Model Checking Coverability Graphs of Vector Addition Systems
Coverability Graphs
CTL Model Checking
Small Model Properties
Model Checking Coverability Graphs of Vector Addition Systems Michel Blockelet and Sylvain Schmitz LSV, ENS Cachan & CNRS, Cachan, France
MFCS 2011, Warsaw, August 25, 2011
Coverability Graphs
CTL Model Checking
Small Model Properties
Outline “coverability-like”-properties known ES-complete properties for VAS: coverability, boundedness, regularity, ...
this talk a unifying view based on VAS coverability graphs and CTL model checking
contents Coverability Graphs CTL Model Checking Small Model Properties
Coverability Graphs
CTL Model Checking
Small Model Properties
Vector Addition Systems S = hV, x0 i I V: a finite set of transitions in Zk , I x0 : an initial configuration in Nk I
a
semantics: for x, x 0 in Nk and a in V, x → − x 0 iff x + a = x0
Example S = h{a, b, c}, h1, 0, 1ii with transitions a = h1, 1, −1i, b = h−1, 0, 1i, and c = h0, −1, 0i: a
a
h1, 0, 1i → − h2, 1, 0i → − /
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Graph I
finite abstraction of the VAS reachability graph
I
allows to decide various properties of the VAS (coverability, boundedness, place boundedness, regularity, reversal boundedness, trace boundedness, LTL model-checking, . . . )
I
but of non-primitive recursive size! (Cardoza et al., 1976)
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Tree (Karp and Miller, 1969) a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 =
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness erability, boundedness, is the someset h1,of5,j:1i lar? k x of>reachable k _ is the set valplace boundedness, reg1,^ , 1of reachable reachable? configurations finite? EF µ(j) > ω EF µ(j) > x(j) EFµ(j) > ω ues on . coordinate 2 fiularity, ..
b
b
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
(Valk and j=1 Vidal-Naquet, 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969) based on Rackoff (1978) ∗ ∗
(no: L ∩ (ab) c (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Tree (Karp and Miller, 1969) a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 =
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness erability, boundedness, is the someset h1,of5,j:1i lar? k x of>reachable k _ is the set valplace boundedness, reg1,^ , 1of reachable reachable? configurations finite? EF µ(j) > ω EF µ(j) > x(j) EFµ(j) > ω ues on . coordinate 2 fiularity, ..
b
b
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
(Valk and j=1 Vidal-Naquet, 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969) based on Rackoff (1978) ∗ ∗
(no: L ∩ (ab) c (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Tree (Karp and Miller, 1969) a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 =
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness erability, boundedness, is the someset h1,of5,j:1i lar? k x of>reachable k _ is the set valplace boundedness, reg1,^ , 1of reachable reachable? configurations finite? EF µ(j) > ω EF µ(j) > x(j) EFµ(j) > ω ues on . coordinate 2 fiularity, ..
b
b
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
(Valk and j=1 Vidal-Naquet, 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969) based on Rackoff (1978) ∗ ∗
(no: L ∩ (ab) c (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Tree (Karp and Miller, 1969) a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 =
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” witness w for covcoverability of x: place boundedness: unboundedness: non-regularity ∃x ∈ Nk , x 0 − → x} regua place unboundedness erability, boundedness, is the some set h1,of5,j:1i lar? k x of>reachable k _ is the set val^ place boundedness, reg1, 1, of 1 reachable reachable? configurations finite? EF µ(j) > ω EF µ(j) > x(j) EFµ(j) > ω ues on . coordinate 2 fiularity, ..
b
b
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
(Valk and j=1 Vidal-Naquet, 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969) based on Rackoff (1978) ∗ ∗
(no: L ∩ (ab) c (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Tree (Karp and Miller, 1969) a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 =
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” witness w for covcoverability of x: place boundedness: unboundedness: non-regularity ∃x ∈ Nk , x 0 − → x} regua place unboundedness erability, boundedness, is the some set h1,of5,j:1i lar? k x of>reachable k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω EF µ(j) > x(j) EFµ(j) > ω ues on . coordinate 2 fiularity, ..
b
b
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
(Valk and j=1 Vidal-Naquet, 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969) based on Rackoff (1978) ∗ ∗
(no: L ∩ (ab) c (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Tree (Karp and Miller, 1969) a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 =
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” witness w for covcoverability of x: place boundedness: unboundedness: non-regularity ∃x ∈ Nk , x 0 − → x} regua place unboundedness erability, boundedness, is the some set h1,of5,j:1i lar? k x of>reachable k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω EF µ(j) > x(j) EFµ(j) > ω ues on . coordinate 2 fiularity, ..
b
b
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
(Valk and j=1 Vidal-Naquet, 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969) based on Rackoff (1978) ∗ ∗
(no: L ∩ (ab) c (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Tree (Karp and Miller, 1969) a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 =
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” witness w for covcoverability of x: place boundedness: unboundedness: non-regularity ∃x ∈ Nk , x 0 − → x} regua place unboundedness erability, boundedness, is the some set h1,of5,j:1i lar? k x of>reachable k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω EF µ(j) > x(j) EFµ(j) > ω ues on . coordinate 2 fiularity, ..
b
b
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
(Valk and j=1 Vidal-Naquet, 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969) based on Rackoff (1978) ∗ ∗
(no: L ∩ (ab) c (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Tree (Karp and Miller, 1969) a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 =
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” witness w for covcoverability of x: place boundedness: unboundedness: non-regularity ∃x ∈ Nk , x 0 − → x} regua place unboundedness erability, boundedness, is the some set h1,of5,j:1i lar? k x of>reachable k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω EF µ(j) > x(j) EFµ(j) > ω ues on . coordinate 2 fiularity, ..
b
b
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
(Valk and j=1 Vidal-Naquet, 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969) based on Rackoff (1978) ∗ ∗
(no: L ∩ (ab) c (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Graph (Valk and Vidal-Naquet, 1981) a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 =
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” witness w for covcoverability of x: place boundedness: unboundedness: non-regularity ∃x ∈ Nk , x 0 − → x} regua place unboundedness erability, boundedness, is the some set h1,of5,j:1i lar? k x of>reachable k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω EF µ(j) > x(j) EFµ(j) > ω ues on . coordinate 2 fiularity, ..
b
b
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
(Valk and j=1 Vidal-Naquet, 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969) based on Rackoff (1978) ∗ ∗
(no: L ∩ (ab) c (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Graph a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 b
b =
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness of5,j:1i erability, boundedness, is the someset x of > reachable h1, lar? k k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω x(j)2 fiues EF onEFµ(j) coordinate ularity, .Vidal-Naquet, . .µ(j)>>ω (Valk and j=1 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969)
based on Rackoff (1978) (no: L ∩ (ab)∗ c∗ (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Graph a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 b
b =
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness of5,j:1i erability, boundedness, is the someset x of > reachable h1, lar? k k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω x(j)2 fiues EF onEFµ(j) coordinate ularity, .Vidal-Naquet, . .µ(j)>>ω (Valk and j=1 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969)
based on Rackoff (1978) (no: L ∩ (ab)∗ c∗ (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Graph a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 b
b =
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness of5,j:1i erability, boundedness, is the someset x of > reachable h1, lar? k k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω x(j)2 fiues EF onEFµ(j) coordinate ularity, .Vidal-Naquet, . .µ(j)>>ω (Valk and j=1 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969)
based on Rackoff (1978) (no: L ∩ (ab)∗ c∗ (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Graph a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 b
b =
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness of5,j:1i erability, boundedness, is the someset x of > reachable h1, lar? k k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω x(j)2 fiues EF onEFµ(j) coordinate ularity, .Vidal-Naquet, . .µ(j)>>ω (Valk and j=1 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969)
based on Rackoff (1978) (no: L ∩ (ab)∗ c∗ (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Graph I
finite abstraction of the VAS reachability graph
I
allows to decide various properties of the VAS (coverability, boundedness, place boundedness, regularity, reversal boundedness, trace boundedness, LTL model-checking, . . . )
I
but of non-primitive recursive size! (Cardoza et al., 1976)
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Graph I
finite abstraction of the VAS reachability graph
I
allows to decide various properties of the VAS (coverability, boundedness, place boundedness, regularity, reversal boundedness, trace boundedness, LTL model-checking, . . . )
I
but of non-primitive recursive size! (Cardoza et al., 1976)
Coverability Graphs
CTL Model Checking
Small Model Properties
Partial Cover a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 b
b =
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness of5,j:1i erability, boundedness, is the someset x of > reachable h1, lar? k k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω x(j)2 fiues EF onEFµ(j) coordinate ularity, .Vidal-Naquet, . .µ(j)>>ω (Valk and j=1 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969)
based on Rackoff (1978) (no: L ∩ (ab)∗ c∗ (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
PrECTL>(F) Syntax ϕ ::= > | ⊥ | ϕ ∨ ϕ | ϕ ∧ ϕ | EFψ ϕ | µ(j) > c with c ∈ N ∪ {ω} and ψ a QFP formula with k free variables
Semantics Over partial covers: s |= EFψ ϕ
a
i=1
s |= µ(j) > c
a
2 1 → · · · ∈ Paths(s), ∃n 6 |π|, → s1 − iff ∃π = s0 − n X ai ) and sn |= ϕ, PA |= ψ(
iff `(s)(j) > c .
Coverability Graphs
CTL Model Checking
Small Model Properties
PrECTL>(F) Syntax ϕ ::= > | ⊥ | ϕ ∨ ϕ | ϕ ∧ ϕ | EFψ ϕ | µ(j) > c with c ∈ N ∪ {ω} and ψ a QFP formula with k free variables
Semantics Over VAS: hV, x0 i |= ϕ if ∃ partial cover C s.t. C |= ϕ
Coverability Graphs
CTL Model Checking
Small Model Properties
Examples a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 b
b =
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness of5,j:1i erability, boundedness, is the someset x of > reachable h1, lar? k k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω x(j)2 fiues EF onEFµ(j) coordinate ularity, .Vidal-Naquet, . .µ(j)>>ω (Valk and j=1 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969)
based on Rackoff (1978) (no: L ∩ (ab)∗ c∗ (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Examples a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 b
b =
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness of5,j:1i erability, boundedness, is the someset x of > reachable h1, lar? k k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω x(j)2 fiues EF onEFµ(j) coordinate ularity, .Vidal-Naquet, . .µ(j)>>ω (Valk and j=1 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969)
based on Rackoff (1978) (no: L ∩ (ab)∗ c∗ (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Examples a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 b
b =
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness of5,j:1i erability, boundedness, is the someset x of > reachable h1, lar? k k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω x(j)2 fiues EF onEFµ(j) coordinate ularity, .Vidal-Naquet, . .µ(j)>>ω (Valk and j=1 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969)
based on Rackoff (1978) (no: L ∩ (ab)∗ c∗ (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Examples non-regularity: EF
_
_
^
I ⊆ {1, . . . , k} I,∅
I⊆J⊆{1,...,k}
ψI,J (x1 , . . . , xk ) =
µ(j) > ω ∧ EFψI,J >
j∈J
^ j∈I
xj < 0 ∧
^ j<J
xj > 0
Coverability Graphs
CTL Model Checking
Small Model Properties
Examples a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 b
b =
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness of5,j:1i erability, boundedness, is the someset x of > reachable h1, lar? k k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω x(j)2 fiues EF onEFµ(j) coordinate ularity, .Vidal-Naquet, . .µ(j)>>ω (Valk and j=1 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969)
based on Rackoff (1978) (no: L ∩ (ab)∗ c∗ (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
(Eventually) Increasing Formulæ EFx1 >0 (µ(2) > ω ∧ EFx1 >0∧x2 ∧ EFµ(1) > ω) 1, 0 < 6 h−1, −1i
0, ω h1, 0i
1, ω
I
h0, 1i
1, ω
(F) formulæ have finite tree models increasing formulæ eventually increasing formulæ (eiPrECTL> (F)): EFϕ where ϕ is increasing
Coverability Graphs
CTL Model Checking
Small Model Properties
(Eventually) Increasing Formulæ EFx1 >0 (µ(2) > ω ∧ EFx1 >0∧x2 ∧ EFµ(1) > ω) 1, 0 < 6 h−1, −1i
0, ω h1, 0i
1, ω
I
h0, 1i
1, ω
(F) formulæ have finite tree models increasing formulæ eventually increasing formulæ (eiPrECTL> (F)): EFϕ where ϕ is increasing
Coverability Graphs
CTL Model Checking
Small Model Properties
(Eventually) Increasing Formulæ EFx1 >0 (µ(2) > ω ∧ EFx1 >0∧x2 ∧ EFµ(1) > ω) 1, 0 < 6 h−1, −1i
0, ω h1, 0i
1, ω
I
h0, 1i
1, ω
(F) formulæ have finite tree models increasing formulæ eventually increasing formulæ (eiPrECTL> (F)): EFϕ where ϕ is increasing
Coverability Graphs
CTL Model Checking
Small Model Properties
Complexity Theorem The VAS model-checking problem for eiPrECTL> (F) formulæ is ES-complete. I
lower bound: coverability (Cardoza et al., 1976),
I
upper bound: small model (∼ 22
O(k)
·|V|·|ϕ|
)
Coverability Graphs
CTL Model Checking
Small Model Properties
Proof Idea (based on Rackoff, 1978)
Construct a small model by induction on i, 0 6 i 6 k: I allow negative values in coordinates j > i in models, I ignore coverability constraints µ(j) > c for j > i and c < ω (noted ϕ|i ) I called i-admissible models.
Coverability Graphs
CTL Model Checking
Small Model Properties
Small Bounded Models (based on Rackoff, 1978)
(i, r)-bounded partial cover: all finite values on coordinates 6 i are < r.
Lemma
C |= ϕ|i and C (i, r)-bounded imply ∃C 0 , C 0 |= ϕ|i with d |C 0 | 6 (2|V| r)(k+|ϕ|) for some constant d. (based on small solutions to QFP/LIP instances, e.g. Papadimitriou, 1981)
Coverability Graphs
CTL Model Checking
Small Model Properties
Main Induction (using ideas from Rackoff, 1978; Atig and Habermehl, 2011)
Small i-admissible model of size 6 g(i) regardless of initial state: I base i = 0: g(0) by reduction to LIP, I ind. step i + 1: set r = 2|V| · g(i) + 2|ϕ| I I
(i + 1, r)-bounded: use small bounded model, not (i + 1, r)-bounded kd
finally: g(k) 6 22
·|V|·|ϕ|
.
Coverability Graphs
CTL Model Checking
Small Model Properties
Main Induction (using ideas from Rackoff, 1978; Atig and Habermehl, 2011)
Small i-admissible model of size 6 g(i) regardless of initial state: I base i = 0: g(0) by reduction to LIP, I ind. step i + 1: set r = 2|V| · g(i) + 2|ϕ| I I
(i + 1, r)-bounded: use small bounded model, not (i + 1, r)-bounded kd
finally: g(k) 6 22
·|V|·|ϕ|
.
Coverability Graphs
CTL Model Checking
Small Model Properties
Main Induction (using ideas from Rackoff, 1978; Atig and Habermehl, 2011)
Small i-admissible model of size 6 g(i) regardless of initial state: I base i = 0: g(0) by reduction to LIP, I ind. step i + 1: set r = 2|V| · g(i) + 2|ϕ| I I
(i + 1, r)-bounded: use small bounded model, not (i + 1, r)-bounded kd
finally: g(k) 6 22
·|V|·|ϕ|
.
Coverability Graphs
CTL Model Checking
Case Not (i + 1, r)-Bounded |= EFϕ (i + 1, r)-bounded < r 6 `(s)(i + 1) < ω < 6
CTL Model Checking
Small Model Properties
Model Checking Coverability Graphs of Vector Addition Systems Michel Blockelet and Sylvain Schmitz LSV, ENS Cachan & CNRS, Cachan, France
MFCS 2011, Warsaw, August 25, 2011
Coverability Graphs
CTL Model Checking
Small Model Properties
Outline “coverability-like”-properties known ES-complete properties for VAS: coverability, boundedness, regularity, ...
this talk a unifying view based on VAS coverability graphs and CTL model checking
contents Coverability Graphs CTL Model Checking Small Model Properties
Coverability Graphs
CTL Model Checking
Small Model Properties
Vector Addition Systems S = hV, x0 i I V: a finite set of transitions in Zk , I x0 : an initial configuration in Nk I
a
semantics: for x, x 0 in Nk and a in V, x → − x 0 iff x + a = x0
Example S = h{a, b, c}, h1, 0, 1ii with transitions a = h1, 1, −1i, b = h−1, 0, 1i, and c = h0, −1, 0i: a
a
h1, 0, 1i → − h2, 1, 0i → − /
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Graph I
finite abstraction of the VAS reachability graph
I
allows to decide various properties of the VAS (coverability, boundedness, place boundedness, regularity, reversal boundedness, trace boundedness, LTL model-checking, . . . )
I
but of non-primitive recursive size! (Cardoza et al., 1976)
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Tree (Karp and Miller, 1969) a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 =
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness erability, boundedness, is the someset h1,of5,j:1i lar? k x of>reachable k _ is the set valplace boundedness, reg1,^ , 1of reachable reachable? configurations finite? EF µ(j) > ω EF µ(j) > x(j) EFµ(j) > ω ues on . coordinate 2 fiularity, ..
b
b
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
(Valk and j=1 Vidal-Naquet, 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969) based on Rackoff (1978) ∗ ∗
(no: L ∩ (ab) c (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Tree (Karp and Miller, 1969) a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 =
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness erability, boundedness, is the someset h1,of5,j:1i lar? k x of>reachable k _ is the set valplace boundedness, reg1,^ , 1of reachable reachable? configurations finite? EF µ(j) > ω EF µ(j) > x(j) EFµ(j) > ω ues on . coordinate 2 fiularity, ..
b
b
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
(Valk and j=1 Vidal-Naquet, 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969) based on Rackoff (1978) ∗ ∗
(no: L ∩ (ab) c (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Tree (Karp and Miller, 1969) a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 =
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness erability, boundedness, is the someset h1,of5,j:1i lar? k x of>reachable k _ is the set valplace boundedness, reg1,^ , 1of reachable reachable? configurations finite? EF µ(j) > ω EF µ(j) > x(j) EFµ(j) > ω ues on . coordinate 2 fiularity, ..
b
b
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
(Valk and j=1 Vidal-Naquet, 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969) based on Rackoff (1978) ∗ ∗
(no: L ∩ (ab) c (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Tree (Karp and Miller, 1969) a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 =
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” witness w for covcoverability of x: place boundedness: unboundedness: non-regularity ∃x ∈ Nk , x 0 − → x} regua place unboundedness erability, boundedness, is the some set h1,of5,j:1i lar? k x of>reachable k _ is the set val^ place boundedness, reg1, 1, of 1 reachable reachable? configurations finite? EF µ(j) > ω EF µ(j) > x(j) EFµ(j) > ω ues on . coordinate 2 fiularity, ..
b
b
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
(Valk and j=1 Vidal-Naquet, 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969) based on Rackoff (1978) ∗ ∗
(no: L ∩ (ab) c (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Tree (Karp and Miller, 1969) a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 =
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” witness w for covcoverability of x: place boundedness: unboundedness: non-regularity ∃x ∈ Nk , x 0 − → x} regua place unboundedness erability, boundedness, is the some set h1,of5,j:1i lar? k x of>reachable k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω EF µ(j) > x(j) EFµ(j) > ω ues on . coordinate 2 fiularity, ..
b
b
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
(Valk and j=1 Vidal-Naquet, 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969) based on Rackoff (1978) ∗ ∗
(no: L ∩ (ab) c (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Tree (Karp and Miller, 1969) a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 =
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” witness w for covcoverability of x: place boundedness: unboundedness: non-regularity ∃x ∈ Nk , x 0 − → x} regua place unboundedness erability, boundedness, is the some set h1,of5,j:1i lar? k x of>reachable k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω EF µ(j) > x(j) EFµ(j) > ω ues on . coordinate 2 fiularity, ..
b
b
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
(Valk and j=1 Vidal-Naquet, 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969) based on Rackoff (1978) ∗ ∗
(no: L ∩ (ab) c (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Tree (Karp and Miller, 1969) a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 =
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” witness w for covcoverability of x: place boundedness: unboundedness: non-regularity ∃x ∈ Nk , x 0 − → x} regua place unboundedness erability, boundedness, is the some set h1,of5,j:1i lar? k x of>reachable k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω EF µ(j) > x(j) EFµ(j) > ω ues on . coordinate 2 fiularity, ..
b
b
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
(Valk and j=1 Vidal-Naquet, 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969) based on Rackoff (1978) ∗ ∗
(no: L ∩ (ab) c (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Tree (Karp and Miller, 1969) a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 =
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” witness w for covcoverability of x: place boundedness: unboundedness: non-regularity ∃x ∈ Nk , x 0 − → x} regua place unboundedness erability, boundedness, is the some set h1,of5,j:1i lar? k x of>reachable k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω EF µ(j) > x(j) EFµ(j) > ω ues on . coordinate 2 fiularity, ..
b
b
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
(Valk and j=1 Vidal-Naquet, 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969) based on Rackoff (1978) ∗ ∗
(no: L ∩ (ab) c (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Graph (Valk and Vidal-Naquet, 1981) a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 =
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” witness w for covcoverability of x: place boundedness: unboundedness: non-regularity ∃x ∈ Nk , x 0 − → x} regua place unboundedness erability, boundedness, is the some set h1,of5,j:1i lar? k x of>reachable k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω EF µ(j) > x(j) EFµ(j) > ω ues on . coordinate 2 fiularity, ..
b
b
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
(Valk and j=1 Vidal-Naquet, 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969) based on Rackoff (1978) ∗ ∗
(no: L ∩ (ab) c (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Graph a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 b
b =
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness of5,j:1i erability, boundedness, is the someset x of > reachable h1, lar? k k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω x(j)2 fiues EF onEFµ(j) coordinate ularity, .Vidal-Naquet, . .µ(j)>>ω (Valk and j=1 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969)
based on Rackoff (1978) (no: L ∩ (ab)∗ c∗ (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Graph a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 b
b =
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness of5,j:1i erability, boundedness, is the someset x of > reachable h1, lar? k k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω x(j)2 fiues EF onEFµ(j) coordinate ularity, .Vidal-Naquet, . .µ(j)>>ω (Valk and j=1 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969)
based on Rackoff (1978) (no: L ∩ (ab)∗ c∗ (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Graph a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 b
b =
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness of5,j:1i erability, boundedness, is the someset x of > reachable h1, lar? k k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω x(j)2 fiues EF onEFµ(j) coordinate ularity, .Vidal-Naquet, . .µ(j)>>ω (Valk and j=1 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969)
based on Rackoff (1978) (no: L ∩ (ab)∗ c∗ (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Graph a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 b
b =
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness of5,j:1i erability, boundedness, is the someset x of > reachable h1, lar? k k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω x(j)2 fiues EF onEFµ(j) coordinate ularity, .Vidal-Naquet, . .µ(j)>>ω (Valk and j=1 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969)
based on Rackoff (1978) (no: L ∩ (ab)∗ c∗ (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Graph I
finite abstraction of the VAS reachability graph
I
allows to decide various properties of the VAS (coverability, boundedness, place boundedness, regularity, reversal boundedness, trace boundedness, LTL model-checking, . . . )
I
but of non-primitive recursive size! (Cardoza et al., 1976)
Coverability Graphs
CTL Model Checking
Small Model Properties
Coverability Graph I
finite abstraction of the VAS reachability graph
I
allows to decide various properties of the VAS (coverability, boundedness, place boundedness, regularity, reversal boundedness, trace boundedness, LTL model-checking, . . . )
I
but of non-primitive recursive size! (Cardoza et al., 1976)
Coverability Graphs
CTL Model Checking
Small Model Properties
Partial Cover a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 b
b =
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness of5,j:1i erability, boundedness, is the someset x of > reachable h1, lar? k k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω x(j)2 fiues EF onEFµ(j) coordinate ularity, .Vidal-Naquet, . .µ(j)>>ω (Valk and j=1 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969)
based on Rackoff (1978) (no: L ∩ (ab)∗ c∗ (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
PrECTL>(F) Syntax ϕ ::= > | ⊥ | ϕ ∨ ϕ | ϕ ∧ ϕ | EFψ ϕ | µ(j) > c with c ∈ N ∪ {ω} and ψ a QFP formula with k free variables
Semantics Over partial covers: s |= EFψ ϕ
a
i=1
s |= µ(j) > c
a
2 1 → · · · ∈ Paths(s), ∃n 6 |π|, → s1 − iff ∃π = s0 − n X ai ) and sn |= ϕ, PA |= ψ(
iff `(s)(j) > c .
Coverability Graphs
CTL Model Checking
Small Model Properties
PrECTL>(F) Syntax ϕ ::= > | ⊥ | ϕ ∨ ϕ | ϕ ∧ ϕ | EFψ ϕ | µ(j) > c with c ∈ N ∪ {ω} and ψ a QFP formula with k free variables
Semantics Over VAS: hV, x0 i |= ϕ if ∃ partial cover C s.t. C |= ϕ
Coverability Graphs
CTL Model Checking
Small Model Properties
Examples a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 b
b =
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness of5,j:1i erability, boundedness, is the someset x of > reachable h1, lar? k k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω x(j)2 fiues EF onEFµ(j) coordinate ularity, .Vidal-Naquet, . .µ(j)>>ω (Valk and j=1 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969)
based on Rackoff (1978) (no: L ∩ (ab)∗ c∗ (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Examples a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 b
b =
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness of5,j:1i erability, boundedness, is the someset x of > reachable h1, lar? k k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω x(j)2 fiues EF onEFµ(j) coordinate ularity, .Vidal-Naquet, . .µ(j)>>ω (Valk and j=1 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969)
based on Rackoff (1978) (no: L ∩ (ab)∗ c∗ (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Examples a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 b
b =
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness of5,j:1i erability, boundedness, is the someset x of > reachable h1, lar? k k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω x(j)2 fiues EF onEFµ(j) coordinate ularity, .Vidal-Naquet, . .µ(j)>>ω (Valk and j=1 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969)
based on Rackoff (1978) (no: L ∩ (ab)∗ c∗ (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
Examples non-regularity: EF
_
_
^
I ⊆ {1, . . . , k} I,∅
I⊆J⊆{1,...,k}
ψI,J (x1 , . . . , xk ) =
µ(j) > ω ∧ EFψI,J >
j∈J
^ j∈I
xj < 0 ∧
^ j<J
xj > 0
Coverability Graphs
CTL Model Checking
Small Model Properties
Examples a = h1, 1, −1i, b = h−1, 0, 1i, c = h0, −1, 0i:
regularity: 1, 0, 1 b
b =
a
2, 1, 0 b
< a
0, 0, 2
c
2, 0, 0
1, 1,ω, 1, 11 aab
b
2, ω, 0
1, 0, 1 c b
1, ω, 1
c
2, ω, 0
c
c a bb
= c
0, ω, 1 2 1, a
1, ω, 1
c c
Idea is the of set the L = paper: {w ∈ V∗ a| coverability: boundedness: “small” w for covcoverability of x: place boundedness: unboundedness: kwitness non-regularity ∃x ∈ N , x − → x} regu0 a place unboundedness of5,j:1i erability, boundedness, is the someset x of > reachable h1, lar? k k _ is the set of val^ place reg1,boundedness, ω, 1 reachable reachable? configurations finite? EF µ(j) > ω x(j)2 fiues EF onEFµ(j) coordinate ularity, .Vidal-Naquet, . .µ(j)>>ω (Valk and j=1 1981) 1, ω, 1 j=1 nite? (Karp and Miller, 1969)
based on Rackoff (1978) (no: L ∩ (ab)∗ c∗ (ab) c )
0, ω, 2 n 6n
=
Coverability Graphs
CTL Model Checking
Small Model Properties
(Eventually) Increasing Formulæ EFx1 >0 (µ(2) > ω ∧ EFx1 >0∧x2 ∧ EFµ(1) > ω) 1, 0 < 6 h−1, −1i
0, ω h1, 0i
1, ω
I
h0, 1i
1, ω
(F) formulæ have finite tree models increasing formulæ eventually increasing formulæ (eiPrECTL> (F)): EFϕ where ϕ is increasing
Coverability Graphs
CTL Model Checking
Small Model Properties
(Eventually) Increasing Formulæ EFx1 >0 (µ(2) > ω ∧ EFx1 >0∧x2 ∧ EFµ(1) > ω) 1, 0 < 6 h−1, −1i
0, ω h1, 0i
1, ω
I
h0, 1i
1, ω
(F) formulæ have finite tree models increasing formulæ eventually increasing formulæ (eiPrECTL> (F)): EFϕ where ϕ is increasing
Coverability Graphs
CTL Model Checking
Small Model Properties
(Eventually) Increasing Formulæ EFx1 >0 (µ(2) > ω ∧ EFx1 >0∧x2 ∧ EFµ(1) > ω) 1, 0 < 6 h−1, −1i
0, ω h1, 0i
1, ω
I
h0, 1i
1, ω
(F) formulæ have finite tree models increasing formulæ eventually increasing formulæ (eiPrECTL> (F)): EFϕ where ϕ is increasing
Coverability Graphs
CTL Model Checking
Small Model Properties
Complexity Theorem The VAS model-checking problem for eiPrECTL> (F) formulæ is ES-complete. I
lower bound: coverability (Cardoza et al., 1976),
I
upper bound: small model (∼ 22
O(k)
·|V|·|ϕ|
)
Coverability Graphs
CTL Model Checking
Small Model Properties
Proof Idea (based on Rackoff, 1978)
Construct a small model by induction on i, 0 6 i 6 k: I allow negative values in coordinates j > i in models, I ignore coverability constraints µ(j) > c for j > i and c < ω (noted ϕ|i ) I called i-admissible models.
Coverability Graphs
CTL Model Checking
Small Model Properties
Small Bounded Models (based on Rackoff, 1978)
(i, r)-bounded partial cover: all finite values on coordinates 6 i are < r.
Lemma
C |= ϕ|i and C (i, r)-bounded imply ∃C 0 , C 0 |= ϕ|i with d |C 0 | 6 (2|V| r)(k+|ϕ|) for some constant d. (based on small solutions to QFP/LIP instances, e.g. Papadimitriou, 1981)
Coverability Graphs
CTL Model Checking
Small Model Properties
Main Induction (using ideas from Rackoff, 1978; Atig and Habermehl, 2011)
Small i-admissible model of size 6 g(i) regardless of initial state: I base i = 0: g(0) by reduction to LIP, I ind. step i + 1: set r = 2|V| · g(i) + 2|ϕ| I I
(i + 1, r)-bounded: use small bounded model, not (i + 1, r)-bounded kd
finally: g(k) 6 22
·|V|·|ϕ|
.
Coverability Graphs
CTL Model Checking
Small Model Properties
Main Induction (using ideas from Rackoff, 1978; Atig and Habermehl, 2011)
Small i-admissible model of size 6 g(i) regardless of initial state: I base i = 0: g(0) by reduction to LIP, I ind. step i + 1: set r = 2|V| · g(i) + 2|ϕ| I I
(i + 1, r)-bounded: use small bounded model, not (i + 1, r)-bounded kd
finally: g(k) 6 22
·|V|·|ϕ|
.
Coverability Graphs
CTL Model Checking
Small Model Properties
Main Induction (using ideas from Rackoff, 1978; Atig and Habermehl, 2011)
Small i-admissible model of size 6 g(i) regardless of initial state: I base i = 0: g(0) by reduction to LIP, I ind. step i + 1: set r = 2|V| · g(i) + 2|ϕ| I I
(i + 1, r)-bounded: use small bounded model, not (i + 1, r)-bounded kd
finally: g(k) 6 22
·|V|·|ϕ|
.
Coverability Graphs
CTL Model Checking
Case Not (i + 1, r)-Bounded |= EFϕ (i + 1, r)-bounded < r 6 `(s)(i + 1) < ω < 6
