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Probabilistic Modelling, Inference and Learning using Logical Theories
"... This paper provides a study of probabilistic modelling, inference and learning in a logicbased setting. We show how probability densities, being functions, can be represented and reasoned with naturally and directly in higherorder logic, an expressive formalism not unlike the (informal) everyday l ..."
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Cited by 9 (3 self)
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This paper provides a study of probabilistic modelling, inference and learning in a logicbased setting. We show how probability densities, being functions, can be represented and reasoned with naturally and directly in higherorder logic, an expressive formalism not unlike the (informal) everyday language of mathematics. We give efficient inference algorithms and illustrate the general approach with a diverse collection of applications. Some learning issues are also considered.
Terminating tableau systems for hybrid logic with difference and converse
, 2009
"... This paper contributes to the principled construction of tableaubased decision procedures for hybrid logic with global, difference, and converse modalities. We also consider reflexive and transitive relations. For conversefree formulas we present a terminating control that does not rely on the usu ..."
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Cited by 9 (2 self)
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This paper contributes to the principled construction of tableaubased decision procedures for hybrid logic with global, difference, and converse modalities. We also consider reflexive and transitive relations. For conversefree formulas we present a terminating control that does not rely on the usual chainbased blocking scheme. Our tableau systems are based on a new model existence theorem.
Biform theories in Chiron
 Towards Mechanized Mathematical Assistants, volume 4573 of Lecture Notes in Computer Science
, 2007
"... Abstract. An axiomatic theory represents mathematical knowledge declaratively as a set of axioms. An algorithmic theory represents mathematical knowledge procedurally as a set of algorithms. A biform theory is simultaneously an axiomatic theory and an algorithmic theory. It represents mathematical k ..."
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Cited by 8 (5 self)
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Abstract. An axiomatic theory represents mathematical knowledge declaratively as a set of axioms. An algorithmic theory represents mathematical knowledge procedurally as a set of algorithms. A biform theory is simultaneously an axiomatic theory and an algorithmic theory. It represents mathematical knowledge both declaratively and procedurally. Since the algorithms of algorithmic theories manipulate the syntax of expressions, biform theories—as well as algorithmic theories—are difficult to formalize in a traditional logic without the means to reason about syntax. Chiron is a derivative of vonNeumannBernaysGödel (nbg) set theory that is intended to be a practical, generalpurpose logic for mechanizing mathematics. It includes elements of type theory, a scheme for handling undefinedness, and a facility for reasoning about the syntax of expressions. It is an exceptionally wellsuited logic for formalizing biform theories. This paper defines the notion of a biform theory, gives an overview of Chiron, and illustrates how biform theories can be formalized in Chiron. 1
Chiron: A multiparadigm logic
 University of Bialystok
, 2007
"... Abstract. Chiron is a derivative of vonNeumannBernaysGödel (nbg) set theory that is intended to be a practical, generalpurpose logic for mechanizing mathematics. It supports several reasoning paradigms by integrating nbg set theory with elements of type theory, a scheme for handling undefinednes ..."
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Cited by 7 (5 self)
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Abstract. Chiron is a derivative of vonNeumannBernaysGödel (nbg) set theory that is intended to be a practical, generalpurpose logic for mechanizing mathematics. It supports several reasoning paradigms by integrating nbg set theory with elements of type theory, a scheme for handling undefinedness, and a facility for reasoning about the syntax of expressions. This paper gives a quick, informal presentation of the syntax and semantics of Chiron and then discusses some of the benefits Chiron provides as a multiparadigm logic. 1
OpenTheory: Package Management for Higher Order Logic Theories
"... Interactive theorem proving has grown from toy examples to major projects formalizing mathematics and verifying software, and there is now a critical need for theory engineering techniques to support these efforts. This paper introduces the OpenTheory project, which aims to provide an effective pack ..."
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Cited by 5 (3 self)
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Interactive theorem proving has grown from toy examples to major projects formalizing mathematics and verifying software, and there is now a critical need for theory engineering techniques to support these efforts. This paper introduces the OpenTheory project, which aims to provide an effective package management system for logical theories. The OpenTheory article format allows higher order logic theories to be exported from one theorem prover, compressed by a standalone tool, and imported into a different theorem prover. Articles naturally support theory interpretations, which is the mechanism by which theories can be cleanly transferred from one theorem prover context to another, and which also leads to more efficient developments of standard theories.
Declarative Programming for Agent Applications
"... This paper introduces the computational model of a declarative programming language intended for agent applications. Features supported by the language include functional and logic programming idioms, higherorder functions, modal computation, probabilistic computation, and some theoremproving capa ..."
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Cited by 1 (1 self)
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This paper introduces the computational model of a declarative programming language intended for agent applications. Features supported by the language include functional and logic programming idioms, higherorder functions, modal computation, probabilistic computation, and some theoremproving capabilities. The need for these features is motivated and examples are given to illustrate the central ideas.
Probabilities on Sentences in an Expressive Logic
, 2012
"... 1 Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higherorder logic are ideally suited for repre ..."
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1 Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higherorder logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truthvalues. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability of being true, what probability should be ascribed to other (query) sentences? A natural wishlist, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wishlist into technical requirements for a prior probability
Terminating Tableaux for the Basic Fragment of Simple Type Theory
, 2009
"... We consider the basic fragment of simple type theory, which restricts equations to base types and disallows lambda abstractions and quantifiers. We show that this fragment has the finite model property and that satisfiability can be decided with a terminating tableau system. Both results are with re ..."
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Cited by 1 (1 self)
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We consider the basic fragment of simple type theory, which restricts equations to base types and disallows lambda abstractions and quantifiers. We show that this fragment has the finite model property and that satisfiability can be decided with a terminating tableau system. Both results are with respect to standard models. 1
Theoretical Foundations for Practical ‘Totally Functional Programming’
, 2007
"... Interpretation is an implicit part of today’s programming; it has great power but is overused and has
significant costs. For example, interpreters are typically significantly hard to understand and hard
to reason about. The methodology of “Totally Functional Programming” (TFP) is a reasoned
attempt ..."
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Interpretation is an implicit part of today’s programming; it has great power but is overused and has
significant costs. For example, interpreters are typically significantly hard to understand and hard
to reason about. The methodology of “Totally Functional Programming” (TFP) is a reasoned
attempt to redress the problem of interpretation. It incorporates an awareness of the undesirability
of interpretation with observations that definitions and a certain style of programming appear to
offer alternatives to it. Application of TFP is expected to lead to a number of significant outcomes,
theoretical as well as practical. Primary among these are novel programming languages to lessen or
eliminate the use of interpretation in programming, leading to betterquality software. However,
TFP contains a number of lacunae in its current formulation, which hinder development of these
outcomes. Among others, formal semantics and typesystems for TFP languages are yet to be
discovered, the means to reduce interpretation in programs is to be determined, and a detailed
explication is needed of interpretation, definition, and the differences between the two. Most
important of all however is the need to develop a complete understanding of the nature of
interpretation. In this work, suitable typesystems for TFP languages are identified, and guidance
given regarding the construction of appropriate formal semantics. Techniques, based around the
‘fold’ operator, are identified and developed for modifying programs so as to reduce the amount of
interpretation they contain. Interpretation as a means of languageextension is also investigated.
v
Finally, the nature of interpretation is considered. Numerous hypotheses relating to it considered in
detail. Combining the results of those analyses with discoveries from elsewhere in this work leads
to the proposal that interpretation is not, in fact, symbolbased computation, but is in fact something
more fundamental: computation that varies with input. We discuss in detail various implications of
this characterisation, including its practical application. An often moreuseful property, ‘inherent
interpretiveness’, is also motivated and discussed in depth. Overall, our inquiries act to give
conceptual and theoretical foundations for practical TFP.
Composable Packages for Higher Order Logic Theories
"... Interactive theorem proving is tackling ever larger formalization and verification projects, and there is a critical need for theory engineering techniques to support these efforts. One such technique is effective package management, which has the potential to simplify the development of logical the ..."
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Interactive theorem proving is tackling ever larger formalization and verification projects, and there is a critical need for theory engineering techniques to support these efforts. One such technique is effective package management, which has the potential to simplify the development of logical theories by precisely checking dependencies and promoting reuse. This paper introduces a domainspecific language for defining composable packages of higher order logic theories, which is designed to naturally handle the complex dependency structures that often arise in theory development. The package composition language functions as a module system for theories, and the paper presents a welldefined semantics for the supported operations. Preliminary tests of the package language and its toolset have been made by packaging the theories distributed with the HOL Light theorem prover. This experience is described, leading to some initial theory engineering discussion on the ideal properties of a reusable theory. 1