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30
Fresh Logic
 Journal of Applied Logic
, 2007
"... Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with resp ..."
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Cited by 219 (28 self)
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Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with respect to metavariables. We present oneandahalfthorder logic, in which these concepts are made explicit. We exhibit both sequent and algebraic specifications of oneandahalfthorder logic derivability, show them equivalent, show that the derivations satisfy cutelimination, and prove correctness of an interpretation of firstorder logic within it. We discuss the technicalities in a wider context as a casestudy for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation
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 14 (4 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 13 (7 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
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.
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 8 (6 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
Terminating Tableaux for Graded Hybrid Logic with Global Modalities and Role Hierarchies
"... Abstract. We present a terminating tableau calculus for graded hybrid logic with global modalities, reflexivity, transitivity and role hierarchies. Termination of the system is achieved through patternbased blocking. Previous approaches to related logics all rely on chainbased blocking. Besides be ..."
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Cited by 8 (1 self)
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Abstract. We present a terminating tableau calculus for graded hybrid logic with global modalities, reflexivity, transitivity and role hierarchies. Termination of the system is achieved through patternbased blocking. Previous approaches to related logics all rely on chainbased blocking. Besides being conceptually simple and suitable for efficient implementation, the patternbased approach gives us a NExpTime complexity bound for the decision procedure.
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 6 (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.
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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Cited by 5 (2 self)
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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 4 (3 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
A Qualitative Comparison of the Suitability of Four Theorem Provers for Basic Auction Theory
"... Novel auction schemes are constantly being designed. Their design has significant consequences for the allocation of goods and the revenues generated. But how to tell whether a new design has the desired properties, such as efficiency, i.e. allocating goods to those bidders who value them most? We s ..."
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Cited by 3 (3 self)
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Novel auction schemes are constantly being designed. Their design has significant consequences for the allocation of goods and the revenues generated. But how to tell whether a new design has the desired properties, such as efficiency, i.e. allocating goods to those bidders who value them most? We say: by formal, machinechecked proofs. We investigated the suitability of the Isabelle, Theorema, Mizar, and Hets/CASL/ TPTP theorem provers for reproducing a key result of auction theory: Vickrey’s 1961 theorem on the properties of secondprice auctions. Based on our formalisation experience, taking an auction designer’s perspective, we give recommendations on what system to use for formalising auctions, and outline further steps towards a complete auction theory toolbox.