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On the construction of free algebras for equational systems
 IN: SPECIAL ISSUE FOR AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP 2007). VOLUME 410 OF THEORETICAL COMPUTER SCIENCE
, 2009
"... The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary and transfinite construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applica ..."
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The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary and transfinite construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applications.
Cartesian closed 2categories and permutation equivalence in higherorder rewriting
, 2010
"... Vol. 9(3:10)2013, pp. 1–22 www.lmcsonline.org ..."
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Secondorder algebraic theories
, 2010
"... Technical reports published by the University of Cambridge Computer Laboratory are freely available via the Internet: ..."
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Technical reports published by the University of Cambridge Computer Laboratory are freely available via the Internet:
Initial Algebra Semantics for Cyclic Sharing Structures
"... Abstract. Terms are a concise representation of tree structures. Since they can be naturally defined by an inductive type, they offer data structures in functional programming and mechanised reasoning with useful principles such as structural induction and structural recursion. In the case of graphs ..."
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Abstract. Terms are a concise representation of tree structures. Since they can be naturally defined by an inductive type, they offer data structures in functional programming and mechanised reasoning with useful principles such as structural induction and structural recursion. In the case of graphs or ”treelike ” structures – trees involving cycles and sharing – however, it is not clear what kind of inductive structures exists and how we can faithfully assign a term representation of them. In this paper we propose a simple term syntax for cyclic sharing structures that admits structural induction and recursion principles. We show that the obtained syntax is directly usable in the functional language Haskell, as well as ordinary data structures such as lists and trees. To achieve this goal, we use categorical approach to initial algebra semantics in a presheaf category. That approach follows the line of Fiore, Plotkin and Turi’s models of abstract syntax with variable binding. 1
Term Equational Systems and Logics (Extended Abstract)
"... We introduce an abstract general notion of system of equations between terms, called Term Equational System, and develop a sound logical deduction system, called Term Equational Logic, for equational reasoning. Further, we give an analysis of algebraic free constructions that together with an intern ..."
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We introduce an abstract general notion of system of equations between terms, called Term Equational System, and develop a sound logical deduction system, called Term Equational Logic, for equational reasoning. Further, we give an analysis of algebraic free constructions that together with an internal completeness result may be used to synthesise complete equational logics. Indeed, as an application, we synthesise a sound and complete nominal equational logic, called Synthetic Nominal Equational Logic, based on the category of Nominal Sets.
Polymorphic abstract syntax via Grothendieck construction
 In FoSSaCS’11, LNCS3467
, 2011
"... Abstract. Abstract syntax with variable binding is known to be characterised as an initial algebra in a presheaf category. This paper extends it to the case of polymorphic typed abstract syntax with binding. We consider two variations, secondorder and higherorder polymorphic syntax. The central i ..."
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Abstract. Abstract syntax with variable binding is known to be characterised as an initial algebra in a presheaf category. This paper extends it to the case of polymorphic typed abstract syntax with binding. We consider two variations, secondorder and higherorder polymorphic syntax. The central idea is to apply Fiore’s initial algebra characterisation of typed abstract syntax with binding repeatedly, i.e. first to the type structure and secondly to the term structure of polymorphic system. In this process, we use the Grothendieck construction to combine differently staged categories of polymorphic contexts. 1
Multiversal polymorphic algebraic theories
 Proc. LICS
, 2013
"... Abstract—We formalise and study the notion of polymorphic algebraic theory, as understood in the mathematical vernacular as a theory presented by equations between polymorphicallytyped terms with both type and term variable binding. The prototypical example of a polymorphic algebraic theory is Syst ..."
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Abstract—We formalise and study the notion of polymorphic algebraic theory, as understood in the mathematical vernacular as a theory presented by equations between polymorphicallytyped terms with both type and term variable binding. The prototypical example of a polymorphic algebraic theory is System F, but our framework applies more widely. The extra generality stems from a mathematical analysis that has led to a unified theory of polymorphic algebraic theories with the following ingredients: polymorphic signatures that specify arbitrary polymorphic operators (e.g. as in extended λcalculi and algebraic effects); metavariables, both for types and terms, that enable the generic description of metatheories; multiple type universes that allow a notion of translation between theories that is parametric over different type universes; polymorphic structures that provide a general notion of algebraic model (including the PLcategory semantics of System F); a Polymorphic Equational Logic that constitutes a sound and complete logical framework for equational reasoning. Our work is semantically driven, being based on a hierarchical twolevelled algebraic modelling of abstract syntax with variable binding. Index Terms—polymorphism, equational logic, presheaves, categorical semantics, the Grothendieck construction
SecondOrder Equational Logic (Extended Abstract)
"... We provide an extension of universal algebra and its equational logic from first to second order. Conservative extension, soundness, and completeness results are established. ..."
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We provide an extension of universal algebra and its equational logic from first to second order. Conservative extension, soundness, and completeness results are established.
Algebraic MetaTheories and . . .
"... Fiore and Hur [18] recently introduced a novel methodology—henceforth referred to as Sol—for the Synthesis of equational and rewriting logics from mathematical models. In [18], Sol was successfully applied to rationally reconstruct the traditional equational logic for universal algebra of Birkhoff [ ..."
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Fiore and Hur [18] recently introduced a novel methodology—henceforth referred to as Sol—for the Synthesis of equational and rewriting logics from mathematical models. In [18], Sol was successfully applied to rationally reconstruct the traditional equational logic for universal algebra of Birkhoff [3] and its multisorted version [26], and also to synthesise a new version of the Nominal Algebra of Gabbay and Mathijssen [41] and the Nominal Equational Logic of Clouston and Pitts [8] for reasoning about languages with namebinding operators. Based on these case studies and further preliminary investigations, we contend that Sol can make an impact in the problem of engineering logics for modern computational languages. For example, our proposed research on secondorder equational logic will provide foundations for designing a secondorder extension of the Maude system [37], a firstorder semantic and logical framework used in formal software engineering for specification and programming. Our research strategy can be visualised as follows: (I)