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Modular Swinging Types
, 1999
"... . Swinging types [18] provide an integrated framework for specifying software on the basis of manysorted logic in terms of "static" functions and relations as well as "dynamic" transition systems. Swinging types combine equational, Horn and modal logic for the purpose of using evaluation and pr ..."
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Cited by 8 (8 self)
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. Swinging types [18] provide an integrated framework for specifying software on the basis of manysorted logic in terms of "static" functions and relations as well as "dynamic" transition systems. Swinging types combine equational, Horn and modal logic for the purpose of using evaluation and proof rules from all three logics for rapid prototyping and verification. A swinging specification separates from each other visible sorts that denote domains of data identified by their structure; hidden sorts that denote domains of data identified by their behavior in response to observers; predicates (least relations) that represent inductive (ly provable) properties; and predicates (greatest relations) that represent complementary "coinductive" properties. The paper at hand deals with structured specifications with swinging components. Vertical structuring is supported by a deductionoriented refinement criterion that admits, for instance, to implement visible sorts by hidden s...
A.: Nominal algebra
, 2006
"... Abstract. 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 mathem ..."
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Cited by 7 (2 self)
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Abstract. 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. 1.
Combining Algebraic and SetTheoretic Specifications
 Recent Trends in Data Type Specification", 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.
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 "swinging " 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
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 logic ..."
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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.
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.
R nandG nLogics
"... is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Report Series. Copies may be obtained by contacting: BRICS ..."
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is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Report Series. Copies may be obtained by contacting: BRICS
Borrowing Interpolation
"... 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 ..."
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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. 1.