KR 2014 - Montali - State-Boundedness in Data-Aware Dynamic Systems

State-Boundedness in Data-Aware Dynamic Systems
Marco Montali
KRDB Research Centre for Knowledge and Data
Free University of Bozen-Bolzano
Joint work with: B. Bagheri Hariri, D. Calvanese, A. Deutsch
KR 2014
Marco Montali (unibz) State-Boundedness KR 2014 1 / 26

Data-Aware Dynamic System
A dynamic system that manipulate data over time.
Recall the VSL keynote by F. Baader.
Data-aware dynamic system
Data Layer
Process Layer external
world
UpdateRead
• Data layer: maintains data of interest.
Relational database.
(Description logic) ontology.
• Process layer: evolves the extensional part of the data layer.
Control-ﬂow component: determines when actions can be executed.
Actions: atomic units of work that update the data.
Interact with the external world to inject fresh data into the system.

Data-Centric Dynamic Systems (DCDSs)
• Data layer: relational database with FO constraints.
• Process (control-flow): condition-action rules.
• Actions: specified by (parallel) effects that query the current state
and determine the next state.
Service calls can be invoked to get new data.
Example
• Data layer: schema {R/2, Q/1, S/1}, no constraint.
• Process: ∃y.R(x, y) → t(x).
• Action: t(p) :



R(x, y) ∧ x = p R(x, y)
R(p, y) R(p, f (y))
R(p, y) Q(p)




Example
• Action: t(p) :



R(x, y) ∧ x = p R(x, y)
R(p, y) R(p, f (y))
R(p, y) Q(p)



R(a,b), R(a,c),
R(c,d),
Q(d),
S(b)

Example
• Action: t(p) :



R(x, y) ∧ x = p R(x, y)
R(p, y) R(p, f (y))
R(p, y) Q(p)



R(a,b), R(a,c),
R(c,d),
Q(d),
S(b)
t(a)
t(c)

Example
• Action: t(a) :



R(x, y) ∧ x = a R(x, y)
R(a, y) R(a, f (y))
R(a, y) Q(a)



R(a,b), R(a,c),
R(c,d),
Q(d),
S(b)
R(c,d), Q(a),
R(a,f(b)) R(a,f(c))
t(a)

Example
• Action: t(a) :



R(x, y) ∧ x = a R(x, y)
R(a, y) R(a, f (y))
R(a, y) Q(a)



R(a,b), R(a,c),
R(c,d),
Q(d),
S(b)
R(c,d), Q(a),
R(a,f(b)) R(a,f(c))
R(c,d), Q(a),
R(a,a)t(a)
f(b) = f(c) = a

Example
• Action: t(a) :



R(x, y) ∧ x = a R(x, y)
R(a, y) R(a, f (y))
R(a, y) Q(a)



R(a,b), R(a,c),
R(c,d),
Q(d),
S(b)
R(c,d), Q(a),
R(a,f(b)) R(a,f(c)) R(c,d), Q(a),
R(a,c), R(a,u)
t(a) f(b) = c,
f(c) = u

Example
• Action: t(a) :



R(x, y) ∧ x = a R(x, y)
R(a, y) R(a, f (y))
R(a, y) Q(a)



R(a,b), R(a,c),
R(c,d),
Q(d),
S(b)
R(c,d), Q(a),
R(a,f(b)) R(a,f(c))
. . .
t(a)
. . .

Verification
Execution semantics: a relational
transition system, possibly with
infinitely many states.
... .
.
.
... .
.
.
. . .
. . .
Verification Problem
Check whether the dynamic system guarantees a desired property,
expressed in some variant of a first-order temporal logic.

Verification
Execution semantics: a relational
transition system, possibly with
infinitely many states.
... .
.
.
... .
.
.
. . .
. . .
Verification Problem
Check whether the dynamic system guarantees a desired property,
expressed in some variant of a first-order temporal logic.
Undecidable in general!
(Even for propositional temporal logics)

State-boundedness
State-Bounded System
Has an overall bound on the number of
individuals stored in each single state of
the system.
Structurally State-Bounded
State-Bounded for any given initial database.
...
...
. . .
. . .

State-boundedness
State-Bounded System
Has an overall bound on the number of
individuals stored in each single state of
the system.
Structurally State-Bounded
State-Bounded for any given initial database.
...
...
. . .
. . .
Decidability under state-boundedness
Shown in a plethora of recent works, for a variety of dynamic systems:
• Artifact-centric MASs, and FO-CTLK [BelardinelliEtAl-KR12].
• DCDSs and µLp.
• DL-based Dynamic Systems, and µLECQ
p [CalvaneseEtAl-RR13].
• Data-aware MASs with commitments, and µLECQ
p [MontaliEtAl-AAMAS14].
• Bounded situation calculus action theories, and µL [DeGiacomoEtAl-KR12].

Checking State-Boundedness
State-boundedness is a semantic property
Typically, assumed to hold.
Theorem ([BagheriHaririEtAl-PODS13])
Checking whether a DCDS is state-bounded is undecidable.
study of suﬃcient, syntactic conditions guaranteeing state-boundedness.

Central question
Do there exist signiﬁcant classes of data-aware dynamic systems for
which checking state-boundedness is decidable?

Central question
A similar question has been extensively studied in a diﬀerent setting . . .

P/T nets
• Introduced by Carl Adam Petri in his PhD thesis (1962).
• Extensively used for modelling concurrent systems:
Distributed systems, workﬂows, business processes, . . .
• Study of several formal properties: reachability, deadlock
freedom, boundedness.
a b
c
d
e
f
g

P/T nets - The Good
• Introduced by Carl Adam Petri in his PhD thesis (1962).
• Extensively used for modelling concurrent systems:
Distributed systems, workﬂows, business processes, . . .
• Study of several formal properties: reachability, deadlock
freedom, boundedness.
a b
c
d
e
f
g

P/T Nets - The Bad
Not all marked Petri nets are bounded.
a
b c

Forms of Boundedness
Boundedness
A marked Petri net is bounded
if all executions starting from the
given marking do not produce an
unbounded amount of tokens.
Structural boundedness
A Petri net is structurally
bounded if for all possible initial
markings the resulting marked
net is bounded.

Forms of Boundedness
Boundedness
A marked Petri net is bounded
if all executions starting from the
given marking do not produce an
unbounded amount of tokens.
Bounded
a
b c
Structural boundedness
A Petri net is structurally
bounded if for all possible initial
markings the resulting marked
net is bounded.
Structurally unbounded
a
b c

Reset Transfer Nets - The Ugly
P/T nets ⊂ RT nets ⊂ Reset Post G-nets [DufourEtAlICALP98]
P0 a
P1
P2
b P3
c
P4
P2
P2
P2
Reset arc
P t
P
When t ﬁres, all tokens in P are removed.
Transfer pair
P t Q
P P
When t ﬁres, all tokens in P are transferred to Q.

Boundedness Spectrum
boundedness
structural
boundedness
RT nets
T nets
R nets
P/T nets
RT nets
T nets R nets
P/T nets

boundedness
structural
boundedness
RT nets
T nets
R nets
P/T nets
RT nets
T nets R nets
P/T nets
Undec.

boundedness
structural
boundedness
RT nets
T nets
R nets
P/T nets
RT nets
T nets R nets
P/T nets
Undec.
Dec.
ExpSpace
PTime

Understanding State-Boundedness
Goal
Devise a connection between RT nets and DCDSs so as to understand the
state-boundedness spectrum in data-aware dynamic systems.
Main issue: set vs bag semantics.
RT nets
DCDSs

Goal
RT nets
DCDSs
LRT DCDSs

Goal
RT nets
bounded
structurally
bounded
DCDSs
LRT DCDSs
state bounded
structurally
state bounded

From RT Nets to DCDSs
P0 a
P1
P2
b P3
c
P4
P2
P2
P2
Idea
Tokens as distinct identifiers distributed over place relations.
Only cardinalities matter, not the data values.
Data layer.
• Unary relations for places: {Pi/1 | i ∈ {1, . . . , 4}}
• No constraints.
Process layer: each transition becomes an action + condition-action rule.
• Condition: gets tokens from input places; feeds the action with them.
• Action: moves tokens according to the firing semantics of the net.
Service calls to generate identifiers for new tokens.

From RT Nets to DCDSs - The P/T Case
P0 a
P1
P2
b P3
c
P4
P2
P2
P2
P0(x1) → a(x1)

P0 a
P1
P2
b P3
c
P4
P2
P2
P2
P0(x1) → a(x1)
a(x1) :



P0 P0(y) ∧ y = x1 P0(f (y))
P1
P2
P3
P4




P0 a
P1
P2
b P3
c
P4
P2
P2
P2
P0(x1) → a(x1)
a(x1) :



P0 P0(y) ∧ y = x1 P0(f (y))
P1 P1(y) P1(h1(y))
true P1(g1())
P2 P2(y) P2(h2(y))
true P2(g2())
P3
P4




P0 a
P1
P2
b P3
c
P4
P2
P2
P2
P0(x1) → a(x1)
a(x1) :



P0 P0(y) ∧ y = x1 P0(f (y))
P1 P1(y) P1(h1(y))
true P1(g1())
P2 P2(y) P2(h2(y))
true P2(g2())
P3 P3(y) P3(h3(y))
P4 P4(y) P4(h4(y))




From RT Nets to DCDSs - The Reset Case
P0 a
P1
P2
b P3
c
P4
P2
P2
P2
P1(x1) ∧ P3(x2) → b(x1, x2)

P0 a
P1
P2
b P3
c
P4
P2
P2
P2
P1(x1) ∧ P3(x2) → b(x1, x2)
b(x1, x2) :



P0 P0(y) P0(h0(y))
P1
P2
P3
P4




P0 a
P1
P2
b P3
c
P4
P2
P2
P2
P1(x1) ∧ P3(x2) → b(x1, x2)
b(x1, x2) :



P0 P0(y) P0(h0(y))
P1 P1(y) ∧ y = x1 P1(h1(y))
P2
P3 P3(y) ∧ y = x2 P3(h3(y))
P4




P0 a
P1
P2
b P3
c
P4
P2
P2
P2
P1(x1) ∧ P3(x2) → b(x1, x2)
b(x1, x2) :



P0 P0(y) P0(h0(y))
P1 P1(y) ∧ y = x1 P1(h1(y))
P2 –
P3 P3(y) ∧ y = x2 P3(h3(y))
P4




P0 a
P1
P2
b P3
c
P4
P2
P2
P2
P1(x1) ∧ P3(x2) → b(x1, x2)
b(x1, x2) :



P0 P0(y) P0(h0(y))
P1 P1(y) ∧ y = x1 P1(h1(y))
P2 –
P3 P3(y) ∧ y = x2 P3(h3(y))
P4 P4(y) P4(h4(y))
true P4(g4())




P0 a
P1
P2
b P3
c
P4
P2
P2
P2
P1(x1) ∧ P3(x2) → b(x1, x2)
b(x1, x2) :



P0 P0(y) P0(h0(y))
P1 P1(y) ∧ y = x1 P1(h1(y))
P2 –
P3 P3(y) ∧ y = x2 P3(h3(y))
P4 P4(y) P4(h4(y))
true P4(g4())



A relation (P2) does not appear in the action!

From RT Nets to DCDSs - The Transfer Case
P0 a
P1
P2
b P3
c
P4
P2
P2
P2
P1(x1) → c(x1)

P0 a
P1
P2
b P3
c
P4
P2
P2
P2
P1(x1) → c(x1)
c(x1) :



P0 P0(y) P0(h0(y))
P1
P2
P3 P3(y) P3(h3(y))
P4




P0 a
P1
P2
b P3
c
P4
P2
P2
P2
P1(x1) → c(x1)
c(x1) :



P0 P0(y) P0(h0(y))
P1 P1(y) ∧ y = x1 P1(h1(y))
P2
P3 P3(y) P3(h3(y))
P4




P0 a
P1
P2
b P3
c
P4
P2
P2
P2
P1(x1) → c(x1)
c(x1) :



P0 P0(y) P0(h0(y))
P1 P1(y) ∧ y = x1 P1(h1(y))
P2 P2(y) P4(h2(y))
P3 P3(y) P3(h3(y))
P4




P0 a
P1
P2
b P3
c
P4
P2
P2
P2
P1(x1) → c(x1)
c(x1) :



P0 P0(y) P0(h0(y))
P1 P1(y) ∧ y = x1 P1(h1(y))
P2 P2(y) P4(h2(y))
P3 P3(y) P3(h3(y))
P4 P4(y) P4(h4(y))




P0 a
P1
P2
b P3
c
P4
P2
P2
P2
P1(x1) → c(x1)
c(x1) :



P0 P0(y) P0(h0(y))
P1 P1(y) ∧ y = x1 P1(h1(y))
P2 P2(y) P4(h2(y))
P3 P3(y) P3(h3(y))
P4 P4(y) P4(h4(y))



There is an eﬀect involving two diﬀerent relations (P2 and P4)!

Is the Translation Correct?
NO! The resulting DCDS has a lossy behavior.

NO! The resulting DCDS has a lossy behavior.
Petri net
P0
. . . t . . . t
=⇒
P0
. . . t . . .
DCDS
t(. . .) : {. . . , P0(y) P0(h0(y)), . . .}
P0
. . . t . . .a
b c t(...)
=⇒
h0(a) = a, h0(b) = b, h0(c) = c
P0
. . . t . . .a
b c
h0(a) = a, h0(b) = a, h0(c) = a
P0
. . . t . . .a
h0(a) = a, h0(b) = a, h0(c) = c
P0
. . . t . . .a c
. . .

However. . .
The resulting DCDS reproduces all behaviors of the net (and more).

However. . .
The resulting DCDS reproduces all behaviors of the net (and more).
Theorem
An RT net is (structurally) bounded if and only if the corresponding
DCDS is (structurally) state-bounded.

LRT DCDS
Data Layer
Schema R with unary relations only, and no constraint.
Process
Only one rule per action, of the form Q(x) → α(x), where
Q(x1, . . . , xn) =
i∈{1,...,n}, Pi=Pj for i=j
Pi(xi)
Shape of action α(x)
For each Pi ∈ rels(Q), α must contain:
• Pi(y) ∧ y = xi Pi(fi(y))
and may contain:
• true Pi(gi())
For each Pl ∈ R rels(Q), α may contain:
• Pl(y) Pl(hl(y))
• either true Pj(gj()),
or Pj (y) Pj(hj(y))
for some Pj ∈ R (rels(Q) ∪ Pj).

From LRT DCDSs to RT Nets
Consider schema R = {P0, P1, P2, P3, P4}, and action t with:
• process condition-action rule P0(x0) ∧ P1(x1) → t(x0, x1)
• action t(x0, x1) :



P0(y) ∧ y = x0 P0(f0(y))
P1(y) ∧ y = x1 P1(f1(y))
true P1(g1())
true P2(g2())
P3(y) P4(h3(y))
P4(y) P4(h4(y))







P0(y) ∧ y = x0 P0(f0(y))
P1(y) ∧ y = x1 P1(f1(y))
true P1(g1())
true P2(g2())
P3(y) P4(h3(y))
P4(y) P4(h4(y))



P0
P1
t
P2
P3
P4




P0(y) ∧ y = x0 P0(f0(y))
P1(y) ∧ y = x1 P1(f1(y))
true P1(g1())
true P2(g2())
P3(y) P4(h3(y))
P4(y) P4(h4(y))



P0
P1
t
P2
P3
P4
P2




P0(y) ∧ y = x0 P0(f0(y))
P1(y) ∧ y = x1 P1(f1(y))
true P1(g1())
true P2(g2())
P3(y) P4(h3(y))
P4(y) P4(h4(y))



P0
P1
t
P2
P3
P4
P2
P3
P3




P0(y) ∧ y = x0 P0(f0(y))
P1(y) ∧ y = x1 P1(f1(y))
true P1(g1())
true P2(g2())
P3(y) P4(h3(y))
P4(y) P4(h4(y))



P0
P1
t
P2
P3
P4
P2
P3
P3
Theorem
An LRT DCDS is (structurally)
state-bounded if and only if the
corresponding RT net is
(structurally) bounded.

State-Boundedness Spectrum
state
boundedness
structural state
boundedness
LRT DCDSs
LT DCDSs
LR DCDSs
LP DCDSs
LRT DCDSs
LT DCDSs LR DCDSs
LP DCDSs
Undec.
Dec.
ExpSpace
PTime

Take Home Message
LRT DCDSs are weak:
• Only unary relations.
• Only conjunctions without joins in conditions.
• Only atomic queries inside eﬀects (possibly with a value inequality).
• Very limited use of negation (inequalities).
• No direct transfer of values from one state to the other.

Take Home Message
LRT DCDSs are weak:
• Only unary relations.
• Only conjunctions without joins in conditions.
• Only atomic queries inside effects (possibly with a value inequality).
• Very limited use of negation (inequalities).
• No direct transfer of values from one state to the other.
Still, to ensure that (structural) state-boundedness is decidable . . .
Boundedness
All relations must appear on the left-hand side of action effects, i.e.,
contribute to form the new state.
Structural Boundedness
Each action must be such that only a fixed amount of tuples is added
to/removed from each relation in the schema.

Back to our Question
Central question
Answer
NO
Hence, it becomes important to provide signiﬁcant suﬃcient, checkable
syntactic conditions that guarantee structural state-boundedness.

Back to our Question
Central question
Answer
NO
Hence, it becomes important to provide signiﬁcant suﬃcient, checkable
syntactic conditions that guarantee structural state-boundedness.
We follow this line, focusing on DCDSs and starting from
[BagheriHaririEtAl-PODS13].

GR-Acyclicity [BagheriHaririEtAl-PODS13]
Example
Consider a DCDS with process {true → α()}, and
α() :



P(x) P(x)
P(x) Q(f (x))
Q(x) Q(x)




GR-Acyclicity [BagheriHaririEtAl-PODS13]
Example
Consider a DCDS with process {true → α()}, and
α() :



P(x) P(x)
P(x) Q(f (x))
Q(x) Q(x)



We approximate the DCDS data-ﬂow through a dependency graph.
P,1 Q,1*
The system is not state-bounded, due to:
• a generate cycle that continuously feeds a path issuing service calls;
• a recall cycle that accumulates the obtained results.
• (+ the fact that both cycles are simultaneously active)
GR-acycliclity detects exactly these undesired situations.

Our Contribution
GR
GR+
in CoNP in Σp
2

Our Contribution
GR
GR+
GR++
Safe-GR++
Stratiﬁed-GR++
∃GR++
Safe-∃GR++
Stratiﬁed-∃GR++
in CoNP in Σp
2

Conclusion
1.
No significant decidable classes of data-aware dynamic systems for which
state-boundedness is decidable.
2.
It becomes crucial to provide checkable, sufficient conditions.
• We have built on results on chase termination for tuple-generating
dependencies, providing a family of conditions for DCDSs.
Ongoing and future work
• Refine the syntactic conditions to handle if-then-else effects.
• Follow a different approach: provide modelling guidelines towards
systems that are structurally state bounded by design.
Preliminary results in [SolomakhinEtAl-ICSOC13].

KR 2014 - Montali - State-Boundedness in Data-Aware Dynamic Systems

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