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14
Fold and Unfold for Program Semantics
 In Proc. 3rd ACM SIGPLAN International Conference on Functional Programming
, 1998
"... In this paper we explain how recursion operators can be used to structure and reason about program semantics within a functional language. In particular, we show how the recursion operator fold can be used to structure denotational semantics, how the dual recursion operator unfold can be used to str ..."
Abstract

Cited by 22 (4 self)
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In this paper we explain how recursion operators can be used to structure and reason about program semantics within a functional language. In particular, we show how the recursion operator fold can be used to structure denotational semantics, how the dual recursion operator unfold can be used to structure operational semantics, and how algebraic properties of these operators can be used to reason about program semantics. The techniques are explained with the aid of two main examples, the first concerning arithmetic expressions, and the second concerning Milner's concurrent language CCS. The aim of the paper is to give functional programmers new insights into recursion operators, program semantics, and the relationships between them. 1 Introduction Many computations are naturally expressed as recursive programs defined in terms of themselves, and properties proved of such programs using some form of inductive argument. Not surprisingly, many programs will have a similar recursive stru...
Topological and smooth stacks
"... Abstract. We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a “stack over a stack ” and the distinction between small stacks (which are algebraic objects) and large stacks (which are ..."
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Cited by 8 (0 self)
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Abstract. We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a “stack over a stack ” and the distinction between small stacks (which are algebraic objects) and large stacks (which are generalized spaces). 1.
Faith & Falsity: a study of faithful interpretations and false Σ 0 1sentences
 Logic Group Preprint Series 216, Department of Philosophy, Utrecht University, Heidelberglaan 8, 3584 CS
, 2002
"... A theory T is trustworthy iff, whenever a theory U is interpretable in T, then it is faithfully interpretable. In this paper we provide a characterization of trustworthiness. We provide a simple proof of Friedman’s Theorem that finitely axiomatized, sequential, consistent theories are trustworthy. W ..."
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Cited by 6 (4 self)
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A theory T is trustworthy iff, whenever a theory U is interpretable in T, then it is faithfully interpretable. In this paper we provide a characterization of trustworthiness. We provide a simple proof of Friedman’s Theorem that finitely axiomatized, sequential, consistent theories are trustworthy. We provide an example of a theory whose schematic predicate logic is
Dynamic bracketing and discourse representation
 Philosophy Department, Utrecht University, March1995
"... In this paper we describe a framework for the construction of entities, that can serve asinterpretations of arbitrary contiguous chunks of text. An important part of the paper is devoted to describing stacking cells: the proposed meanings for bracketstructures. 1 ..."
Abstract

Cited by 6 (1 self)
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In this paper we describe a framework for the construction of entities, that can serve asinterpretations of arbitrary contiguous chunks of text. An important part of the paper is devoted to describing stacking cells: the proposed meanings for bracketstructures. 1
Categories of Theories and Interpretations
 Logic in Tehran. Proceedings of the workshop and conference on Logic, Algebra and Arithmetic, held October 18–22
"... In this paper we study categories of theories and interpretations. In these categories, notions of sameness of theories, like synonymy, biinterpretability and mutual interpretability, take the form of isomorphism. We study the usual notions like monomorphism and product in the various theories. We ..."
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Cited by 5 (3 self)
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In this paper we study categories of theories and interpretations. In these categories, notions of sameness of theories, like synonymy, biinterpretability and mutual interpretability, take the form of isomorphism. We study the usual notions like monomorphism and product in the various theories. We provide some examples to separate notions across categories. In contrast, we show that, in some cases, notions in different categories do coincide. E.g., we can, under suchandsuch conditions, infer synonymity of two theories from their being equivalent in the sense of a coarser equivalence relation. We illustrate that the categories offer an appropriate framework for conceptual analysis of notions. For example, we provide a ‘coordinate free ’ explication of the notion of axiom scheme. Also we give a closer analysis of the objectlanguage / metalanguage distinction. Our basic category can be enriched with a form of 2structure. We use
The Predicative Frege Hierarchy
, 2006
"... In this paper, we characterize the strength of the predicative Frege hierarchy, P n+1 V, introduced by John Burgess in his book [Bur05]. We show that P n+1 V and Q + con n (Q) are mutually interpretable. It follows that PV: = P 1 V is mutually interpretable with Q. This fact was proved earlier by Mi ..."
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Cited by 2 (0 self)
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In this paper, we characterize the strength of the predicative Frege hierarchy, P n+1 V, introduced by John Burgess in his book [Bur05]. We show that P n+1 V and Q + con n (Q) are mutually interpretable. It follows that PV: = P 1 V is mutually interpretable with Q. This fact was proved earlier by Mihai Ganea in [Gan06] using a different proof. Another consequence of the our main result is that P 2 V is mutually interpretable with Kalmar Arithmetic (a.k.a. EA, EFA, I∆0+EXP, Q3). The fact that P 2 V interprets EA, was proved earlier by Burgess. We provide a different proof. Each of the theories P n+1 V is finitely axiomatizable. Our main result implies that the whole hierarchy taken together, P ω V, is not finitely axiomatizable. What is more: no theory that is mutually locally interpretable
Homological Algebra of Racks and Quandles
"... Contents Introduction 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 Extensions 5 1.1 Extensions and expan ..."
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Cited by 1 (1 self)
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Contents Introduction 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 Extensions 5 1.1 Extensions and expansions . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Factor systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Abelian extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Quandle extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.5 Involutory extensions . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Modules 29 2.1 Rack modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 Beck modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3 A digression on Gmodules . . . . . . . . . . . . . . . . . . . . . 41 2.4 Free modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5 The rack
Selected Papers
, 2007
"... The CALCO Young Researchers Workshop, CALCOjnr, was a satellite event for 2nd Conference on Algebra and Coalgebra in Computer Science, August 2024, 2007, Bergen, Norway (CALCO’07). CALCOjnr was dedicated to presentations by PhD students and by those who completed their doctoral studies within the ..."
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The CALCO Young Researchers Workshop, CALCOjnr, was a satellite event for 2nd Conference on Algebra and Coalgebra in Computer Science, August 2024, 2007, Bergen, Norway (CALCO’07). CALCOjnr was dedicated to presentations by PhD students and by those who completed their doctoral studies within the past few years. This report contains selected, refereed papers from the workshop.
Cardinal Arithmetic in Weak Theories
, 2008
"... In this paper we develop the theory of cardinals in the theory COPY. This is the theory of two total, jointly injective binary predicates in a second order version, where we may quantify over binary relations. The only second order axioms of the theory are the axiom asserting the existence of an emp ..."
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In this paper we develop the theory of cardinals in the theory COPY. This is the theory of two total, jointly injective binary predicates in a second order version, where we may quantify over binary relations. The only second order axioms of the theory are the axiom asserting the existence of an empty relation and the adjunction axiom, which says that we may enrich any relation R with a pair x, y. The theory COPY is strictly weaker than the theory AS, adjunctive set theory. The relevant notion of weaker here is direct interpretability. We will explain and motivate this notion in the paper. A consequence is that our development of cardinals is inherited by stronger theories like AS. We will show that the cardinals satisfy (at least) Robinson’s Arithmetic Q. A curious aspect of our approach is that we develop cardinal multiplication using neither recursion nor pairing, thus diverging both from Frege’s paradigm and from the tradition in set theory. Our development directly uses the universal property characterizing the product that is familiar from category theory. The broader context of this paper is the study of a double degree structure: the degrees of (relative) interpretability and the finer degrees of direct interpretability. Most of the theories studied are in one of two degrees of interpretability: the bottom degree of predicate logic or the degree of Q. The theories will differ significantly if we compare them using direct interpretability.
Fibered category
, 2010
"... Beck modules provide a convenient notion of coefficient module to be used in cohomology theories. The notion makes sense in a broad context, and recovers the usual notions of module in familiar settings, such as groups, rings, commutative ..."
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Beck modules provide a convenient notion of coefficient module to be used in cohomology theories. The notion makes sense in a broad context, and recovers the usual notions of module in familiar settings, such as groups, rings, commutative