Graded consequence: an institution theoretic study

Abstract

We develop a general study of graded consequence (of many-valued logic) in an institution theoretic (in the sense of Goguen and Burstall) style. This means both syntax and semantics are considered fully abstract, as well as the satisfaction between them. Our approach contrasts to other approaches on many-valued logic in that it is a multisignature one, in the spirit of institution theory. We consider graded consequence at three different conceptual levels: entailment, semantic, and closure operators, and explore several interpretations between them. We also study logical connectors and quantifiers both at the entailment and semantic level, compactness and soundness properties.

Many-Valued Institutions for Constraint Specification

“…Building on this formalisation, we have shown how the selection of a best supplier in the context of service discovery and binding can be defined in terms of graded semantic consequence, and we have studied the unpredictability of the evolution of service applications that originates from the change of the truth structures that underlie the service components. In order to facilitate an implementation of our model-theoretical approach to choosing a best supplier, we intend to further examine sound and complete proof systems defined in terms of many-valued rules as in [10]. These could be used in the development of operational semantics for the execution of such serviceoriented applications (i.e.…”

“…2, we first extend the traditional notion of institution along the lines of [10] by replacing the boolean space of truth values with residuated lattices [16], which offer a unifying truth structure for both idempotent c-semirings and valuation structures [3]. Using a simple example, we explain how first-order logic specifications can be extended with soft constraints, and then show how this extension can be generalised to define a logical system of soft constraints as a many-valued institution parameterised by a stratified logic [1].…”

“…Based on this construction, in Sec. 3, we formalise the mechanism of selecting a most promising provider of a needed resource in the context of service discovery and binding on the quantification of the compatibility of two constraint specifications as a value of a residuated lattice; we achieve this by defining a compatibility score using the concept of graded semantic consequence [10]. Lastly, in Sec.…”

“…The operation * captures the accumulation of truth values that result from successive inferences, and → corresponds to the entailment between two degrees of satisfaction. To capture soft csp as a many-valued logical system, we therefore extend the notion of institution in keeping with [10]: …”

Abstract Constraint Data Types

“…Following in his footsteps, we provide an extension of the traditional notion of algebraic data type specification to encompass soft-constraints as formalised in [1]. Finally, we relate this extension with institutions [6] and recent work on graded consequence in institutions [3]. …”