Modeling Contexts with Dependent Types

Abstract

In the area of knowledge representation, a challenging topic is the formalization of context knowledge on the basis of logical foundations and ontological semantics. However, most attempts to provide a formal model of contexts suffer from a number of difficulties, such as limited expressiveness of representation, restricted variable quantification, lack of (meta) reasoning about properties, etc. In addition, type theory originally developed for formal modeling of mathematics has also been successfully applied to the correct specification of programs and in the semantics of natural language. In this paper, we suggest a type theoretical approach to the problem of context and action modeling. Type theory is used both for representing the system's knowledge of the discourse domain and for reasoning about it. For that purpose, we extend an existing dependent type theory having nice properties, with context-based rules and appropriate inductive types. We claim that the resulting theory exploiting the power of dependent types is able to provide a very expressive system together with a unified theory allowing higher-order reasoning.

Applying a Dependency Mechanism for Voting Protocol Models Using Event-B

“…iii )Dependencies: this knowledge is deduced, and expressed as a combination of situations and constraints over time. The use of dependencies was inspired by the work of [17,2,8], and led us to formalize a dependency mechanism in Event-B as a proof of the coherency of the contexts in Event-B.…”

A type-theoretical approach for ontologies: The case of roles

“…Since an explicit context is able to encapsulate heterogeneous types in which any of them can (i) express a relation or a sortal denoting a type of particulars, e.g., U niversity and (ii) depend on previous terms within the context structure, therefore the explicit introduction of context subsumes all approaches. In addition, the concept of context has been formally expressed in a variant of dependent type theory Dapoigny (2010b) with nested sum types arranged in a partonomic hierarchy. Since each context type is indexed on a player, their aggregation (for a given player) into wider context types automatically introduces several possible role types for that player.…”

“…In the case where x i does not refer to a nested sum type, we have obviously τ i = T i . From here on, we only consider relation types without nested sum-types because (i) simpler expressions enhance the readability, (ii) the introduction of nested argument types τ i does not interfere with the lemma below and (iii) people interested in the nested version can refer to Dapoigny (2010b) where nested structures are detailed in context modeling. (i) for non dependent categories, all the properties of each type T i also hold in T i , (ii) for dependent categories, all the properties of each family type T j also hold in T j , that is, the family T j is a subfamily of T j , (iii) the predicate P is a sub-predicate of P (i.e., it is more precise).…”

“…More precisely, in K-DTT, we follow the view already explained in Dapoigny (2010b) which assumes (i) that a context type is related to a single intention (here, a role type) and (ii) that a context type is extensible either to the aggregation of disjoint context types or to a partonomic hierarchy of nested context types (see definition 4). It follows that each time a context is extended with new information then it is related to a distinct role type.…”

“…While type theory is widely used in mathematical logic Barendregt (1997Barendregt ( , 2001; Martin-Lof (1982); Andrews (1986) and in Computer Science Barendregt (1992); Constable (1988); Reus (1999); Oury (2008); Hancock (2000); Luo (1994), there are few works using it for knowledge representation. The proposed theory called K-DTT (Knowledge-based Dependent Type Theory) is derived from Dapoigny (2010b) for the modeling of contexts and from Dapoigny (2012) for the modeling of situations. The logic in the lower layer operates on (names of) types whose meaning is constrained in the upper (ontological) layer.…”