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As the operations of the world become more and more dependent on
highly interconnected, massively complex, networked systems of
computational devices, the need to develop a mathematical
understanding of their properties and behaviours is increasingly
pressing. Our approach, described in this monograph, is to combine
the compositionality of formal specification -- using techniques
from algebra, computation theory, logic, and probability theory --
with the control of level of abstraction afforded by the classical
mathematical modelling method.
This is a monograph about logic. Specifically, it presents the
mathe matical theory of the logic of bunched implications, BI: I
consider Bl's proof theory, model theory and computation theory.
However, the mono graph is also about informatics in a sense which
I explain. Specifically, it is about mathematical models of
resources and logics for reasoning about resources. I begin with an
introduction which presents my (background) view of logic from the
point of view of informatics, paying particular attention to three
logical topics which have arisen from the development of logic
within informatics: * Resources as a basis for semantics; *
Proof-search as a basis for reasoning; and * The theory of
representation of object-logics in a meta-logic. The ensuing
development represents a logical theory which draws upon the
mathematical, philosophical and computational aspects of logic.
Part I presents the logical theory of propositional BI, together
with a computational interpretation. Part II presents a
corresponding devel opment for predicate BI. In both parts, I
develop proof-, model- and type-theoretic analyses. I also provide
semantically-motivated compu tational perspectives, so beginning a
mathematical theory of resources. I have not included any analysis,
beyond conjecture, of properties such as decidability, finite
models, games or complexity. I prefer to leave these matters to
other occasions, perhaps in broader contexts.
This is a monograph about logic. Specifically, it presents the
mathe matical theory of the logic of bunched implications, BI: I
consider Bl's proof theory, model theory and computation theory.
However, the mono graph is also about informatics in a sense which
I explain. Specifically, it is about mathematical models of
resources and logics for reasoning about resources. I begin with an
introduction which presents my (background) view of logic from the
point of view of informatics, paying particular attention to three
logical topics which have arisen from the development of logic
within informatics: * Resources as a basis for semantics; *
Proof-search as a basis for reasoning; and * The theory of
representation of object-logics in a meta-logic. The ensuing
development represents a logical theory which draws upon the
mathematical, philosophical and computational aspects of logic.
Part I presents the logical theory of propositional BI, together
with a computational interpretation. Part II presents a
corresponding devel opment for predicate BI. In both parts, I
develop proof-, model- and type-theoretic analyses. I also provide
semantically-motivated compu tational perspectives, so beginning a
mathematical theory of resources. I have not included any analysis,
beyond conjecture, of properties such as decidability, finite
models, games or complexity. I prefer to leave these matters to
other occasions, perhaps in broader contexts.
This book is a specialized monograph on the development of the
mathematical and computational metatheory of reductive logic and
proof-search, areas of logic that are becoming important in
computer science. A systematic foundational text on these emerging
topics, it includes proof-theoretic, semantic/model-theoretic and
algorithmic aspects. The scope ranges from the conceptual
background to reductive logic, through its mathematical metatheory,
to its modern applications in the computational sciences. Suitable
for researchers and graduate students in mathematical,
computational and philosophical logic, and in theoretical computer
science and artificial intelligence, this is the latest in the
prestigous world-renowned Oxford Logic Guides, which contains
Michael Dummet's Elements of intuitionism (2nd Edition), Dov M.
Gabbay, Mark A. Reynolds, and Marcelo Finger's Temporal Logic
Mathematical Foundations and Computational Aspects , J. M. Dunn and
G. Hardegree's Algebraic Methods in Philosophical Logic, H. Rott's
Change, Choice and Inference: A Study of Belief Revision and
Nonmonotonic Reasoning , and P. T. Johnstone's Sketches of an
Elephant: A Topos Theory Compendium: Volumes 1 and 2 .
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