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12
Nominal algebra
, 2006
"... Nominal terms are a termlanguage used to accurately and expressively represent systems with binding. We present Nominal Algebra (NA), a theory of algebraic equality on nominal terms. Builtin support for binding in the presence of metavariables allows NA to closely mirror informal mathematical us ..."
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Nominal terms are a termlanguage used to accurately and expressively represent systems with binding. We present Nominal Algebra (NA), a theory of algebraic equality on nominal terms. Builtin support for binding in the presence of metavariables allows NA to closely mirror informal mathematical usage and notation, where expressions such as λa.t or ∀a.φ are common, in which metavariables t and φ explicitly occur in the scope of a variable a. We describe the syntax and semantics of NA, and provide a sound and complete proof system for it. We also give some examples of axioms; other work has considered sets of axioms of particular interest in some detail.
Borrowing Interpolation
, 2011
"... We present a generic method for establishing interpolation properties by ‘borrowing ’ across logical systems. The framework used is that of the socaled ‘institution theory’ which is a categorical abstract model theory providing a formal definition for the informal concept of ‘logical system’ and a ..."
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We present a generic method for establishing interpolation properties by ‘borrowing ’ across logical systems. The framework used is that of the socaled ‘institution theory’ which is a categorical abstract model theory providing a formal definition for the informal concept of ‘logical system’ and a mathematical concept of ‘homomorphism’ between logical systems. We develop three different styles or patterns to apply the proposed borrowing interpolation method. These three ways are illustrated by the development of a series of concrete interpolation results for logical systems that are used in mathematical logic or in computing science, most of these interpolation properties apparently being new results. These logical systems include fragments of (classical many sorted) first order logic with equality, preordered algebra and its Horn fragment, partial algebra, higher order logic. Applications are also expected for many other logical systems, including membership algebra, various types of order sorted algebra, the logic of predefined types, etc., and various combinations of the logical systems discussed here.
Combining Algebraic and SetTheoretic Specifications
 RECENT TRENDS IN DATA TYPE SPECIFICATION&QUOT;, PROC. 11TH WORKSHOP ON SPECIFICATION OF ABSTRACT DATA TYPES JOINT WITH THE 9TH GENERAL COMPASS WORKSHOP
, 1996
"... Specification frameworks such as B and Z provide power sets and cartesian products as builtin type constructors, and employ a rich notation for defining (among other things) abstract data types using formulae of predicate logic and lambdanotation. In contrast, the socalled algebraic specificat ..."
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Cited by 3 (2 self)
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Specification frameworks such as B and Z provide power sets and cartesian products as builtin type constructors, and employ a rich notation for defining (among other things) abstract data types using formulae of predicate logic and lambdanotation. In contrast, the socalled algebraic specification frameworks often limit the type structure to sort constants and firstorder functionalities, and restrict formulae to (conditional) equations. Here, we
Swinging Data Types: The Dielectic between Actions and Constructors
 REPORT, FB INFORMATIK, UNIVERSITÄT DORTMUND
, 1998
"... Initial structures are good for modelling constructorbased data types because they fit the intuition about these types and admit resolution and rewriteoriented inductive theorem proving. The corresponding specification and verification methods do not comply so well with nonfree or permutative ty ..."
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Cited by 1 (1 self)
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Initial structures are good for modelling constructorbased data types because they fit the intuition about these types and admit resolution and rewriteoriented inductive theorem proving. The corresponding specification and verification methods do not comply so well with nonfree or permutative types such as sets, bags and maps and are still less appropriate when infinite structures like streams or processes come into play. Nonfree and infinite structure are better modelled as dynamic objects, which are identified through reactions upon actions (methods, messages, state transitions) rather than through constructors they might be built of. Extensional, contextual, behavioural, observational or bisimilarity relations model object equality and the suitable domains are final structures that are conservative with respect to visible subtypes. Consequently, a collection of data types and programs should be designed hierarchically as a &quot;swinging &quot; chain of specifications each of which extends its predecessor by either constructor types or action types. Constructor types introduce the visible domains and come with inductively defined total functions, structural equality and safety predicates with Horn clause axioms, while action types provide the hidden domains together with coinductively defined partial functions, behavioural equality and liveness predicates with liveness axioms that are dual to Horn clauses. A swinging specification is interpreted as a sequence of initial and final models. General proof
Primitive Inductive Theorems Bridge Implicit Induction Methods and Inductive Theorems in HigherOrder Rewriting
"... Abstract. Automated reasoning of inductive theorems is considered important in program verification. To verify inductive theorems automatically, several implicit induction methods like the inductionless induction and the rewriting induction methods have been proposed. In studying inductive theorems ..."
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Abstract. Automated reasoning of inductive theorems is considered important in program verification. To verify inductive theorems automatically, several implicit induction methods like the inductionless induction and the rewriting induction methods have been proposed. In studying inductive theorems on higherorder rewritings, we found that the class of the theorems shown by known implicit induction methods does not coincide with that of inductive theorems, and the gap between them is a barrier in developing mechanized methods for disproving inductive theorems. This paper fills this gap by introducing the notion of primitive inductive theorems, and clarifying the relation between inductive theorems and primitive inductive theorems. Based on this relation, we achieve mechanized methods for proving and disproving inductive theorems.
Algebraic System Specification and Development: Survey and Annotated Bibliography  Second Edition 
, 1997
"... Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.5.4 Special Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.6 Semantics of Programming Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6.1 Semantics of Ada . . . ..."
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Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.5.4 Special Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.6 Semantics of Programming Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6.1 Semantics of Ada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6.2 Action Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.7 Specification Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.7.1 Early Algebraic Specification Languages . . . . . . . . . . . . . . . . . . . . . . . . 53 4.7.2 Recent Algebraic Specification Languages . . . . . . . . . . . . . . . . . . . . . . . 55 4.7.3 The Common Framework Initiative. . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5 Methodology 57 5.1 Development Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.1.1 Applica...
R^n and G^nLogics
 HigherOrder Algebra, Logic, and Term Rewriting, volume 1074 of Lecture Notes in Computer Science
, 1996
"... This paper proposes a simple, settheoretic framework providing expressive typing, higherorder functions and initial models at the same time. Building upon Russell's ramified theory of types, we develop the theory of R logics, which are axiomatisable by an ordersorted equational Horn ..."
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This paper proposes a simple, settheoretic framework providing expressive typing, higherorder functions and initial models at the same time. Building upon Russell's ramified theory of types, we develop the theory of R logics, which are axiomatisable by an ordersorted equational Horn logic with a membership predicate, and of G logics, that provide in addition partial functions. The latter are therefore more adapted to the use in the program specification domain, while sharing interesting properties, like existence of an initial model, with R logics. Operational semantics of R logics presentations is obtained through ordersorted conditional rewriting.
From Conventional to InstitutionIndependent Logic Programming
, 2014
"... We propose a logicindependent approach to logic programming through which the paradigm as we know it for Hornclause logic can be explored for other formalisms. Our investigation is based on abstractions of notions such as logic program, clause, query, solution, and computed answer, which we develo ..."
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We propose a logicindependent approach to logic programming through which the paradigm as we know it for Hornclause logic can be explored for other formalisms. Our investigation is based on abstractions of notions such as logic program, clause, query, solution, and computed answer, which we develop over Goguen and Burstall’s theory of institutions. These give rise to a series of concepts that formalize the interplay between the denotational and the operational semantics of logic programming. We examine properties concerning the satisfaction of quantified sentences, discuss a variant of Herbrand’s theorem that is not limited in scope to any particular logical system or construction of logic programs, and describe a general resolutionbased procedure for computing solutions to queries. We prove that this procedure is sound; moreover, under additional hypotheses that reflect faithfully properties of actual logicprogramming languages, we show that it is also complete.
2 Algebraic Preliminaries
"... The purpose of this chapter is to present the basic definitions and results on which the following chapters rely. Most of this material is quite standard and for that reason the presentation will be concise. More detailed presentations with greater emphasis on motivation, exercises, and examples may ..."
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The purpose of this chapter is to present the basic definitions and results on which the following chapters rely. Most of this material is quite standard and for that reason the presentation will be concise. More detailed presentations with greater emphasis on motivation, exercises, and examples may be found in [EM85, Wir90,LEW96,ST]. The most basic assumption of work on algebraic specification is that a program is modeled as an algebra, that is, a set of data together with a number of functions over this set. The branch of mathematics which deals with algebras in a general sense (as opposed to the study of specific classes of algebras, such as groups and rings) is called universal algebra or sometimes general algebra. This chapter presents the basics of universal algebra, generalized to the manysorted case as required to model programs which manipulate several kinds or sorts of data. Some extensions useful for modeling more complex programs are sketched at the end of the chapter. 2.1 Manysorted sets When using an algebra to model a program which manipulates several sorts of data, it is natural to partition the underlying set of values in the algebra so that there is one set of values for each sort of data. It is often convenient to manipulate such a family of sets as a unit in such a way that operations on this unit respect the “typing ” of data values. Let S be a set (of sorts). An Ssorted set is an Sindexed family of sets X = 〈Xs〉s∈S,whichisempty if Xs is empty for all s ∈ S. The empty Ssorted set is written ∅. Let X = 〈Xs〉s∈S and Y = 〈Ys〉s∈S be Ssorted sets. Union, intersection, Cartesian product, disjoint union, inclusion (subset), and equality of X and Y are defined as follows: