A Logical Theory of Concurrent Objects and Its Realization in the Maude Language∗
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Initial Algebra Semantics in Matching Logic
Technical Report: Initial Algebra Semantics in Matching Logic Xiaohong Chen1, Dorel Lucanu2, and Grigore Ro¸su1 1University of Illinois at Urbana-Champaign, Champaign, USA 2Alexandru Ioan Cuza University, Ia¸si,Romania xc3@illinois, [email protected], [email protected] July 24, 2020 Abstract Matching logic is a unifying foundational logic for defining formal programming language semantics, which adopts a minimalist design with few primitive constructs that are enough to express all properties within a variety of logical systems, including FOL, separation logic, (dependent) type systems, modal µ-logic, and more. In this paper, we consider initial algebra semantics and show how to capture it by matching logic specifications. Formally, given an algebraic specification E that defines a set of sorts (of data) and a set of operations whose behaviors are defined by a set of equational axioms, we define a corresponding matching logic specification, denoted INITIALALGEBRA(E), whose models are exactly the initial algebras of E. Thus, we reduce initial E-algebra semantics to the matching logic specifications INITIALALGEBRA(E), and reduce extrinsic initial E-algebra reasoning, which includes inductive reasoning, to generic, intrinsic matching logic reasoning. 1 Introduction Initial algebra semantics is a main approach to formal programming language semantics based on algebraic specifications and their initial models. It originated in the 1970s, when algebraic techniques started to be applied to specify basic data types such as lists, trees, stacks, etc. The original paper on initial algebra semantics [Goguen et al., 1977] reviewed various existing algebraic specification techniques and showed that they were all initial models, making the concept of initiality explicit for the first time in formal language semantics. -
PDF (Dissertation.Pdf)
Kind Theory Thesis by Joseph R. Kiniry In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2002 (Defended 10 May 2002) ii © 2002 Joseph R. Kiniry All Rights Reserved iii Preface This thesis describes a theory for representing, manipulating, and reasoning about structured pieces of knowledge in open collaborative systems. The theory's design is motivated by both its general model as well as its target user commu- nity. Its model is structured information, with emphasis on classification, relative structure, equivalence, and interpretation. Its user community is meant to be non-mathematicians and non-computer scientists that might use the theory via computational tool support once inte- grated with modern design and development tools. This thesis discusses a new logic called kind theory that meets these challenges. The core of the work is based in logic, type theory, and universal algebras. The theory is shown to be efficiently implementable, and several parts of a full realization have already been constructed and are reviewed. Additionally, several software engineering concepts, tools, and technologies have been con- structed that take advantage of this theoretical framework. These constructs are discussed as well, from the perspectives of general software engineering and applied formal methods. iv Acknowledgements I am grateful to my initial primary adviser, Prof. K. Mani Chandy, for bringing me to Caltech and his willingness to let me explore many unfamiliar research fields of my own choosing. I am also appreciative of my second adviser, Prof. Jason Hickey, for his support, encouragement, feedback, and patience through the later years of my work. -
MONADS and ALGEBRAS I I
\chap10" 2009/6/1 i i page 223 i i 10 MONADSANDALGEBRAS In the foregoing chapter, the adjoint functor theorem was seen to imply that the category of algebras for an equational theory T always has a \free T -algebra" functor, left adjoint to the forgetful functor into Sets. This adjunction describes the notion of a T -algebra in a way that is independent of the specific syntactic description given by the theory T , the operations and equations of which are rather like a particular presentation of that notion. In a certain sense that we are about to make precise, it turns out that every adjunction describes, in a \syntax invariant" way, a notion of an \algebra" for an abstract \equational theory." Toward this end, we begin with yet a third characterization of adjunctions. This one has the virtue of being entirely equational. 10.1 The triangle identities Suppose we are given an adjunction, - F : C D : U: with unit and counit, η : 1C ! UF : FU ! 1D: We can take any f : FC ! D to φ(f) = U(f) ◦ ηC : C ! UD; and for any g : C ! UD we have −1 φ (g) = D ◦ F (g): FC ! D: This we know gives the isomorphism ∼ HomD(F C; D) =φ HomC(C; UD): Now put 1UD : UD ! UD in place of g : C ! UD in the foregoing. We −1 know that φ (1UD) = D, and so 1UD = φ(D) = U(D) ◦ ηUD: i i i i \chap10" 2009/6/1 i i page 224 224 MONADS AND ALGEBRAS i i And similarly, φ(1FC ) = ηC , so −1 1FC = φ (ηC ) = FC ◦ F (ηC ): Thus we have shown that the two diagrams below commute. -
On Coalgebras Over Algebras
On coalgebras over algebras Adriana Balan1 Alexander Kurz2 1University Politehnica of Bucharest, Romania 2University of Leicester, UK 10th International Workshop on Coalgebraic Methods in Computer Science A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 1 / 31 Outline 1 Motivation 2 The final coalgebra of a continuous functor 3 Final coalgebra and lifting 4 Commuting pair of endofunctors and their fixed points A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 2 / 31 Category with no extra structure Set: final coalgebra L is completion of initial algebra I [Barr 1993] Deficit: if H0 = 0, important cases not covered (as A × (−)n, D, Pκ+) Locally finitely presentable categories: Hom(B; L) completion of Hom(B; I ) for all finitely presentable objects B [Adamek 2003] Motivation Starting data: category C, endofunctor H : C −! C Among fixed points: final coalgebra, initial algebra Categories enriched over complete metric spaces: unique fixed point [Adamek, Reiterman 1994] Categories enriched over cpo: final coalgebra L coincides with initial algebra I [Plotkin, Smyth 1983] A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 3 / 31 Locally finitely presentable categories: Hom(B; L) completion of Hom(B; I ) for all finitely presentable objects B [Adamek 2003] Motivation Starting data: category C, endofunctor H : C −! C Among fixed points: final coalgebra, initial algebra Categories enriched over complete metric spaces: unique fixed point [Adamek, Reiterman 1994] Categories enriched over cpo: final coalgebra L coincides with initial algebra I [Plotkin, Smyth 1983] Category with no extra structure Set: final coalgebra L is completion of initial algebra I [Barr 1993] Deficit: if H0 = 0, important cases not covered (as A × (−)n, D, Pκ+) A. -
A Note on the Under-Appreciated For-Loop
A Note on the Under-Appreciated For-Loop Jos´eN. Oliveira Techn. Report TR-HASLab:01:2020 Oct 2020 HASLab - High-Assurance Software Laboratory Universidade do Minho Campus de Gualtar – Braga – Portugal http://haslab.di.uminho.pt TR-HASLab:01:2020 A Note on the Under-Appreciated For-Loop by Jose´ N. Oliveira Abstract This short research report records some thoughts concerning a simple algebraic theory for for-loops arising from my teaching of the Algebra of Programming to 2nd year courses at the University of Minho. Interest in this so neglected recursion- algebraic combinator grew recently after reading Olivier Danvy’s paper on folding over the natural numbers. The report casts Danvy’s results as special cases of the powerful adjoint-catamorphism theorem of the Algebra of Programming. A Note on the Under-Appreciated For-Loop Jos´eN. Oliveira Oct 2020 Abstract This short research report records some thoughts concerning a sim- ple algebraic theory for for-loops arising from my teaching of the Al- gebra of Programming to 2nd year courses at the University of Minho. Interest in this so neglected recursion-algebraic combinator grew re- cently after reading Olivier Danvy’s paper on folding over the natural numbers. The report casts Danvy’s results as special cases of the pow- erful adjoint-catamorphism theorem of the Algebra of Programming. 1 Context I have been teaching Algebra of Programming to 2nd year courses at Minho Uni- versity since academic year 1998/99, starting just a few days after AFP’98 took place in Braga, where my department is located. -
Philosophia Scientiæ, 18-3 | 2014, « Logic and Philosophy of Science in Nancy (I) » [Online], Online Since 01 October 2014, Connection on 05 November 2020
Philosophia Scientiæ Travaux d'histoire et de philosophie des sciences 18-3 | 2014 Logic and Philosophy of Science in Nancy (I) Selected Contributed Papers from the 14th International Congress of Logic, Methodology and Philosophy of Science Pierre Édouard Bour, Gerhard Heinzmann, Wilfrid Hodges and Peter Schroeder-Heister (dir.) Electronic version URL: http://journals.openedition.org/philosophiascientiae/957 DOI: 10.4000/philosophiascientiae.957 ISSN: 1775-4283 Publisher Éditions Kimé Printed version Date of publication: 1 October 2014 ISBN: 978-2-84174-689-7 ISSN: 1281-2463 Electronic reference Pierre Édouard Bour, Gerhard Heinzmann, Wilfrid Hodges and Peter Schroeder-Heister (dir.), Philosophia Scientiæ, 18-3 | 2014, « Logic and Philosophy of Science in Nancy (I) » [Online], Online since 01 October 2014, connection on 05 November 2020. URL : http://journals.openedition.org/ philosophiascientiae/957 ; DOI : https://doi.org/10.4000/philosophiascientiae.957 This text was automatically generated on 5 November 2020. Tous droits réservés 1 This issue collects a selection of contributed papers presented at the 14th International Congress of Logic, Methodology and Philosophy of Science in Nancy, July 2011. These papers were originally presented within three of the main sections of the Congress. They deal with logic, philosophy of mathematics and cognitive science, and philosophy of technology. A second volume of contributed papers, dedicated to general philosophy of science, and other topics in the philosophy of particular sciences, will appear in the next issue of Philosophia Scientiæ (19-1), 2015. Philosophia Scientiæ, 18-3 | 2014 2 TABLE OF CONTENTS Logic and Philosophy of Science in Nancy (I) Preface Pierre Edouard Bour, Gerhard Heinzmann, Wilfrid Hodges and Peter Schroeder-Heister Copies of Classical Logic in Intuitionistic Logic Jaime Gaspar A Critical Remark on the BHK Interpretation of Implication Wagner de Campos Sanz and Thomas Piecha Gödel’s Incompleteness Phenomenon—Computationally Saeed Salehi Meinong and Husserl on Existence. -
Dualising Initial Algebras
Math. Struct. in Comp. Science (2003), vol. 13, pp. 349–370. c 2003 Cambridge University Press DOI: 10.1017/S0960129502003912 Printed in the United Kingdom Dualising initial algebras NEIL GHANI†,CHRISTOPHLUTH¨ ‡, FEDERICO DE MARCHI†¶ and J O H NPOWER§ †Department of Mathematics and Computer Science, University of Leicester ‡FB 3 – Mathematik und Informatik, Universitat¨ Bremen §Laboratory for Foundations of Computer Science, University of Edinburgh Received 30 August 2001; revised 18 March 2002 Whilst the relationship between initial algebras and monads is well understood, the relationship between final coalgebras and comonads is less well explored. This paper shows that the problem is more subtle than might appear at first glance: final coalgebras can form monads just as easily as comonads, and, dually, initial algebras form both monads and comonads. In developing these theories we strive to provide them with an associated notion of syntax. In the case of initial algebras and monads this corresponds to the standard notion of algebraic theories consisting of signatures and equations: models of such algebraic theories are precisely the algebras of the representing monad. We attempt to emulate this result for the coalgebraic case by first defining a notion of cosignature and coequation and then proving that the models of such coalgebraic presentations are precisely the coalgebras of the representing comonad. 1. Introduction While the theory of coalgebras for an endofunctor is well developed, less attention has been given to comonads.Wefeel this is a pity since the corresponding theory of monads on Set explains the key concepts of universal algebra such as signature, variables, terms, substitution, equations,and so on. -
Corecursive Algebras: a Study of General Structured Corecursion (Extended Abstract)
Corecursive Algebras: A Study of General Structured Corecursion (Extended Abstract) Venanzio Capretta1, Tarmo Uustalu2, and Varmo Vene3 1 Institute for Computer and Information Science, Radboud University Nijmegen 2 Institute of Cybernetics, Tallinn University of Technology 3 Department of Computer Science, University of Tartu Abstract. We study general structured corecursion, dualizing the work of Osius, Taylor, and others on general structured recursion. We call an algebra of a functor corecursive if it supports general structured corecur- sion: there is a unique map to it from any coalgebra of the same functor. The concept of antifounded algebra is a statement of the bisimulation principle. We show that it is independent from corecursiveness: Neither condition implies the other. Finally, we call an algebra focusing if its codomain can be reconstructed by iterating structural refinement. This is the strongest condition and implies all the others. 1 Introduction A line of research started by Osius and Taylor studies the categorical foundations of general structured recursion. A recursive coalgebra (RCA) is a coalgebra of a functor F with a unique coalgebra-to-algebra morphism to any F -algebra. In other words, it is an algebra guaranteeing unique solvability of any structured recursive diagram. The notion was introduced by Osius [Osi74] (it was also of interest to Eppendahl [Epp99]; we studied construction of recursive coalgebras from coalgebras known to be recursive with the help of distributive laws of functors over comonads [CUV06]). Taylor introduced the notion of wellfounded coalgebra (WFCA) and proved that, in a Set-like category, it is equivalent to RCA [Tay96a,Tay96b],[Tay99, Ch. -
Cognitive Systems Research
Cognitive Systems Research A Uniform Model of Computational Conceptual Blending --Manuscript Draft-- Manuscript Number: COGSYS-D-20-00015 Article Type: Research Paper Keywords: conceptual blending; computational creativity; amalgams; category theory; Case- based reasoning Abstract: We present a mathematical model for the cognitive operation of conceptual blending that aims at being uniform across different representation formalisms, while capturing the relevant structure of this operation. The model takes its inspiration from amalgams as applied in case-base reasoning, but lifts them into the theory of categories so as to follow Joseph Goguen's intuition for a mathematically precise characterisation of conceptual blending at a representation-independent level of abstraction. We prove that our amalgam-based category-theoretical model of conceptual blending is essentially equivalent to the pushout model in the ordered category of partial maps as put forward by Joseph Goguen. But unlike Goguen's approach, our model is more suitable for a computational realisation of conceptual blending, and we exemplify this by concretising our model to computational conceptual blends for various representation formalisms and application domains. Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation Manuscript File A Uniform Model of Computational Conceptual Blending Marco Schorlemmer and Enric Plaza Artificial Intelligence Reserach Institute, IIIA-CSIC Bellaterra (Barcelona), Catalonia, Spain Abstract We present a mathematical model for the cognitive operation of conceptual blending that aims at being uniform across di↵erent representation formalisms, while capturing the relevant structure of this operation. The model takes its in- spiration from amalgams as applied in case-based reasoning, but lifts them into the theory of categories so as to follow Joseph Goguen’s intuition for a math- ematically precise characterisation of conceptual blending at a representation- independent level of abstraction. -
A Presentation of the Initial Lift-Algebra'
JOURNAL OF PURE AND APPLIED ALGEBRA ELSBVlER Journal of pure and Applied Algebra 116 (1997) 185-198 A presentation of the initial lift-algebra’ Mamuka Jibladze Universiteit Utrecht, Vakgroep Wiskeinde, Postbus 80010, 3508 TA Utrecht, Netherlandr Received 5 February 1996; revised 23 July 1996 Abstract The object of study of the present paper may be considered as a model, in an elementary topos with a natural numbers object, of a non-classical variation of the Peano arithmetic. The new feature consists in admitting, in addition to the constant (zero) SOE N and the unary operation (the successor map) SI : N + N, arbitrary operations s,, : N” --f N of arities u ‘between 0 and 1’. That is, u is allowed to range over subsets of a singleton set. @ 1997 Elsevier Science B.V. 1991 Math. Subj. Class.: Primary 18B25, 03G30; Secondary 68Q55 In view of the Peano axioms, the set of natural numbers can be considered as a solution of the ‘equation’ X + 1 = X. Lawvere with his notion of natural numbers object (NNO) gave precise meaning to this statement: here X varies over objects of some category S, 1 is a terminal object of this category, and X + 1 (let us call it the decidable lift of X) denotes coproduct of X and 1 in S; this obviously defines an endofunctor of S. Now for any E : S + S whatsoever, one solves the ‘equation’ E(X) = X by considering E-algebras: an E-algebra structure on an object X is a morphism E(X) 4 X, and one can form the category of these by defining a morphism from E(X) ---fX to another algebra E(Y) + Y to be an X + Y making E(X) -E(Y) J 1 X-Y ’ Supported by ISF Grant MXH200, INTAS Grant # 93-2618, Fellows’ Research Fund of St John’s College and Leverhulme Fund of the Isaac Newton Institute, Cambridge, UK. -
Final Coalgebras from Corecursive Algebras
Final Coalgebras from Corecursive Algebras Paul Blain Levy School of Computer Science, University of Birmingham, UK [email protected] Abstract We give a technique to construct a final coalgebra in which each element is a set of formulas of modal logic. The technique works for both the finite and the countable powerset functors. Starting with an injectively structured, corecursive algebra, we coinductively obtain a suitable subalgebra called the “co-founded part”. We see – first with an example, and then in the general setting of modal logic on a dual adjunction – that modal theories form an injectively structured, corecursive algebra, so that this construction may be applied. We also obtain an initial algebra in a similar way. We generalize the framework beyond Set to categories equipped with a suitable factorization op system, and look at the examples of Poset and Set . 1998 ACM Subject Classification F.1.1 Models of Computation, F.1.2 Modes of Computation, F.4.1 Mathematical Logic – Modal logic Keywords and phrases coalgebra, modal logic, bisimulation, category theory, factorization sys- tem Digital Object Identifier 10.4230/LIPIcs.CALCO.2015.221 1 Introduction 1.1 The Problem Consider image-countable labelled transition systems, i.e. coalgebras for the Set functor A B : X 7→ (Pc X) . Here A is a fixed set (not necessarily countable) of labels and Pc X is the set of countable subsets of X. It is well-known [25] that, in order to distinguish all pairs of non-bisimilar states, Hennessy-Milner logic with finitary conjunction is not sufficiently expressive, and we instead require infinitary conjunction. -
Constructively Characterizing Fold and Unfold*
Constructively Characterizing Fold and Unfold? Tjark Weber1 and James Caldwell2 1 Institut fur¨ Informatik, Technische Universit¨at Munchen¨ Boltzmannstr. 3, D-85748 Garching b. Munchen,¨ Germany [email protected] 2 Department of Computer Science, University of Wyoming Laramie, Wyoming 82071-3315 [email protected] Abstract. In this paper we formally state and prove theorems charac- terizing when a function can be constructively reformulated using the recursion operators fold and unfold, i.e. given a function h, when can a function g be constructed such that h = fold g or h = unfold g? These results are refinements of the classical characterization of fold and unfold given by Gibbons, Hutton and Altenkirch in [6]. The proofs presented here have been formalized in Nuprl's constructive type theory [5] and thereby yield program transformations which map a function h (accom- panied by the evidence that h satisfies the required conditions), to a function g such that h = fold g or, as the case may be, h = unfold g. 1 Introduction Under the proofs-as-programs interpretation, constructive proofs of theorems re- lating programs yield \correct-by-construction" program transformations. In this paper we formally prove constructive theorems characterizing when a function can be formulated using the recursion operators fold and unfold, i.e. given a func- tion h, when does there exist (constructively) a function g such that h = fold g or h = unfold g? The proofs have been formalized in Nuprl's constructive type theory [1, 5] and thereby yield program transformations which map a function h { accompanied by the evidence that h satisfies the required conditions { to a function g such that h = fold g or, as the case may be, h = unfold g.