352 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
Size, Cost and Capacity: A Semantic Technique for Hard Random QBFs
As a natural extension of the SAT problem, an array of proof systems for quantified Boolean formulas (QBF) have been proposed, many of which extend a propositional proof system to handle universal quantification. By formalising the construction of the QBF proof system obtained from a propositional proof system by adding universal reduction (Beyersdorff, Bonacina & Chew, ITCS'16), we present a new technique for proving proof-size lower bounds in these systems. The technique relies only on two semantic measures: the cost of a QBF, and the capacity of a proof. By examining the capacity of proofs in several QBF systems, we are able to use the technique to obtain lower bounds based on cost alone. As applications of the technique, we first prove exponential lower bounds for a new family of simple QBFs representing equality. The main application is in proving exponential lower bounds with high probability for a class of randomly generated QBFs, the first 'genuine' lower bounds of this kind, which apply to the QBF analogues of resolution, Cutting Planes, and Polynomial Calculus. Finally, we employ the technique to give a simple proof of hardness for a prominent family of QBFs
Subclasses of Presburger Arithmetic and the Weak EXP Hierarchy
It is shown that for any fixed , the -fragment of
Presburger arithmetic, i.e., its restriction to quantifier alternations
beginning with an existential quantifier, is complete for
, the -th level of the weak EXP
hierarchy, an analogue to the polynomial-time hierarchy residing between
and . This result completes the
computational complexity landscape for Presburger arithmetic, a line of
research which dates back to the seminal work by Fischer & Rabin in 1974.
Moreover, we apply some of the techniques developed in the proof of the lower
bound in order to establish bounds on sets of naturals definable in the
-fragment of Presburger arithmetic: given a -formula
, it is shown that the set of non-negative solutions is an ultimately
periodic set whose period is at most doubly-exponential and that this bound is
tight.Comment: 10 pages, 2 figure
Reasons for Hardness in QBF Proof Complexity
Quantified Boolean Formulas (QBF) extend the canonical NP-complete satisfiability problem by including Boolean quantifiers. Determining the truth of a QBF is PSPACE-complete; this is expected to be a harder problem than satisfiability, and hence QBF solving has much wider
applications in practice. QBF proof complexity forms the theoretical basis for understanding QBF solving, as well as providing insights into more general complexity theory, but is less well understood than propositional proof complexity.
We begin this thesis by looking at the reasons underlying QBF hardness, and in particular when the hardness is propositional in nature, rather than arising due to the quantifiers. We introduce relaxing QU-Res, a previous model for identifying such propositional hardness, and construct an example where relaxing QU-Res is unsuccessful in this regard. We then provide a new model for identifying such hardness which we prove captures this concept.
Now equipped with a means of identifying ‘genuine’ QBF hardness, we prove a new lower bound technique for tree-like QBF proof systems. Lower bounds using this technique allows us to show a new separation between tree-like and dag-like systems. We give a characterisation of lower bounds for a large class of tree-like proof systems, in which such lower bounds play a prominent role.
Further to the tree-like bound, we provide a new lower bound technique for QBF proof systems in general. This technique has some similarities to the above technique for tree-like systems, but requires some refinement to provide bounds for dag-like systems. We give applications of this
new technique by proving lower bounds across several systems. The first such lower bounds are for a very simple family of QBFs. We then provide a construction to combine false QBFs to give formulas for which we can show lower bounds in this way, allowing the generation of the first random QBF proof complexity lower bounds
Exponential Lower Bounds and Separation for Query Rewriting
We establish connections between the size of circuits and formulas computing
monotone Boolean functions and the size of first-order and nonrecursive Datalog
rewritings for conjunctive queries over OWL 2 QL ontologies. We use known lower
bounds and separation results from circuit complexity to prove similar results
for the size of rewritings that do not use non-signature constants. For
example, we show that, in the worst case, positive existential and nonrecursive
Datalog rewritings are exponentially longer than the original queries;
nonrecursive Datalog rewritings are in general exponentially more succinct than
positive existential rewritings; while first-order rewritings can be
superpolynomially more succinct than positive existential rewritings
Circuit Complexity Meets Ontology-Based Data Access
Ontology-based data access is an approach to organizing access to a database
augmented with a logical theory. In this approach query answering proceeds
through a reformulation of a given query into a new one which can be answered
without any use of theory. Thus the problem reduces to the standard database
setting.
However, the size of the query may increase substantially during the
reformulation. In this survey we review a recently developed framework on
proving lower and upper bounds on the size of this reformulation by employing
methods and results from Boolean circuit complexity.Comment: To appear in proceedings of CSR 2015, LNCS 9139, Springe
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