Propositional Dynamic Logic
with Converse and Repeat
for Message-Passing Systems
Roy Mennicke
Technische Universität Ilmenau, Germany
Message Sequence Charts (MSCs)
MSCs model the behaviour of a finite set of parallel processes P
communicating using FIFO channels.
p
q
p!q
q?p
p!q
q!p
p!q
q?p
p?q
q!p
p?q
q?p
p?q
q!p
In our MSCs, events are labeled by letters from the alphabet
Σ := {p!q, p?q | p, q ∈ P, p 6= q} .
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PDL with Converse and Repeat (CRPDL)
Formulas are evaluated at individual events.
Syntax:
(a ranges over Σ)
π ::= proc | proc−1 | msg | msg−1 | π; π | π + π | π ∗ | {ϕ}
ϕ ::= a | ¬ϕ | ϕ ∨ ϕ | ϕ ∧ ϕ | hπi tt | hπiω
hπiω is called repeat operator
proc−1 and msg−1 form converse operator
Examples:
p!q → hproc∗ i p?q
W
ψp = q∈P,p6=q (p!q ∨ p?q) where p ∈ P
ω
ϕ = π1 ; π2 ; . . . ; π|P| where πr = ((proc + msg)∗ ; {ψr })
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Satisfiability
The (unrestricted) satisfiability problem is undecidable.
Let b ∈ N. An MSC is existentially b-bounded if it admits an
execution with b-bounded channels.
Bounded Satisfiability Problem
Input:
formula ϕ and bound b ∈ N (given in unary)
Question: Is there an existentially b-bounded MSC satisfying ϕ?
PSPACE-hardness: follows from the PSPACE-hardness of the
satisfiablity problem of LTL
It remains to show that the above problem is in PSPACE:
as usual, we transform formulas into automata
for this purpose, we introduce a new automaton model
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Message Sequence Chart Automata (MSCAs)
MSCAs are multi-way alternating parity automata:
started at an individual event of an MSC
capable of (not) moving into any direction (multi-way)
can change into different states and move into different
directions at the same time (alternation), i.e., its runs are trees
a run tree is accepting if all its branches are accepting;
a finite branch is accepting if it ends in a state of even rank;
an infinite branch is accepting if the minimum of the ranks
occuring infinitely often is even (parity acceptance condition)
a pointed MSC (M, v) is accepted if there exists an accepting
run for it
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Example MSCA
Consider the MSCA A:
Σ
ι|1
proc
s|0
an event v is accepted by A ⇐⇒ there exists a successor of v
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From Formulas to MSCAs
Theorem
If ϕ is a formula, then one can compute an MSCA Aϕ in polynomial time which exactly accepts the set of models of ϕ.
The size of Aϕ is linear in ϕ.
Ahprocitt is the example MSCA from the last slide.
Σ
proc
ι|1
s|0
The state s is called concatenation state.
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Inductive Definition of Ahπ1 ;π2 itt
ι | rι
Ahπ1 itt
Σ
s | rs
stay
ι0 | rι0
Ahπ2 itt
s0 | rs0
Ahπ1 ;π2 itt accepts the correct language since
every run of an MSCA is maximal (by definition)and
every accepting run of an MSCA Ahπitt exhibits exactly one
branch ending in the concatenation state of Ahπitt .
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Inductive Definition of Ah{ϕ}itt
ι0 | rι0
stay
Σ
ι|1
Aϕ
stay
s|0
Consider the formula ψ = h{ϕ}; proci tt. We cannot concatenate
the MSCAs Aϕ and Ahprocitt directly in order to obtain an MSCA
for ψ due to the following fact:
The MSCA Ahprocitt needs to be started in the same event in
which Aϕ was started. However, the test automaton Aϕ may move
away from the event it was started in.
Therefore, we construct the above intermediate automaton.
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Bounded Satisfiability Problem
Theorem
The following problem is PSPACE-complete:
Input:
formula ϕ and bound b ∈ N (given in unary)
Question: Is there an existentially b-bounded MSC M
satisfying ϕ?
It remained to show that the above problem is in PSPACE.
Sketch of proof: We transform Aϕ into a Büchi automaton B
whose number of states is exponential in ϕ and b such that
L(B) 6= ∅ ⇐⇒
there exists an existentially b-bounded
MSC M satisfying ϕ
Note that B is not constructed explicitly.
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Open Questions
Are CRPDL and MSCAs expressively equivalent?
What is the expressive power of MSCAs and CRPDL,
especially in comparison with EMSO?
Is the bounded satisfiability problem for CRPDL enriched with
the intersection operator still in PSPACE?
Is bi-directional PDL [Bollig, Kuske, Meinecke ’07] a proper
fragment of CRPDL?
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