313 research outputs found
Path Checking for MTL and TPTL over Data Words
Metric temporal logic (MTL) and timed propositional temporal logic (TPTL) are
quantitative extensions of linear temporal logic, which are prominent and
widely used in the verification of real-timed systems. It was recently shown
that the path checking problem for MTL, when evaluated over finite timed words,
is in the parallel complexity class NC. In this paper, we derive precise
complexity results for the path-checking problem for MTL and TPTL when
evaluated over infinite data words over the non-negative integers. Such words
may be seen as the behaviours of one-counter machines. For this setting, we
give a complete analysis of the complexity of the path-checking problem
depending on the number of register variables and the encoding of constraint
numbers (unary or binary). As the two main results, we prove that the
path-checking problem for MTL is P-complete, whereas the path-checking problem
for TPTL is PSPACE-complete. The results yield the precise complexity of model
checking deterministic one-counter machines against formulae of MTL and TPTL
The First-Order Theory of Ground Tree Rewrite Graphs
We prove that the complexity of the uniform first-order theory of ground tree
rewrite graphs is in ATIME(2^{2^{poly(n)}},O(n)). Providing a matching lower
bound, we show that there is some fixed ground tree rewrite graph whose
first-order theory is hard for ATIME(2^{2^{poly(n)}},poly(n)) with respect to
logspace reductions. Finally, we prove that there exists a fixed ground tree
rewrite graph together with a single unary predicate in form of a regular tree
language such that the resulting structure has a non-elementary first-order
theory.Comment: accepted for Logical Methods in Computer Scienc
Processing Succinct Matrices and Vectors
We study the complexity of algorithmic problems for matrices that are
represented by multi-terminal decision diagrams (MTDD). These are a variant of
ordered decision diagrams, where the terminal nodes are labeled with arbitrary
elements of a semiring (instead of 0 and 1). A simple example shows that the
product of two MTDD-represented matrices cannot be represented by an MTDD of
polynomial size. To overcome this deficiency, we extended MTDDs to MTDD_+ by
allowing componentwise symbolic addition of variables (of the same dimension)
in rules. It is shown that accessing an entry, equality checking, matrix
multiplication, and other basic matrix operations can be solved in polynomial
time for MTDD_+-represented matrices. On the other hand, testing whether the
determinant of a MTDD-represented matrix vanishes PSPACE$-complete, and the
same problem is NP-complete for MTDD_+-represented diagonal matrices. Computing
a specific entry in a product of MTDD-represented matrices is #P-complete.Comment: An extended abstract of this paper will appear in the Proceedings of
CSR 201
Branching-time model checking of one-counter processes
One-counter processes (OCPs) are pushdown processes which operate only on a
unary stack alphabet. We study the computational complexity of model checking
computation tree logic (CTL) over OCPs. A PSPACE upper bound is inherited from
the modal mu-calculus for this problem. First, we analyze the periodic
behaviour of CTL over OCPs and derive a model checking algorithm whose running
time is exponential only in the number of control locations and a syntactic
notion of the formula that we call leftward until depth. Thus, model checking
fixed OCPs against CTL formulas with a fixed leftward until depth is in P. This
generalizes a result of the first author, Mayr, and To for the expression
complexity of CTL's fragment EF. Second, we prove that already over some fixed
OCP, CTL model checking is PSPACE-hard. Third, we show that there already
exists a fixed CTL formula for which model checking of OCPs is PSPACE-hard. To
obtain the latter result, we employ two results from complexity theory: (i)
Converting a natural number in Chinese remainder presentation into binary
presentation is in logspace-uniform NC^1 and (ii) PSPACE is AC^0-serializable.
We demonstrate that our approach can be used to obtain further results. We show
that model-checking CTL's fragment EF over OCPs is hard for P^NP, thus
establishing a matching lower bound and answering an open question of the first
author, Mayr, and To. We moreover show that the following problem is hard for
PSPACE: Given a one-counter Markov decision process, a set of target states
with counter value zero each, and an initial state, to decide whether the
probability that the initial state will eventually reach one of the target
states is arbitrarily close to 1. This improves a previously known lower bound
for every level of the Boolean hierarchy by Brazdil et al
Memoization for Unary Logic Programming: Characterizing PTIME
We give a characterization of deterministic polynomial time computation based
on an algebraic structure called the resolution semiring, whose elements can be
understood as logic programs or sets of rewriting rules over first-order terms.
More precisely, we study the restriction of this framework to terms (and logic
programs, rewriting rules) using only unary symbols. We prove it is complete
for polynomial time computation, using an encoding of pushdown automata. We
then introduce an algebraic counterpart of the memoization technique in order
to show its PTIME soundness. We finally relate our approach and complexity
results to complexity of logic programming. As an application of our
techniques, we show a PTIME-completeness result for a class of logic
programming queries which use only unary function symbols.Comment: Soumis {\`a} LICS 201
Efficient Parallel Path Checking for Linear-Time Temporal Logic With Past and Bounds
Path checking, the special case of the model checking problem where the model
under consideration is a single path, plays an important role in monitoring,
testing, and verification. We prove that for linear-time temporal logic (LTL),
path checking can be efficiently parallelized. In addition to the core logic,
we consider the extensions of LTL with bounded-future (BLTL) and past-time
(LTL+Past) operators. Even though both extensions improve the succinctness of
the logic exponentially, path checking remains efficiently parallelizable: Our
algorithm for LTL, LTL+Past, and BLTL+Past is in AC^1(logDCFL) \subseteq NC
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