=Paper= {{Paper |id=None |storemode=property |title=Multi Context Logics: a Formal Framework for Structuring Knowledge |pdfUrl=https://ceur-ws.org/Vol-875/invited_paper_2.pdf |volume=Vol-875 |dblpUrl=https://dblp.org/rec/conf/womo/Serafini12 }} ==Multi Context Logics: a Formal Framework for Structuring Knowledge== https://ceur-ws.org/Vol-875/invited_paper_2.pdf 
   Multi context logics: a formal framework for
              structuring knowledge

                                 Luciano Serafini

                            Fondazione Bruno Kessler
                     via Sommarive 18, I-38123, Trento, Italy
                               serafini@fbk.eu

    Multi context logics (MCL) is a formalism that allow to integrate multiple
logical theories (contexts) in a more complex structure called multi context sys-
tem. In the past 20 years multi context logics have been developed for contexts
in propositional logics, first order logics, description logics and temporal logic.
The two principles MCL are locality and compatibility. The principle of local-
ity states that a context axiomatizes in a logical theory a portion of the world,
and that every statement entailed by such a theory is intended to hold within
such a portion of the world. The principle of compatibility instead states that,
since different contexts can describe overlapping portions of world, the theories
they contain must be constrained so that they describe compatible situations.
Following these two principles, the formal semantics of an MCL is the result of
a suitable composition of the semantics associated to each single context. This
takes the name of Local models semantics. The effects of the two principles above
on the inference engines that can be defined on a multi contextual knowledge
base are the following: Locality principle implies that inference rules applied
to knowledge inside a context (aka, local inference rules) allow to infer local
truths; Compatibility principle instead implies that certain facts in a context
can be inferred on the base of other facts present on other compatible contexts.
This information propagation is formalized via a special type of inter contextual
inference rules called bridge rules
    In general terms, a multi context logic is defined on a family of logical lan-
guages {Li }i∈I where each Li is used to specify what holds in the i-th context.
The set I of context indexes (aka context names) can be either a simple set,
or a set equipped with some algebraic structure, like total or partial order, and
operations on context indices. The relations and functions defined on I can be
used to specify the organization of contexts in terms of an algebraic structure.
For instance a partial order ≺ no I, can be used to represent that a context is
wider (more general) than another context, e.g., football ≺ sport, means that the
context of sport is more specific that the context of football. To represent what
is true in a scene in different time stamps, we can enrich I with a total order ≺
so that CV Luciano 2010 ≺ CV Luciano 2011 represents the fact that the context
describing Luciano’s curriculum vitæat 2010 precedes the context describing his
CV at 2011.
    A model for a multi context logic {Li }i∈I is a pair hMI , Ci composed of a
family of local models MI = {Mi }i∈I , where each Mi is a model of Li , and
a compatibility relation C among the local models. The formal structure of C
can vary depending on the type of local models and the type of constraints it
is necessary to impose on the local models of different contexts. For this reason
we don’t give a general definition of C, which will be completely defined for
each specific multi context logic. Satisfiability of formulas in Li is defined w.r.t1.
the local models. Namely If φ is a formula of the language Li , then the a multi
context model satisfies i : φ iff Mi |=i φ, where |=i is the satisfiability relation
associated to the local logic Li .
    A Multi context theory in a multi context logic LI = {Li }i∈I is a family of
theories {Ti }i∈I , where each Ti is a set of statements in the logics Li , and a set
BR of bridge rules. Intuitively each Ti axiomatizes the constraints on the local
models Mi , while the bridge rules BR axiomatizes the compatibility relations.
Bridge rules are cross logical axioms and their syntactic form depends on the
local logics, so as in the case of the compatibility relation their syntactic depends
on the syntax of each Li .
    In my invited talk, I will go through the many possible examples of multi
context logics, starting from the simplest one, the propositional multi context
logics, going through hierarchical meta logics, multi context logics for beliefs and
propositional attitudes, non-monotonic propositional context logics, distributed
first order logics, distributed description logics, and logics for semantic import
and contextualized knowledge repository for the semantic web.