449 research outputs found
The Logic of Counting Query Answers
We consider the problem of counting the number of answers to a first-order
formula on a finite structure. We present and study an extension of first-order
logic in which algorithms for this counting problem can be naturally and
conveniently expressed, in senses that are made precise and that are motivated
by the wish to understand tractable cases of the counting problem
Backdoors to Normality for Disjunctive Logic Programs
Over the last two decades, propositional satisfiability (SAT) has become one
of the most successful and widely applied techniques for the solution of
NP-complete problems. The aim of this paper is to investigate theoretically how
Sat can be utilized for the efficient solution of problems that are harder than
NP or co-NP. In particular, we consider the fundamental reasoning problems in
propositional disjunctive answer set programming (ASP), Brave Reasoning and
Skeptical Reasoning, which ask whether a given atom is contained in at least
one or in all answer sets, respectively. Both problems are located at the
second level of the Polynomial Hierarchy and thus assumed to be harder than NP
or co-NP. One cannot transform these two reasoning problems into SAT in
polynomial time, unless the Polynomial Hierarchy collapses. We show that
certain structural aspects of disjunctive logic programs can be utilized to
break through this complexity barrier, using new techniques from Parameterized
Complexity. In particular, we exhibit transformations from Brave and Skeptical
Reasoning to SAT that run in time O(2^k n^2) where k is a structural parameter
of the instance and n the input size. In other words, the reduction is
fixed-parameter tractable for parameter k. As the parameter k we take the size
of a smallest backdoor with respect to the class of normal (i.e.,
disjunction-free) programs. Such a backdoor is a set of atoms that when deleted
makes the program normal. In consequence, the combinatorial explosion, which is
expected when transforming a problem from the second level of the Polynomial
Hierarchy to the first level, can now be confined to the parameter k, while the
running time of the reduction is polynomial in the input size n, where the
order of the polynomial is independent of k.Comment: A short version will appear in the Proceedings of the Proceedings of
the 27th AAAI Conference on Artificial Intelligence (AAAI'13). A preliminary
version of the paper was presented on the workshop Answer Set Programming and
Other Computing Paradigms (ASPOCP 2012), 5th International Workshop,
September 4, 2012, Budapest, Hungar
A Goal-Directed Implementation of Query Answering for Hybrid MKNF Knowledge Bases
Ontologies and rules are usually loosely coupled in knowledge representation
formalisms. In fact, ontologies use open-world reasoning while the leading
semantics for rules use non-monotonic, closed-world reasoning. One exception is
the tightly-coupled framework of Minimal Knowledge and Negation as Failure
(MKNF), which allows statements about individuals to be jointly derived via
entailment from an ontology and inferences from rules. Nonetheless, the
practical usefulness of MKNF has not always been clear, although recent work
has formalized a general resolution-based method for querying MKNF when rules
are taken to have the well-founded semantics, and the ontology is modeled by a
general oracle. That work leaves open what algorithms should be used to relate
the entailments of the ontology and the inferences of rules. In this paper we
provide such algorithms, and describe the implementation of a query-driven
system, CDF-Rules, for hybrid knowledge bases combining both (non-monotonic)
rules under the well-founded semantics and a (monotonic) ontology, represented
by a CDF Type-1 (ALQ) theory. To appear in Theory and Practice of Logic
Programming (TPLP
Philosophy and the practice of Bayesian statistics
A substantial school in the philosophy of science identifies Bayesian
inference with inductive inference and even rationality as such, and seems to
be strengthened by the rise and practical success of Bayesian statistics. We
argue that the most successful forms of Bayesian statistics do not actually
support that particular philosophy but rather accord much better with
sophisticated forms of hypothetico-deductivism. We examine the actual role
played by prior distributions in Bayesian models, and the crucial aspects of
model checking and model revision, which fall outside the scope of Bayesian
confirmation theory. We draw on the literature on the consistency of Bayesian
updating and also on our experience of applied work in social science.
Clarity about these matters should benefit not just philosophy of science,
but also statistical practice. At best, the inductivist view has encouraged
researchers to fit and compare models without checking them; at worst,
theorists have actively discouraged practitioners from performing model
checking because it does not fit into their framework.Comment: 36 pages, 5 figures. v2: Fixed typo in caption of figure 1. v3:
Further typo fixes. v4: Revised in response to referee
Induction of Interpretable Possibilistic Logic Theories from Relational Data
The field of Statistical Relational Learning (SRL) is concerned with learning
probabilistic models from relational data. Learned SRL models are typically
represented using some kind of weighted logical formulas, which make them
considerably more interpretable than those obtained by e.g. neural networks. In
practice, however, these models are often still difficult to interpret
correctly, as they can contain many formulas that interact in non-trivial ways
and weights do not always have an intuitive meaning. To address this, we
propose a new SRL method which uses possibilistic logic to encode relational
models. Learned models are then essentially stratified classical theories,
which explicitly encode what can be derived with a given level of certainty.
Compared to Markov Logic Networks (MLNs), our method is faster and produces
considerably more interpretable models.Comment: Longer version of a paper appearing in IJCAI 201
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A cognitive model of the roles of diagrammatic representation in supporting unpractised reasoning about probability
Cognitive process accounts of the advantages conferred by diagrams in problem solving and reasoning have typically attempted to explain an idealised user or a reasoning system that has equivalent to practised knowledge of the task with the target representation. The thesis investigates the question of how diagrams support users in the process of solving unpractised problems in the domain of probability. The research question is addressed by the design and analysis of an empirical study and cognitive model.
The main experiment required participants (N=8) to solve a set of unpractised probability problems presented by combined text and diagram. Think-aloud and eye-movement protocols together with given solutions were used to infer the content and process of problem interpretation, solution interpretation and task execution strategies employed by participants. The data suggested that the diagram was used to facilitate problem solving in three different ways by: (a) supporting sub-problem identification, (b) supporting prior knowledge of diagrammatic sub-schemes used for interpreting a solution and (c) supporting the process of interpreting and testing the specific meaning of given problem instructions and self-generated solution instructions.
These empirical data were used to develop cognitive models of canonical strategies of the three identified phenomena:
• Sub-problem identification advantages are accounted for by proposing that the spatial semantics of diagrams coupled with competences of the visual-spatial processing system and opportunities for demonstrative interpretation strategies increase the probability of goal-relevant data being made available to central cognition for further processing.
• Framing advantages are accounted for by proposing that represented diagrammatic sub-schemes (e.g. part-whole portions, icon-arrays, 2D containers etc.) facilitate access to existing prior knowledge used to frame, derive, and reason about information analogically within that scheme.
• Advantages in instruction interpretation are related to the specificity of diagrams which support the opportunity to demonstratively test and evaluate the referential meaning of an instruction.
The cognitive model also investigates and evaluates assumptions about the prior knowledge for solving unpractised probability problems; a representational scheme for addressing the co-ordination of sub-goals; a deictic problem representation to support online processing of environmental information, a meta-cognitive processing scheme to address self-argumentation and intention tracking and visual and spatial competences to address the requirements of diagrammatic reasoning. The implications of the cognitive model are discussed with regard to existing accounts of diagrammatic reasoning, probability problem solving (PPS), and unpractised problem solving
Quantified Constraints in Twenty Seventeen
I present a survey of recent advances in the algorithmic and computational complexity theory of non-Boolean Quantified Constraint Satisfaction Problems, incorporating some more modern research directions
Query Answering in Probabilistic Data and Knowledge Bases
Probabilistic data and knowledge bases are becoming increasingly important in academia and industry. They are continuously extended with new data, powered by modern information extraction tools that associate probabilities with knowledge base facts. The state of the art to store and process such data is founded on probabilistic database systems, which are widely and successfully employed. Beyond all the success stories, however, such systems still lack the fundamental machinery to convey some of the valuable knowledge hidden in them to the end user, which limits their potential applications in practice. In particular, in their classical form, such systems are typically based on strong, unrealistic limitations, such as the closed-world assumption, the closed-domain assumption, the tuple-independence assumption, and the lack of commonsense knowledge. These limitations do not only lead to unwanted consequences, but also put such systems on weak footing in important tasks, querying answering being a very central one. In this thesis, we enhance probabilistic data and knowledge bases with more realistic data models, thereby allowing for better means for querying them. Building on the long endeavor of unifying logic and probability, we develop different rigorous semantics for probabilistic data and knowledge bases, analyze their computational properties and identify sources of (in)tractability and design practical scalable query answering algorithms whenever possible. To achieve this, the current work brings together some recent paradigms from logics, probabilistic inference, and database theory
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