Sep 15, 2009 - RTL: Properties and Decidability. Conclusions and Future Work. Nils Bulling, Berndt Farwer v Properties of Resource-Bounded Systems. FAUSt ...
Expressing Properties of Resource-Bounded Systems: The Logics RTL and RTL∗ Nils Bulling1
Berndt Farwer2
Department of Informatics, Clausthal University of Technology, Germany School of Engineering and Computing Sciences, Durham University, UK
FAUSt 2009, London, 15 september 2009
Nils Bulling, Berndt Farwer · Properties of Resource-Bounded Systems
FAUSt 2009 1/24
Outline Introduction Modelling Resource-Bounded Systems RTL∗ : Properties and Decidability RTL: Properties and Decidability Conclusions and Future Work
Nils Bulling, Berndt Farwer · Properties of Resource-Bounded Systems
FAUSt 2009 2/24
1. Introduction
Motivation �
Ubiquitous systems need a notion of resources and location �
Here: focus on resources
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Formal approaches to verification, e.g. of agent systems.
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Resource-Bounded Tree Logics RTL and RTL∗ , based on Computation Tree Logics CTL and CTL∗
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Reasoning about computations in the presence of resources
Nils Bulling, Berndt Farwer · Properties of Resource-Bounded Systems
FAUSt 2009 4/24
1. Introduction
Main Idea
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Conservative extension of Computation Tree Logic CTL∗
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Replace Eγ by �ρ�γ �
ρ: set of available resources
Nils Bulling, Berndt Farwer · Properties of Resource-Bounded Systems
FAUSt 2009 5/24
2. Modelling Resource-Bounded Systems Resource-Bounded Agents
Resource-Bounded Agents
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A resource-bounded agent has at its disposal a (limited) repository of resources.
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Performing actions reduces some resources and may produce others; �
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an agent might not always be able to perform all of its available actions.
Single agent: activation or deactivation of transitions.
Nils Bulling, Berndt Farwer · Properties of Resource-Bounded Systems
FAUSt 2009 7/24
2. Modelling Resource-Bounded Systems Resource-Bounded Agents
Definition (Resource-bounded model) A resource-bounded model (RBM): M = (Q, →, Props, π, R, t) �
Q, R, and Props are finite sets of states, resources, and propositions, respectively;
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(Q, →, Props, π) is a Kripke model; and
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t : Q × Q → R⊕ × R⊕ is a (partial) resource function.
Instead of t(q, q � ) we sometimes write tq,q� and for tq,q� = (c, p) we use • tq,q� (resp. t •q,q� ) to refer to c (resp. p).
Nils Bulling, Berndt Farwer · Properties of Resource-Bounded Systems
FAUSt 2009 8/24
2. Modelling Resource-Bounded Systems Resource-Bounded Agents
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A path is ρ-feasible if each transition in the sequence can be executed with the available resources.
Definition (ρ-feasible path) Let M be an RBM and ρ ∈ R± be a resource-quantity set. A path λ = q1 q2 q3 · · · ∈ ΛM (q) is called ρ-feasible if for all i ∈ N the resource-quantity set � � • • • ρ + Σi−1 j=1 (t qj qj+1 − tqj qj+1 ) |• tqi qi+1 − tqi qi+1 is feasible.
Nils Bulling, Berndt Farwer · Properties of Resource-Bounded Systems
FAUSt 2009 9/24
2. Modelling Resource-Bounded Systems Resource-Bounded Tree Logic
The Logic RTL∗ Definition ((Full) Resource-Bounded Tree Logic RTL∗ ) Let R be a set of resources and Props a set of propositions. Formulae of RTL∗ are defined by the following grammar: ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | �ρ�γ where
γ ::= ϕ | ¬γ | γ ∧ γ | ϕ U ϕ | ❣ϕ
with p ∈ Props and ρ ∈ R± .
�ρ�γ means that there is a ρ-feasible path λ on which γ holds.
Nils Bulling, Berndt Farwer · Properties of Resource-Bounded Systems
FAUSt 2009 10/24
2. Modelling Resource-Bounded Systems Resource-Bounded Tree Logic
The Logic RTL RTL is the fragment of RTL∗ in which each temporal operator is immediately preceded by a path quantifier.
Definition (Resource-Bounded Tree Logic RTL) ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | �ρ� ❣ϕ | �ρ�� ϕ | �ρ�ϕ U ϕ
where p ∈ Props and ρ ∈ R± .
Nils Bulling, Berndt Farwer · Properties of Resource-Bounded Systems
FAUSt 2009 11/24
2. Modelling Resource-Bounded Systems Resource-Bounded Tree Logic
Semantics of RTL∗ Let M be an RBM, q a state in M, and λ ∈ ΛM . M, q |= p iff λ[0] ∈ π(p) and p ∈ Props M, q |= ϕ ∧ ψ iff M, q |= ϕ and M, q |= ψ M, q |= �ρ�ϕ iff there is a ρ-feasible path λ ∈ Λ(q) s.t. M, λ |= ϕ M, λ |= ϕ iff M, λ[0] |= ϕ M, λ |= ¬γ iff not M, λ |= γ M, λ |= γ ∧ ψ iff M, λ |= γ and M, λ |= ψ M, λ |= � ϕ iff for all i ∈ N we have that λ[i, ∞] |= ϕ M, λ |= ❢ϕ iff λ[1, ∞] |= ϕ M, λ |= ϕ U ψ iff there is i such that M, λ[i, ∞] |= ψ and M, λ[j, ∞] |= ϕ for all 0 ≤ j < i
Nils Bulling, Berndt Farwer · Properties of Resource-Bounded Systems
FAUSt 2009 12/24
2. Modelling Resource-Bounded Systems Resource-Bounded Tree Logic
An Example (1, 1), (0, 2)
(0, 3), (0, 0)
t � � �
q0 (0, 0), (0, 0)
p
(0, 0), (0, 2)
,0 (0 (3
,3
q1 r
)
)
q2
Can the system run forever given specific resources? Yes: M, q0 |= �(∞, 1)��
M, q0 �|= �(1, 1)�� � since there is no (1, 1)-feasible path. The formula �(1, ∞)�� (p ∨ t) holds in q0 .
Nils Bulling, Berndt Farwer · Properties of Resource-Bounded Systems
FAUSt 2009 13/24
2. Modelling Resource-Bounded Systems Resource-Bounded Tree Logic
Conservative Extension �
Our logics conservatively extend CTL∗ and CTL.
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Define the path quantifier E as �∅� and set tqq� = (∅, ∅) for all states q and q � .
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Every Kripke model has a canonical RBM.
Proposition (Expressiveness) CTL∗ and CTL can be embedded into RTL∗ and RTL over all Kripke models, respectively.
Nils Bulling, Berndt Farwer · Properties of Resource-Bounded Systems
FAUSt 2009 14/24
2. Modelling Resource-Bounded Systems Cover Graphs and Cover Models
The Cover Graph Is there a “good” ρ-feasible path for γ? (0), (0)
(2) (0)
q2 s
q0
(q2 , ω) s (0), (0)
r
(0), (1)
q1
RBM
r
(q0 , 0)
⇒
(q1 , 0)
r
(q0 , ω)
(q1 , ω)
Cover Model for M, q0 , ∅
Theorem (Finiteness of the cover graph) Let ρ ∈ R± , let M be an RBM, and let q be a state in M. Then the (ρ, q)-cover graph of M is finite. Nils Bulling, Berndt Farwer · Properties of Resource-Bounded Systems
FAUSt 2009 15/24
2. Modelling Resource-Bounded Systems Cover Graphs and Cover Models
Cover models as models for RTL formulae? (0), (0)
(2) (0)
q2 s
q0
(q2 , ω) s (0), (0)
r
(0), (1)
q1
r
(q0 , 0)
�ρ�γ � � �
�⇐
(q1 , 0)
r
(q0 , ω)
(q1 , ω)
Eγ
Consider (q0 , 0)(q1 , 0)(q0 , ω)(q2 , ω)(q2 , ω) . . . in C q0 q1 q0 q2 q2 · · · = (q0 , 0)(q1 , 0)(q0 , ω)(q2 , ω)(q2 , ω) . . . |Q not ∅-feasible in M. For γ = ♦ s ∧ ❣ ❣� ¬r we have M, q0 �|= �∅�γ
but C, (q0 , 0) |= Eγ
Nils Bulling, Berndt Farwer · Properties of Resource-Bounded Systems
FAUSt 2009 16/24
3. RTL∗ : Properties and Decidability Bounded Models
Bounded Models Decidability for the general problem is still open. We identify 1. subclasses of RBMs and 2. fragments of RTL∗ which have a decidable model checking property.
Nils Bulling, Berndt Farwer · Properties of Resource-Bounded Systems
FAUSt 2009 18/24
3. RTL∗ : Properties and Decidability Bounded Models
Properties of RBMs �
zero free: non-consuming transitions only as self-loops
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∞-free: no infinite consumption/production
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production free: no producing transitions
k-bounded: resources of ρ-feasible path bounded by k.
Proposition k-boundedness of M for ρ is decidable.
Proposition The model checking problem for RTL∗ R± over k-bounded RBMs is �∞ decidable and PSPACE-hard. The model checking problem for RTL∗ R± over production- and zero-free �∞ RBMs is decidable.
Nils Bulling, Berndt Farwer · Properties of Resource-Bounded Systems
FAUSt 2009 19/24
4. RTL: Properties and Decidability RTL and Cover Models
Model Checking RTL
Claim (Model Checking RTL: Decidability) The model-checking problem for RTL over RBMs is decidable and P-hard. Proof by a combinatorial argument and the decidability of reachability in Petri nets.
Nils Bulling, Berndt Farwer · Properties of Resource-Bounded Systems
FAUSt 2009 21/24
5. Conclusions and Future Work
Conclusion Summary �
Introduced resources into Kripke models and extended CTL∗ .
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Showed decidability results in the presence of some limiting constraints.
Ongoing research �
Model checking complexity and decidability for the general case is part of our current research.
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Are there restrictions that make the model checking problem efficiently decidable for a relevant class of MAS? Extend the resource-bounded setting to the multi-agent case
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Resource-bounded Agent Logic (RAL): an extension of ATL.
Nils Bulling, Berndt Farwer · Properties of Resource-Bounded Systems
FAUSt 2009 23/24
