188 research outputs found
Reasoning with global assumptions in arithmetic modal logics
We establish a generic upper bound ExpTime for reasoning with global assumptions in coalgebraic modal logics. Unlike earlier results of this kind, we do not require a tractable set of tableau rules for the in- stance logics, so that the result applies to wider classes of logics. Examples are Presburger modal logic, which extends graded modal logic with linear inequalities over numbers of successors, and probabilistic modal logic with polynomial inequalities over probabilities. We establish the theoretical upper bound using a type elimination algorithm. We also provide a global caching algorithm that offers potential for practical reasoning
Coalgebraic Reasoning with Global Assumptions in Arithmetic Modal Logics
We establish a generic upper bound ExpTime for reasoning with global
assumptions (also known as TBoxes) in coalgebraic modal logics. Unlike earlier
results of this kind, our bound does not require a tractable set of tableau
rules for the instance logics, so that the result applies to wider classes of
logics. Examples are Presburger modal logic, which extends graded modal logic
with linear inequalities over numbers of successors, and probabilistic modal
logic with polynomial inequalities over probabilities. We establish the
theoretical upper bound using a type elimination algorithm. We also provide a
global caching algorithm that potentially avoids building the entire
exponential-sized space of candidate states, and thus offers a basis for
practical reasoning. This algorithm still involves frequent fixpoint
computations; we show how these can be handled efficiently in a concrete
algorithm modelled on Liu and Smolka's linear-time fixpoint algorithm. Finally,
we show that the upper complexity bound is preserved under adding nominals to
the logic, i.e. in coalgebraic hybrid logic.Comment: Extended version of conference paper in FCT 201
Logics for Unranked Trees: An Overview
Labeled unranked trees are used as a model of XML documents, and logical
languages for them have been studied actively over the past several years. Such
logics have different purposes: some are better suited for extracting data,
some for expressing navigational properties, and some make it easy to relate
complex properties of trees to the existence of tree automata for those
properties. Furthermore, logics differ significantly in their model-checking
properties, their automata models, and their behavior on ordered and unordered
trees. In this paper we present a survey of logics for unranked trees
On two-variable guarded fragment logic with expressive local Presburger constraints
We consider the extension of two-variable guarded fragment logic with local
Presburger quantifiers. These are quantifiers that can express properties such
as ``the number of incoming blue edges plus twice the number of outgoing red
edges is at most three times the number of incoming green edges'' and captures
various description logics with counting, but without constant symbols. We show
that the satisfiability of this logic is EXP-complete. While the lower bound
already holds for the standard two-variable guarded fragment logic, the upper
bound is established by a novel, yet simple deterministic graph theoretic based
algorithm
On Classical Decidable Logics Extended with Percentage Quantifiers and Arithmetics
During the last decades, a lot of effort was put into identifying decidable fragments of first-order logic. Such efforts gave birth, among the others, to the two-variable fragment and the guarded fragment, depending on the type of restriction imposed on formulae from the language. Despite the success of the mentioned logics in areas like formal verification and knowledge representation, such first-order fragments are too weak to express even the simplest statistical constraints, required for modelling of influence networks or in statistical reasoning.
In this work we investigate the extensions of these classical decidable logics with percentage quantifiers, specifying how frequently a formula is satisfied in the indented model. We show, surprisingly, that all the mentioned decidable fragments become undecidable under such extension, sharpening the existing results in the literature. Our negative results are supplemented by decidability of the two-variable guarded fragment with even more expressive counting, namely Presburger constraints. Our results can be applied to infer decidability of various modal and description logics, e.g. Presburger Modal Logics with Converse or ALCI, with expressive cardinality constraints
A Tree Logic with Graded Paths and Nominals
Regular tree grammars and regular path expressions constitute core constructs
widely used in programming languages and type systems. Nevertheless, there has
been little research so far on reasoning frameworks for path expressions where
node cardinality constraints occur along a path in a tree. We present a logic
capable of expressing deep counting along paths which may include arbitrary
recursive forward and backward navigation. The counting extensions can be seen
as a generalization of graded modalities that count immediate successor nodes.
While the combination of graded modalities, nominals, and inverse modalities
yields undecidable logics over graphs, we show that these features can be
combined in a tree logic decidable in exponential time
Global Numerical Constraints on Trees
We introduce a logical foundation to reason on tree structures with
constraints on the number of node occurrences. Related formalisms are limited
to express occurrence constraints on particular tree regions, as for instance
the children of a given node. By contrast, the logic introduced in the present
work can concisely express numerical bounds on any region, descendants or
ancestors for instance. We prove that the logic is decidable in single
exponential time even if the numerical constraints are in binary form. We also
illustrate the usage of the logic in the description of numerical constraints
on multi-directional path queries on XML documents. Furthermore, numerical
restrictions on regular languages (XML schemas) can also be concisely described
by the logic. This implies a characterization of decidable counting extensions
of XPath queries and XML schemas. Moreover, as the logic is closed under
negation, it can thus be used as an optimal reasoning framework for testing
emptiness, containment and equivalence
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