223 research outputs found
A Note on Parameterised Knowledge Operations in Temporal Logic
We consider modeling the conception of knowledge in terms of temporal logic.
The study of knowledge logical operations is originated around 1962 by
representation of knowledge and belief using modalities. Nowadays, it is very
good established area. However, we would like to look to it from a bit another
point of view, our paper models knowledge in terms of linear temporal logic
with {\em past}. We consider various versions of logical knowledge operations
which may be defined in this framework. Technically, semantics, language and
temporal knowledge logics based on our approach are constructed. Deciding
algorithms are suggested, unification in terms of this approach is commented.
This paper does not offer strong new technical outputs, instead we suggest new
approach to conception of knowledge (in terms of time).Comment: 10 page
Almost structural completeness; an algebraic approach
A deductive system is structurally complete if its admissible inference rules
are derivable. For several important systems, like modal logic S5, failure of
structural completeness is caused only by the underivability of passive rules,
i.e. rules that can not be applied to theorems of the system. Neglecting
passive rules leads to the notion of almost structural completeness, that
means, derivablity of admissible non-passive rules. Almost structural
completeness for quasivarieties and varieties of general algebras is
investigated here by purely algebraic means. The results apply to all
algebraizable deductive systems.
Firstly, various characterizations of almost structurally complete
quasivarieties are presented. Two of them are general: expressed with finitely
presented algebras, and with subdirectly irreducible algebras. One is
restricted to quasivarieties with finite model property and equationally
definable principal relative congruences, where the condition is verifiable on
finite subdirectly irreducible algebras.
Secondly, examples of almost structurally complete varieties are provided
Particular emphasis is put on varieties of closure algebras, that are known to
constitute adequate semantics for normal extensions of S4 modal logic. A
certain infinite family of such almost structurally complete, but not
structurally complete, varieties is constructed. Every variety from this family
has a finitely presented unifiable algebra which does not embed into any free
algebra for this variety. Hence unification in it is not unitary. This shows
that almost structural completeness is strictly weaker than projective
unification for varieties of closure algebras
Closure algebras of depth two with extremal relations: Their frames, logics, and structural completeness
We consider varieties generated by finite closure algebras whose canonical
relations have two levels, and whose restriction to a level is an "extremal"
relation, i.e. the identity or the universal relation. The corresponding logics
have frames of depth two, in which a level consists of a set of simple clusters
or of one cluster with one or more elements
Unification and Finite Model Property for Linear Step-Like Temporal Multi-Agent Logic with the Universal Modality
This paper proposes a semantic description of the linear step-like temporal multi-agent logic with the universal modality based on the idea of non-reflexive non-transitive nature of time. We proved a finite model property and projective unification for this logic
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