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72
Theory Interpretation in Simple Type Theory
 HIGHERORDER ALGEBRA, LOGIC, AND TERM REWRITING, VOLUME 816 OF LECTURE NOTES IN COMPUTER SCIENCE
, 1993
"... Theory interpretation is a logical technique for relating one axiomatic theory to another with important applications in mathematics and computer science as well as in logic itself. This paper presents a method for theory interpretation in a version of simple type theory, called lutins, which admit ..."
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Cited by 36 (16 self)
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Theory interpretation is a logical technique for relating one axiomatic theory to another with important applications in mathematics and computer science as well as in logic itself. This paper presents a method for theory interpretation in a version of simple type theory, called lutins, which admits partial functions and subtypes. The method is patterned on the standard approach to theory interpretation in rstorder logic. Although the method is based on a nonclassical version of simple type theory, it is intended as a guide for theory interpretation in classical simple type theories as well as in predicate logics with partial functions.
Interpretability logic
 Mathematical Logic, Proceedings of the 1988 Heyting Conference
, 1990
"... Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbertstyle programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength ..."
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Cited by 33 (9 self)
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Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbertstyle programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength of theories, or better to prove
A New Method for Undecidability Proofs of First Order Theories
 Journal of Symbolic Computation
, 1992
"... this paper is to define a framework for such reduction proofs. The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering. 1. Introduction ..."
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Cited by 29 (6 self)
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this paper is to define a framework for such reduction proofs. The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering. 1. Introduction
On the Difference between Bridge Rules and Lifting Axioms
 Modeling and Using Context, number 2680 in Lecture Notes in Artificial Intelligence
, 2003
"... In a previous paper, we proposed a first formal and conceptual comparison between the two most important formalizations of context in AI: Propositional Logic of Context (PLC) and Local Models Semantics/MultiContext Systems (LMS/MCS). The result was that LMS/MCS is at least as general as PLC, as i ..."
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Cited by 10 (0 self)
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In a previous paper, we proposed a first formal and conceptual comparison between the two most important formalizations of context in AI: Propositional Logic of Context (PLC) and Local Models Semantics/MultiContext Systems (LMS/MCS). The result was that LMS/MCS is at least as general as PLC, as it can be embedded into a particular class of MCS, called MPLC. In this paper we go beyond that result, and prove that, under some important restrictions (including the hypothesis that each context has finite and homogeneous propositional languages), MCS can be embedded in PLC with generic axioms. To prove this theorem, we prove that MCS cannot be embedded in PLC using only lifting axioms to encode bridge rules. This is an important result for a general theory of context and contextual reasoning, as it proves that lifting axioms and entering context are not enough to capture all forms of contextual reasoning that can be captured via bridge rules in LMS/MCS.
Effectively Closed Sets
 ASL Lecture Notes in Logic
"... Abstract. We investigate notions of randomness in the space C[2 IN] of nonempty closed subsets of {0, 1} IN. A probability measure is given and a version of the MartinLöf Test for randomness is defined. Π 0 2 random closed sets exist but there are no random Π 0 1 closed sets. It is shown that a ran ..."
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Cited by 9 (5 self)
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Abstract. We investigate notions of randomness in the space C[2 IN] of nonempty closed subsets of {0, 1} IN. A probability measure is given and a version of the MartinLöf Test for randomness is defined. Π 0 2 random closed sets exist but there are no random Π 0 1 closed sets. It is shown that a random closed set is perfect, has measure 0, and has no computable elements. A closed subset of 2 IN may be defined as the set of infinite paths through a tree and so the problem of compressibility of trees is explored. This leads to some results on a Chaitinstyle notion of randomness for closed sets. 1
The Interpretability Logic of all Reasonable Arithmetical Theories
 ERKENNTNIS
, 1999
"... This paper is a presentation of a status quaestionis, to wit of the problem of the interpretability logic of all reasonable arithmetical theories. We present both the arithmetical side and the modal side of the question. ..."
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Cited by 9 (5 self)
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This paper is a presentation of a status quaestionis, to wit of the problem of the interpretability logic of all reasonable arithmetical theories. We present both the arithmetical side and the modal side of the question.
Hierarchies of Decidable Extensions of Bounded Quantification
 IN 22ND ACM SYMP. ON PRINCIPLES OF PROGRAMMING LANGUAGES
, 1994
"... The system F , the wellknown secondorder polymorphic typed calculus with subtyping and bounded universal type quantification [CW85, BL90, CG92, Pie92, CMMS94], appears to be undecidable [Pie92] because of undecidability of its subtyping component. Attempts were made to obtain decidable type sys ..."
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Cited by 7 (5 self)
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The system F , the wellknown secondorder polymorphic typed calculus with subtyping and bounded universal type quantification [CW85, BL90, CG92, Pie92, CMMS94], appears to be undecidable [Pie92] because of undecidability of its subtyping component. Attempts were made to obtain decidable type systems with subtyping by weakening F [CP94, KS92], and also by reinforcing or extending it [Vor94a, Vor94b, Vor95]. However, for the moment, these extensions lack the important prooftheoretic minimum type property, which holds for F and guarantees that each typable term has the minimum type, being a subtype of any other type of the term in the same context [CG92, Vor94c]. As a preparation step to introducing the extensions of F with the minimum type property and the decidable term typing relation (which we do in [Vor94e]), we define and study here the hierarchies of decidable extensions of the F subtyping relation. We demonstrate conditions providing that each theory in a hierarchy: 1. ext...
On the Use of Reduction Relations to Relate Different Types of Agent Models
"... This paper focuses on relationships between agent models and their physical realisations. Approaches on reduction from philosophical literature are analysed in a formalised manner, and extended by incorporating contextdependency w.r.t. specific makeups for their realisations. It is shown how thes ..."
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Cited by 7 (4 self)
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This paper focuses on relationships between agent models and their physical realisations. Approaches on reduction from philosophical literature are analysed in a formalised manner, and extended by incorporating contextdependency w.r.t. specific makeups for their realisations. It is shown how these contextdependent reduction approaches can be translated into each other and how they can be applied to relate agent models.
Does Reductive Proof Theory Have A Viable Rationale?
 Erkenntnis
, 2000
"... The goals of reduction and reductionism in the natural sciences are mainly explanatory in character, while those in mathematics are primarily foundational. In contrast to global reductionist programs which aim to reduce all of mathematics to one supposedly "universal " system or founda ..."
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Cited by 5 (0 self)
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The goals of reduction and reductionism in the natural sciences are mainly explanatory in character, while those in mathematics are primarily foundational. In contrast to global reductionist programs which aim to reduce all of mathematics to one supposedly "universal " system or foundational scheme, reductive proof theory pursues local reductions of one formal system to another which is more justified in some sense. In this direction, two specific rationales have been proposed as aims for reductive proof theory, the constructive consistencyproof rationale and the foundational reduction rationale. However, recent advances in proof theory force one to consider the viability of these rationales. Despite the genuine problems of foundational significance raised by that work, the paper concludes with a defense of reductive proof theory at a minimum as one of the principal means to lay out what rests on what in mathematics. In an extensive appendix to the paper, various reducti...