Model checking of systems employing commutative functions
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This talk is about how you can find lots of bugs in real code by making compilers aggressively system specific. Model Checking of Systems Employing Commutative Functions. A.Prasad Sistla, Min Zhou, Xiaodong Wang presented by Min Zhou University of Illinois at Chicago. Outline.

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Model checking of systems employing commutative functions

This talk is about how you can find lots of bugs in real code by making compilers aggressively system specific

Model Checking of Systems Employing Commutative Functions

A.Prasad Sistla, Min Zhou, Xiaodong Wang

presented by Min Zhou

University of Illinois at Chicago


Outline
Outline code by making compilers aggressively system specific

  • Transition Diagram(TD) and Symbolic State Graph

  • Predicate Template and bisimulation ~0

  • Extended Predicate Template and bisimulation ~k

  • Experiment Results

  • Conclusion


Transition diagram td

We only consider such TDs who only have assignments: code by making compilers aggressively system specific

x:=c

c is a constant

x:=y

y is another variable

x:=φ(x)

φ is a unary function;

each such φ in a TD is commutative with each other:

φ1 φ2 = φ2φ1

Transition Diagram(TD)

a≤y y++

1

a:=x,x++

0

b:=x,x++

2

b≤y x:=0,y:=0

Variables:a,b,x,y


Symbolic state graph
Symbolic State Graph code by making compilers aggressively system specific

  • Sym_Reach(G, u) = (S0,R0, L0)

s.lc

s.val

s.exp

location

  • variables  expressions

  • s.exp(x): the composition of functions that were applied to x since last time a constant was assigned

  • variables  values

  • s.val(x): the latest constant assigned to variable x


How to construct symbolic state graph
How to construct Symbolic State Graph code by making compilers aggressively system specific

TD

x:=c

x:=φ(x)

x:=y

s1.exp(x)=φ(s0.exp(x))

s1.exp(x)=x

s1.exp(x)=s0.exp(y)[x/y]

q1

q1

q1

s1

s1

s1

s0

q0

q0

s0

s0

q0

s1.val(x)=s0.val(x)

s1.val(x)=c

s1.val(x)=s0.val(y)


Symbolic states
Symbolic States code by making compilers aggressively system specific

  • act_state(s) = (s.lc, h) where h(x) = s.exp(x){s.val(x)}

Symbolic

States

actual

states


Symbolic state graph1

a code by making compilers aggressively system specific y y++

1

1,(0,0,0,0)

(a,b,x+1,y)

s1

0,(0,0,0,0)

(a,b,x,y)

a:=x,x++

0

b:=x,x++

s0

s2

2

0,(0,0,0,0)

(a,b,x+1,y+1)

b  y x:=0,y:=0

s3

2,(0,0,0,0)

(a,b,x+1,y)

Symbolic State Graph

val

TD:

lc

exp


Our goal
Our Goal code by making compilers aggressively system specific

  • Define a bisimulation relation over symbolic states

  • For every location q, define a predicate template ptemplates(q)

  • s ~0 t require they are equivalent w.r.t ptemplates(s.lc)


Predicate template

p code by making compilers aggressively system specific

f

Predicate Template

var(p) X*

predicate, derived from guards and correctness formula


What should be in ptemplates q

code by making compilers aggressively system specific

x:=y

x:=c

p(x)

p(x)

q1

q1

q

q

qi

qi

What should be in ptemplates(q)

  • (AP,fid) U (guard(q), fid)Є ptemplates(q)

x:= φ1(x)

x:=φ2(x)

(P, fid) Є ptemplates(q)

p(x)

q1

q

qi

x:=φ(x)

(P,f(x) = y) Є ptemplates(q)

(P, f(x)=) Є ptemplates(q)


Example

p code by making compilers aggressively system specific0: x  y

p1: a  y

p2: b  y

Formula: (x  y)

Ptemplates(1)={

(p0, fid),

(p1, fid),

(p1, a  x),

(p2, b  x)}

a  y y++

1

a:=x,x++

0

b:=x,x++

2

b  y x:=0,y:=0

Example


Bisimulation 0

Instantiate predicate templates in states: code by making compilers aggressively system specific

(p(xi), xi yi) [s] = p [(s.exp (yi) /xi ) { xi/ yi }], where yi  

Eg:

Define ~0as follows: for any two states s and t, s ~0 t iff

s.lc = t.lc, s.val = t.val

(p, f) Є ptemplates(s.lc), (p, f)[s] (p, f)[t]

Bisimulation ~0

  • p: x1 < c

  • s.exp(x1): x1+1

  • s.exp(x2): x2+2

  • (p,x1  x2 ) [s] =

  • (s.exp (x2) < c){ x1/x2 } =

  • (x1+2 < c)

  • an implicit universal quantifier over the free variables


Theorem 1 0 is a bi simulation on the symbolic state graph sym reach g u

Proof idea: code by making compilers aggressively system specific

assume s0 ~0t0 (p,fid) Є ptemplates(q1)

In this case, (p,fid) Є ptemplates(q0)

so we have (p,fid)[s0] (p,fid)[t0]

x:=φ(x)

s1.exp(x)=φ(s0.exp(x))

q1

q0

s1.val(x)=s0.val(x)

s1

t1

s0

t0

Theorem 1 ~0is a bi-simulation on the symbolic state graph Sym_Reach(G, u).


Theorem 1 0 is a bi simulation on the symbolic state graph sym reach g u1

Now We show code by making compilers aggressively system specific

(p,fid)[s1] (p,fid)[t1]

(p,fid)[s0] (p,fid)[t0] 

x P[s0.exp(x)]  P[t0.exp(x)] 

x P[s0.exp(φ (x))]  P[t0.exp(φ (x))]

x P[φ (s0.exp(x))]  P[φ (t0.exp(x))] 

x P[s1.exp(x)]  P[t1.exp(x)]

x:=φ(x)

s1.exp(x)=φ(s0.exp(x))

q1

q0

s1.val(x)=s0.val(x)

s1

t1

s0

t0

Theorem 1 ~0is a bi-simulation on the symbolic state graph Sym_Reach(G, u).

By commutation


Extension of bisimulation 0

code by making compilers aggressively system specific

P(x)

P(x)

P(x)

si

ti

qi

s

q

t

Extension of Bisimulation ~0

  • If (p,f) Є ptemplates(q), we require (p,f)[s]  (p,f) [t] even in above case.

  • Not necessary. Only need when this path is feasible for these two states

TD:

X

X


Bisimulation k

code by making compilers aggressively system specific

P(x)

qi

q

P(x)

ti

t

ti-k

Bisimulation ~k

  • Only in this case, we require (p,f)[s]  (p,f) [t]

P(x)

si

s

si-k

feasible, length = k


Model checking of systems employing commutative functions
~ code by making compilers aggressively system specifick

  • In ~k , we require a conditional equivalence

  • Lemma ~k+1  ~k,

  • but ~k+1 need more computation


Example of a td for which 0 1

q code by making compilers aggressively system specific2

x2=0 

x1++

x1 20 

x1++, x2++

q0

q1

q4

x1=0 

x2 ++

q3

x2  20 

Example of a TD for which ~0  ~1

any two states of the form

(q1,(0,0), (x1 + c 0, x2 + c’ 0))

are bisimular w.r.t ~1


Experiment results

taken from T.Bultan 1999 code by making compilers aggressively system specific

Experiment Results


Circular queue

taken from T.Bultan 1999 code by making compilers aggressively system specific

Circular Queue


Sliding window
Sliding Window code by making compilers aggressively system specific


Conclusion and future work
Conclusion and future work code by making compilers aggressively system specific

  • Defined a non decreasing chain of bisimulation

  • Can be used in a class of infinite systems

  • ~k can be checked on-the-fly

  • Need investigate how to combine with static analysis