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Modal Logic is a branch of logic with applications in many related
disciplines such as computer science, philosophy, linguistics and
artificial intelligence. Over the last twenty years, in all of
these neighbouring fields, modal systems have been developed that
we call multi-dimensional. (Our definition of multi-dimensionality
in modal logic is a technical one: we call a modal formalism
multi-dimensional if, in its intended semantics, the universe of a
model consists of states that are tuples over some more basic set.)
This book treats such multi-dimensional modal logics in a uniform
way, linking their mathematical theory to the research tradition in
algebraic logic. We will define and discuss a number of systems in
detail, focusing on such aspects as expressiveness, definability,
axiomatics, decidability and interpolation. Although the book will
be mathematical in spirit, we take care to give motivations from
the disciplines mentioned earlier on.
Finite model theory,as understoodhere, is an areaof
mathematicallogic that has developed in close connection with
applications to computer science, in particular the theory of
computational complexity and database theory. One of the
fundamental insights of mathematical logic is that our
understanding of mathematical phenomena is enriched by elevating
the languages we use to describe mathematical structures to objects
of explicit study. If mathematics is the science of patterns, then
the media through which we discern patterns, as well as the
structures in which we discern them, command our attention. It
isthis aspect oflogicwhichis mostprominentin model
theory,"thebranchof mathematical logic which deals with the
relation between a formal language and its interpretations". No
wonder, then, that mathematical logic, and ?nite model theory in
particular, should ?nd manifold applications in computer science:
from specifying programs to querying databases, computer science is
rife with phenomena whose understanding requires close attention to
the interaction between language and structure. This volume gives a
broadoverviewof some central themes of ?nite model theory:
expressive power, descriptive complexity, and zero-one laws,
together with selected applications to database theory and
arti?cial intelligence, es- cially constraint databases and
constraint satisfaction problems. The ?nal chapter provides a
concise modern introduction to modal logic,which emp- sizes the
continuity in spirit and technique with ?nite model theory.
Modal Logic is a branch of logic with applications in many related
disciplines such as computer science, philosophy, linguistics and
artificial intelligence. Over the last twenty years, in all of
these neighbouring fields, modal systems have been developed that
we call multi-dimensional. (Our definition of multi-dimensionality
in modal logic is a technical one: we call a modal formalism
multi-dimensional if, in its intended semantics, the universe of a
model consists of states that are tuples over some more basic set.)
This book treats such multi-dimensional modal logics in a uniform
way, linking their mathematical theory to the research tradition in
algebraic logic. We will define and discuss a number of systems in
detail, focusing on such aspects as expressiveness, definability,
axiomatics, decidability and interpolation. Although the book will
be mathematical in spirit, we take care to give motivations from
the disciplines mentioned earlier on.
Finite model theory,as understoodhere, is an areaof
mathematicallogic that has developed in close connection with
applications to computer science, in particular the theory of
computational complexity and database theory. One of the
fundamental insights of mathematical logic is that our
understanding of mathematical phenomena is enriched by elevating
the languages we use to describe mathematical structures to objects
of explicit study. If mathematics is the science of patterns, then
the media through which we discern patterns, as well as the
structures in which we discern them, command our attention. It
isthis aspect oflogicwhichis mostprominentin model
theory,"thebranchof mathematical logic which deals with the
relation between a formal language and its interpretations". No
wonder, then, that mathematical logic, and ?nite model theory in
particular, should ?nd manifold applications in computer science:
from specifying programs to querying databases, computer science is
rife with phenomena whose understanding requires close attention to
the interaction between language and structure. This volume gives a
broadoverviewof some central themes of ?nite model theory:
expressive power, descriptive complexity, and zero-one laws,
together with selected applications to database theory and
arti?cial intelligence, es- cially constraint databases and
constraint satisfaction problems. The ?nal chapter provides a
concise modern introduction to modal logic,which emp- sizes the
continuity in spirit and technique with ?nite model theory.
Conceived by Johan van Benthem and Yde Venema, arrow logic started
as an attempt to give a general account of the logic of
transitions. The generality of the approach provided a wide
application area ranging from philosophy to computer science. The
book gives a comprehensive survey of logical research within and
around arrow logic. Since the natural operations on transitions
include composition, inverse and identity, their logic, arrow logic
can be studied from two different perspectives, and by two
(complementary) methodologies: modal logic and the algebra of
relations. Some of the results in this volume can be interpreted as
price tags. They show what the prices of desirable properties, such
as decidability, (finite) axiomatisability, Craig interpolation
property, Beth definability etc. are in terms of semantic
properties of the logic. The research program of arrow logic has
considerably broadened in the last couple of years and recently
also covers the enterprise to explore the border between decidable
and undecidable versions of other applied logics. The content of
this volume reflects this broadening. The editors included a number
of papers which are in the spirit of this generalised research
program.
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