specification of processes and the formulation of statements about them, together with calculi for the verification of these statements. [Van Glabbeek, 1987] Describes the system in a way similar to a programming language

Promela Asynchronous composition of independent processes Communication using channels and global variables Non-deterministic choices and interleavings

An Example mtype = { NONCRITICAL, TRYING, CRITICAL }; show mtype state[2]; proctype process(int id) { beginning: noncritical: state[id] = NONCRITICAL; if :: goto noncritical; :: true; fi; trying: state[id] = TRYING; :: goto trying; critical: state[id] = CRITICAL; :: goto critical; goto beginning;} init { run process(0); run process(1); } NC C T

An Example mtype = { NONCRITICAL, TRYING, CRITICAL }; show mtype state[2]; proctype process(int id) { beginning: noncritical: state[id] = NONCRITICAL; if :: goto noncritical; :: true; fi; trying: state[id] = TRYING; :: goto trying; critical: state[id] = CRITICAL; :: goto critical; goto beginning;} init { run process(0); run process(1); }

An Example mtype = { NONCRITICAL, TRYING, CRITICAL }; show mtype state[2]; proctype process(int id) { beginning: noncritical: state[id] = NONCRITICAL; if :: goto noncritical; :: true; fi; trying: state[id] = TRYING; :: goto trying; critical: state[id] = CRITICAL; :: goto critical; goto beginning;} init { run process(0); run process(1); }

An Example mtype = { NONCRITICAL, TRYING, CRITICAL }; show mtype state[2]; proctype process(int id) { beginning: noncritical: state[id] = NONCRITICAL; if :: goto noncritical; :: true; fi; trying: state[id] = TRYING; :: goto trying; critical: state[id] = CRITICAL; :: goto critical; goto beginning;} init { run process(0); run process(1); }

An Example mtype = { NONCRITICAL, TRYING, CRITICAL }; show mtype state[2]; proctype process(int id) { beginning: noncritical: state[id] = NONCRITICAL; if :: goto noncritical; :: true; fi; trying: state[id] = TRYING; :: goto trying; critical: state[id] = CRITICAL; :: goto critical; goto beginning;} init { run process(0); run process(1); }

An Example mtype = { NONCRITICAL, TRYING, CRITICAL }; show mtype state[2]; proctype process(int id) { beginning: noncritical: state[id] = NONCRITICAL; if :: goto noncritical; :: true; fi; trying: state[id] = TRYING; :: goto trying; critical: state[id] = CRITICAL; :: goto critical; goto beginning;} init { run process(0); run process(1); } NC T C

Enabled Statements A statement needs to be enabled for the process to be scheduled. bool a, b; proctype p1() { a = true; a & b; a = false; } proctype p2() b = false; b = true; init { a = false; b = false; run p1(); run p2(); }

Enabled Statements A statement needs to be enabled for the process to be scheduled. bool a, b; proctype p1() { a = true; a & b; a = false; } proctype p2() b = false; b = true; init { a = false; b = false; run p1(); run p2(); } These statements are enabled only if both a and b are true.

Enabled Statements A statement needs to be enabled for the process to be scheduled. bool a, b; proctype p1() { a = true; a & b; a = false; } proctype p2() b = false; b = true; init { a = false; b = false; run p1(); run p2(); } These statements are enabled only if both a and b are true. In this case b is always false and therefore there is a deadlock.

Other constructs Do loops do :: count = count + 1; :: (count == 0) -> break od

Other constructs Do loops Communication over channels proctype sender(chan out) { int x; if ::x=0; ::x=1; fi out ! x; }

Other constructs Do loops Communication over channels Assertions proctype receiver(chan in) { int value; in ? value; assert(value == 0 || value == 1) }

Other constructs Do loops Communication over channels Assertions Atomic Steps int value; proctype increment() { atomic { x = value; x = x + 1; value = x; } }

On-The-Fly System is the asynchronous composition of processes The global transition relation is never build For each state the successor states are enumerated using the transition relation of each process

On-The-Fly 1 0 0

On-The-Fly 1 0 0 1 0

On-The-Fly 1 0 0 1 0

On-The-Fly 1 0 0 1 0 1 1

On-The-Fly 1 0 0 1 0 0 1 1 1

On-The-Fly 1 0 0 1 0 0 1 1 1

On-The-Fly 1 0 0 1 0 0 1 1 1

On-The-Fly 1 0 0 1 0 0 1 1 1

On-The-Fly 1 0 0 1 0 0 1 1 1

On-The-Fly 1 0 0 1 0 0 1 1 1

On-The-Fly 1 0 0 1 0 0 1 1 1

On-The-Fly 1 0 0 1 0 0 1 1 1

On-The-Fly 1 0 0 1 0 0 1 1 1

Visited Set Represents all the states that have been reached so far Will eventually become the set of all reachable state (state space) Test of presence of a state in the set must be efficient It is performed for each reached state procedure DFS(s) visited = visited  {s}; for each successor s’ of s if s’  visited then DFS(s’); end if end for end procedure

Visited Set Hash table Efficient for testing even if the number of elements in it is very big (≥ 106)

Visited Set Hash table Reduce memory usage Efficient for testing even if the number of elements in it is very big (≥ 106) Reduce memory usage Compress each state When a transition is executed only a limited part of the state is modified

Visited Set Hash table Reduce memory usage Reduce the number of states Efficient for testing even if the number of elements in it is very big (≥ 106) Reduce memory usage Compress each state Reduce the number of states Partial Order Reduction

State Representation Global variables Processes and local variables Queues Global Variables Processes Queues

Compression Each transition changes only a small part of the state Assign a code to each element dynamically Encoded states + basic elements use considerably less spaces than the uncompressed states

Compression i=0 j=0 P0 x=0 P0 x=1 Q0 {1} P0 x=0 P1 y=0 1 2 3 2 1 3 2 1 1 2 3 2 1 3 2 1 3 2 P1 y=0 1 P0 x=1 i=0 j=0 P0 x=0 Q0 {1}

Compression i=0 j=0 P0 x=0 P0 x=1 Q0 {} Q0 {1} P0 x=1 P0 x=0 P1 y=0 1 1 1 1 2 q ? x 3 2 1 3 2 1 3 2 P1 y=0 1 P0 x=1 Q0 {} i=0 j=0 P0 x=0 Q0 {1}

Hash Compaction Uses a hashing function to store each state using only 2 bits

Hash Compaction Uses a hashing function to store each state using only 2 bits There is an non-zero probability that two states are mapped into the same bits

Hash Compaction Uses a hashing function to store each state using only 2 bits There is an non-zero probability that two states are mapped into the same bits If the number of states is quite smaller than the number of bits available there is a pretty good chance of not having conflicts

Hash Compaction Uses a hashing function to store each state using only 2 bits There is an non-zero probability that two states are mapped into the same bits If the number of states is quite smaller than the number of bits available there is a pretty good chance of not having conflicts The result is not (always) 100% correct!

Minimized Automata Reduction Turns the state in a sequence of integers

Minimized Automata Reduction Turns the state in a sequence of integers Constructs an automata which accepts the states in the visited set

Minimized Automata Reduction Turns the state in a sequence of integers Constructs an automata which accepts the states in the visited set Works like a BDD but on non-binary variables (MDD)

Minimized Automata Reduction Turns the state in a sequence of integers Constructs an automata which accepts the states in the visited set Works like a BDD but on non-binary variables (MDD) The variables are the components of the state

Minimized Automata Reduction Turns the state in a sequence of integers Constructs an automata which accepts the states in the visited set Works like a BDD but on non-binary variables (MDD) The variables are the components of the state The automata is the minimal automata

Minimized Automata Reduction Turns the state in a sequence of integers Constructs an automata which accepts the states in the visited set Works like a BDD but on non-binary variables (MDD) The variables are the components of the state The automata is the minimal automata The automata is updated efficiently

Partial Order Reduction Some interleavings of processes are equivalent x=0 y=0 x++ y++ x=1 y=0 x=1 y=0 x=0 y=1 x=0 y=1 y++ x++ x=1 y=1

Partial Order Reduction Some interleavings of processes are equivalent Computing such interleavings and storing the intermediate states is expensive

Partial Order Reduction Some interleavings of processes are equivalent Computing such interleavings and storing the intermediate states is expensive Partial order reduction defines a reduced system which is equivalent to the original system but contains less states and transitions Defines an equivalent relation between states and computes the quotient of the state transition graph to obtain a reduced state transition graph. Properties are true of the reduced state transition graph if and only if are true of the original graph.

Partial Order Reduction Optimal partial order reduction is as difficult as model checking!

Partial Order Reduction Optimal partial order reduction is as difficult as model checking! Compute an approximation based on syntactical information

Partial Order Reduction Access to local variables Receive on exclusive receive-access queues Send on exclusive send-access queues Optimal partial order reduction is as difficult as model checking! Compute an approximation based on syntactical information Independent Invisible Check (at run-time) for actions postponed at infinitum Not mentioned in the property So called stack proviso

References http://spinroot.com/ Design and Validation of Computer Protocols by Gerard Holzmann The Spin Model Checker by Gerard Holzmann An automata-theoretic approach to automatic program verification, by Moshe Y. Vardi, and Pierre Wolper An analysis of bitstate hashing, by G.J. Holzmann An Improvement in Formal Verification, by G.J. Holzmann and D. Peled Simple on-the-fly automatic verification of linear temporal logic, by Rob Gerth, Doron Peled, Moshe Vardi, and Pierre Wolper A Minimized automaton representation of reachable states, by A. Puri and G.J. Holzmann