4,215 research outputs found
The Complexity of Model Checking Higher-Order Fixpoint Logic
Higher-Order Fixpoint Logic (HFL) is a hybrid of the simply typed
\lambda-calculus and the modal \lambda-calculus. This makes it a highly
expressive temporal logic that is capable of expressing various interesting
correctness properties of programs that are not expressible in the modal
\lambda-calculus.
This paper provides complexity results for its model checking problem. In
particular we consider those fragments of HFL built by using only types of
bounded order k and arity m. We establish k-fold exponential time completeness
for model checking each such fragment. For the upper bound we use fixpoint
elimination to obtain reachability games that are singly-exponential in the
size of the formula and k-fold exponential in the size of the underlying
transition system. These games can be solved in deterministic linear time. As a
simple consequence, we obtain an exponential time upper bound on the expression
complexity of each such fragment.
The lower bound is established by a reduction from the word problem for
alternating (k-1)-fold exponential space bounded Turing Machines. Since there
are fixed machines of that type whose word problems are already hard with
respect to k-fold exponential time, we obtain, as a corollary, k-fold
exponential time completeness for the data complexity of our fragments of HFL,
provided m exceeds 3. This also yields a hierarchy result in expressive power.Comment: 33 pages, 2 figures, to be published in Logical Methods in Computer
Scienc
Probabilistic Interval Temporal Logic and Duration Calculus with Infinite Intervals: Complete Proof Systems
The paper presents probabilistic extensions of interval temporal logic (ITL)
and duration calculus (DC) with infinite intervals and complete Hilbert-style
proof systems for them. The completeness results are a strong completeness
theorem for the system of probabilistic ITL with respect to an abstract
semantics and a relative completeness theorem for the system of probabilistic
DC with respect to real-time semantics. The proposed systems subsume
probabilistic real-time DC as known from the literature. A correspondence
between the proposed systems and a system of probabilistic interval temporal
logic with finite intervals and expanding modalities is established too.Comment: 43 page
Model Checking Probabilistic Pushdown Automata
We consider the model checking problem for probabilistic pushdown automata
(pPDA) and properties expressible in various probabilistic logics. We start
with properties that can be formulated as instances of a generalized random
walk problem. We prove that both qualitative and quantitative model checking
for this class of properties and pPDA is decidable. Then we show that model
checking for the qualitative fragment of the logic PCTL and pPDA is also
decidable. Moreover, we develop an error-tolerant model checking algorithm for
PCTL and the subclass of stateless pPDA. Finally, we consider the class of
omega-regular properties and show that both qualitative and quantitative model
checking for pPDA is decidable
Simplifying proofs of linearisability using layers of abstraction
Linearisability has become the standard correctness criterion for concurrent
data structures, ensuring that every history of invocations and responses of
concurrent operations has a matching sequential history. Existing proofs of
linearisability require one to identify so-called linearisation points within
the operations under consideration, which are atomic statements whose execution
causes the effect of an operation to be felt. However, identification of
linearisation points is a non-trivial task, requiring a high degree of
expertise. For sophisticated algorithms such as Heller et al's lazy set, it
even is possible for an operation to be linearised by the concurrent execution
of a statement outside the operation being verified. This paper proposes an
alternative method for verifying linearisability that does not require
identification of linearisation points. Instead, using an interval-based logic,
we show that every behaviour of each concrete operation over any interval is a
possible behaviour of a corresponding abstraction that executes with
coarse-grained atomicity. This approach is applied to Heller et al's lazy set
to show that verification of linearisability is possible without having to
consider linearisation points within the program code
Three notes on the complexity of model checking fixpoint logic with chop
This paper analyses the complexity of model checking fixpoint logic with Chop â an extension of the
modal Ό-calculus with a sequential composition operator. It uses two known game-based characterisations
to derive the following results: the combined model checking complexity as well as the data complexity
of FLC are EXPTIME-complete. This is already the case for its alternation-free fragment. The expression
complexity of FLC is trivially P-hard and limited from above by the complexity of solving a
parity game, i.e. in UP â© co-UP. For any fragment of fixed alternation depth, in particular alternation-
free formulas it is P-complete
Temporal Logic with Recursion
We introduce extensions of the standard temporal logics CTL and LTL with a recursion operator that takes propositional arguments. Unlike other proposals for modal fixpoint logics of high expressive power, we obtain logics that retain some of the appealing pragmatic advantages of CTL and LTL, yet have expressive power beyond that of the modal ?-calculus or MSO. We advocate these logics by showing how the recursion operator can be used to express interesting non-regular properties. We also study decidability and complexity issues of the standard decision problems
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