262 research outputs found
Towards Automated Reasoning in Herbrand Structures
Herbrand structures have the advantage, computationally speaking, of being guided by the definability of all elements in them. A salient feature of the logics induced by them is that they internally
exhibit the induction scheme, thus providing a congenial, computationally-oriented framework for
formal inductive reasoning. Nonetheless, their enhanced expressivity renders any effective proof
system for them incomplete. Furthermore, the fact that they are not compact poses yet another prooftheoretic challenge. This paper offers several layers for coping with the inherent incompleteness and
non-compactness of these logics. First, two types of infinitary proof system are introduced—one
of infinite width and one of infinite height—which manipulate infinite sequents and are sound and
complete for the intended semantics. The restriction of these systems to finite sequents induces a
completeness result for finite entailments. Then, in search of effectiveness, two finite approximations
of these systems are presented and explored. Interestingly, the approximation of the infinite-width
system via an explicit induction scheme turns out to be weaker than the effective cyclic fragment of the
infinite-height system
Reasoning about Independence in Probabilistic Models of Relational Data
We extend the theory of d-separation to cases in which data instances are not
independent and identically distributed. We show that applying the rules of
d-separation directly to the structure of probabilistic models of relational
data inaccurately infers conditional independence. We introduce relational
d-separation, a theory for deriving conditional independence facts from
relational models. We provide a new representation, the abstract ground graph,
that enables a sound, complete, and computationally efficient method for
answering d-separation queries about relational models, and we present
empirical results that demonstrate effectiveness.Comment: 61 pages, substantial revisions to formalisms, theory, and related
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Uniform Inductive Reasoning in Transitive Closure Logic via Infinite Descent
Transitive closure logic is a known extension of first-order logic obtained by introducing a
transitive closure operator. While other extensions of first-order logic with inductive definitions
are a priori parametrized by a set of inductive definitions, the addition of the transitive closure
operator uniformly captures all finitary inductive definitions. In this paper we present an
infinitary proof system for transitive closure logic which is an infinite descent-style counterpart
to the existing (explicit induction) proof system for the logic. We show that, as for similar
systems for first-order logic with inductive definitions, our infinitary system is complete for the
standard semantics and subsumes the explicit system. Moreover, the uniformity of the transitive
closure operator allows semantically meaningful complete restrictions to be defined using simple
syntactic criteria. Consequently, the restriction to regular infinitary (i.e. cyclic) proofs provides
the basis for an effective system for automating inductive reasoning
Identifying Independence in Relational Models
The rules of d-separation provide a framework for deriving conditional
independence facts from model structure. However, this theory only applies to
simple directed graphical models. We introduce relational d-separation, a
theory for deriving conditional independence in relational models. We provide a
sound, complete, and computationally efficient method for relational
d-separation, and we present empirical results that demonstrate effectiveness.Comment: This paper has been revised and expanded. See "Reasoning about
Independence in Probabilistic Models of Relational Data"
http://arxiv.org/abs/1302.438
Causal Discovery for Relational Domains: Representation, Reasoning, and Learning
Many domains are currently experiencing the growing trend to record and analyze massive, observational data sets with increasing complexity. A commonly made claim is that these data sets hold potential to transform their corresponding domains by providing previously unknown or unexpected explanations and enabling informed decision-making. However, only knowledge of the underlying causal generative process, as opposed to knowledge of associational patterns, can support such tasks.
Most methods for traditional causal discovery—the development of algorithms that learn causal structure from observational data—are restricted to representations that require limiting assumptions on the form of the data. Causal discovery has almost exclusively been applied to directed graphical models of propositional data that assume a single type of entity with independence among instances. However, most real-world domains are characterized by systems that involve complex interactions among multiple types of entities. Many state-of-the-art methods in statistics and machine learning that address such complex systems focus on learning associational models, and they are oftentimes mistakenly interpreted as causal. The intersection between causal discovery and machine learning in complex systems is small.
The primary objective of this thesis is to extend causal discovery to such complex systems. Specifically, I formalize a relational representation and model that can express the causal and probabilistic dependencies among the attributes of interacting, heterogeneous entities. I show that the traditional method for reasoning about statistical independence from model structure fails to accurately derive conditional independence facts from relational models. I introduce a new theory—relational d-separation—and a novel, lifted representation—the abstract ground graph—that supports a sound, complete, and computationally efficient method for algorithmically deriving conditional independencies from probabilistic models of relational data. The abstract ground graph representation also presents causal implications that enable the detection of causal direction for bivariate relational dependencies without parametric assumptions. I leverage these implications and the theoretical framework of relational d-separation to develop a sound and complete algorithm—the relational causal discovery (RCD) algorithm—that learns causal structure from relational data
Infinitary and Cyclic Proof Systems for Transitive Closure Logic
Transitive closure logic is a known extension of first-order logic obtained by introducing a transitive closure operator. While other extensions of first-order logic with inductive definitions are a priori parametrized by a set of inductive definitions, the addition of the transitive closure operator uniformly captures all finitary inductive definitions. In this paper we present an infinitary proof system for transitive closure logic which is an infinite descent-style counterpart to the existing (explicit induction) proof system for the logic. We show that, as for similar systems for first-order logic with inductive definitions, our infinitary system is complete for the standard semantics and subsumes the explicit system. Moreover, the uniformity of the transitive closure operator allows semantically meaningful complete restrictions to be defined using simple syntactic criteria. Consequently, the restriction to regular infinitary (i.e. cyclic) proofs provides the basis for an effective system for automating inductive reasoning
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A defence of predicativism as a philosophy of mathematics
A specification of a mathematical object is impredicative if it essentially involves quantification over a domain which includes the object being specified (or sets which contain that object, or similar). The basic worry is that we have no non-circular way of
understanding such a specification. Predicativism is the view that mathematics should be limited to the study of objects which can be specified predicatively.
There are two parts to predicativism. One is the criticism of the impredicative aspects of classical mathematics. The other is the
positive project, begun by Weyl in Das Kontinuum (1918), to reconstruct as much as possible of classical mathematics on the basis of a predicatively acceptable set theory, which accepts only countably infinite objects. This is a revisionary project, and certain parts of mathematics will not be saved.
Chapter 2 contains an account of the historical background to the predicativist project. The rigorization of analysis led to Dedekind's and Cantor's theories of the real numbers, which relied on the new notion of abitrary infinite sets; this became a central part of modern classical set theory. Criticism began with Kronecker; continued in the debate about the acceptability of Zermelo's Axiom of Choice; and was somewhat clarified by Poincaré and Russell. In the
light of this, chapter 3 examines the formulation of, and motivations behind the predicativist position.
Chapter 4 begins the critical task by detailing the epistemological problems with the classical account of the continuum. Explanations of classicism which appeal to second-order logic, set theory, and
primitive intuition are examined and are found wanting.
Chapter 5 aims to dispell the worry that predicativism might collapses into mathematical intuitionism. I assess some of the arguments for intuitionism, especially the Dummettian argument from indefinite
extensibility. I argue that the natural numbers are not indefinitely extensible, and that, although the continuum is, we can nonetheless make some sense of classical quantification over it. We need not reject the Law of Excluded Middle.
Chapter 6 begins the positive work by outlining a predicatively acceptable account of mathematical objects which justifies the Vicious Circle Principle. Chapter 7 explores the appropriate shape of formalized predicative mathematics, and the question of just how much mathematics is predicatively acceptable.
My conclusion is that all of the mathematics which we need can be predicativistically justified, and that such mathematics is
particularly transparent to reason. This calls into question one currently prevalent view of the nature of mathematics, on which
mathematics is justified by quasi-empirical means.Supported by the Arts and Humanities Research Council [grant number 111315]
Parallel execution of horn claus programs
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