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For some years, specification of software and hardware systems has
been influenced not only by algebraic methods but also by new
developments in logic. These new developments in logic are partly
based on the use of algorithmic techniques in deduction and proving
methods, but are alsodue to new theoretical advances, to a great
extent stimulated by computer science, which have led to new types
of logic and new logical calculi. The new techniques, methods and
tools from logic, combined with algebra-based ones, offer very
powerful and useful tools for the computer scientist, which may
soon become practical for commercial use, where, in particular,
more powerful specification tools are needed for concurrent and
distributed systems. This volume contains papers based on lectures
by leading researchers which were originally given at an
international summer school held in Marktoberdorf in 1991. The
papers aim to give a foundation for combining logic and algebra for
the purposes of specification under the aspects of automated
deduction, proving techniques, concurrency and logic, abstract data
types and operational semantics, and constructive methods.
Recent developments in computer science clearly show the need for a
better theoretical foundation for some central issues. Methods and
results from mathematical logic, in particular proof theory and
model theory, are of great help here and will be used much more in
future than previously. This book provides an excellent
introduction to the interplay of mathematical logic and computer
science. It contains extensively reworked versions of the lectures
given at the 1997 Marktoberdorf Summer School by leading
researchers in the field.
Topics covered include: proof theory and specification of
computation (J.-Y. Girard, D. Miller), complexity of proofs and
programs (S. R. Buss, S. S. Wainer), computational content of
proofs (H. Schwichtenberg), constructive type theory (P. Aczel, H.
Barendregt, R. L. Constable), computational mathematics, (U.
Martin), rewriting logic (J. Meseguer), and game semantics (S.
Abramski).
The Marktoberdorf Summer School 1995 'Logic of Computation' was the
16th in a series of Advanced Study Institutes under the sponsorship
of the NATO Scientific Affairs Division held in Marktoberdorf. Its
scientific goal was to survey recent progress on the impact of
logical methods in software development. The courses dealt with
many different aspects of this interplay, where major progress has
been made. Of particular importance were the following. * The
proofs-as-programs paradigm, which makes it possible to extract
verified programs directly from proofs. Here a higher order logic
or type theoretic setup of the underlying language has developed
into a standard. * Extensions of logic programming, e.g. by
allowing more general formulas and/or higher order languages. *
Proof theoretic methods, which provide tools to deal with questions
of feasibility of computations and also to develop a general
mathematical understanding of complexity questions. * Rewrite
systems and unification, again in a higher order context. Closely
related is the now well-established Grabner basis theory, which
recently has found interesting applications. * Category theoretic
and more generally algebraic methods and techniques to analyze the
semantics of programming languages. All these issues were covered
by a team of leading researchers. Their courses were grouped under
the following headings.
Logical concepts and methods are of growing importance in many
areas of computer science. The proofs-as-programs paradigm and the
wide acceptance of Prolog show this clearly. The logical notion of
a formal proof in various constructive systems can be viewed as a
very explicit way to describe a computation procedure. Also
conversely, the development of logical systems has been influenced
by accumulating knowledge on rewriting and unification techniques.
This volume contains a series of lectures by leading researchers
giving a presentation of new ideas on the impact of the concept of
a formal proof on computation theory. The subjects covered are:
specification and abstract data types, proving techniques,
constructive methods, linear logic, and concurrency and logic.
As society comes to rely increasingly on software for its welfare
and prosperity there is an urgent need to create systems in which
it can trust. Experience has shown that confidence can only come
from a more profound understanding of the issues, which in turn can
come only if it is based on logically sound foundations.
This volume contains contributions from leading researchers in the
critical disciplines of computing and information science,
mathematics, logic, and complexity. All contributions are
self-contained, aiming at comprehensibility as well as
comprehensiveness. The volume also contains introductory hints to
technical issues, concise surveys, introductions, and various fresh
results and new perspectives.
CSL is the annual conference of the European Association for
Computer Science Logic (EACSL). CSL2000 is the 14th such annual
conference, thus witnessing the importance and sustained
international interest in the application of me- ods from
mathematical logic to computer science. The current conference was
organized by the Mathematics Institute and the Computer Science
Institute of the Ludwig-Maximilians-Universit] at Munc ] hen (LMU),
with generous ?nancial
supportfromtheDeutscheForschungsgemeinschaft, Forschungsinstitutfur
] an- wandte Softwaretechnologie (FAST e.V.), Munc ] hener
Universit] atsgesellschaft e.V., and Siemens AG. Our sponsors
generosity enabled, among other things, stipends for the ?nancial
support of students as well as of researchers from Ea- ern Europe.
Topics in the call for papers for CSL2000 included: automated
deduction andinteractivetheoremproving,
categoricallogicandtopologicalsemantics, c-
structivemathematicsandtypetheory, domaintheory,
equationallogicandterm rewriting, ?nite model theory, database
theory, higher order logic, lambda and combinatory calculi, logical
aspects of computational complexity, logical fo- dations of
programming paradigms, logic programming and constraints, linear
logic, modal and temporal logics, model checking, program
extraction, program logicsandsemantics, programspeci?cation,
transformationandveri?cation.The invited speakers were: Moshe Vardi
(Houston), Paul Beame (Washington), - dreas Blass (Ann Arbor), Egon
B] orger (Pisa), Yuri Gurevich (Redmond), Bruno Poizat (Lyons),
Wolfram Schulte (Redmond), Saharon Shelah (Jerusalem), and Colin
Sterling (Edinburgh). Special thanks to Moshe Vardi for being
willing to speakintheplaceofMikl osAjtai(Almaden),
whocouldnotattendthemeeting. The day of 24 August 2000, during the
week-long CSL2000 meeting, was reserved for theGurevichSymposium, a
special, one-day tribute to the scienti?c contributions of
Professor Yuri Gurevich, at the occasion of his 60th birthday."
Recent developments in computer science clearly show the need for a
better theoretical foundation for some central issues. Methods and
results from mathematical logic, in particular proof theory and
model theory, are of great help here and will be used much more in
future than previously. This book provides an excellent
introduction to the interplay of mathematical logic and computer
science. It contains extensively reworked versions of the lectures
given at the 1997 Marktoberdorf Summer School by leading
researchers in the field.
Topics covered include: proof theory and specification of
computation (J.-Y. Girard, D. Miller), complexity of proofs and
programs (S. R. Buss, S. S. Wainer), computational content of
proofs (H. Schwichtenberg), constructive type theory (P. Aczel, H.
Barendregt, R. L. Constable), computational mathematics, (U.
Martin), rewriting logic (J. Meseguer), and game semantics (S.
Abramski).
For some years, specification of software and hardware systems has
been influenced not only by algebraic methods but also by new
developments in logic. These new developments in logic are partly
based on the use of algorithmic techniques in deduction and proving
methods, but are alsodue to new theoretical advances, to a great
extent stimulated by computer science, which have led to new types
of logic and new logical calculi. The new techniques, methods and
tools from logic, combined with algebra-based ones, offer very
powerful and useful tools for the computer scientist, which may
soon become practical for commercial use, where, in particular,
more powerful specification tools are needed for concurrent and
distributed systems. This volume contains papers based on lectures
by leading researchers which were originally given at an
international summer school held in Marktoberdorf in 1991. The
papers aim to give a foundation for combining logic and algebra for
the purposes of specification under the aspects of automated
deduction, proving techniques, concurrency and logic, abstract data
types and operational semantics, and constructive methods.
This book is for graduate students and researchers, introducing
modern foundational research in mathematics, computer science, and
philosophy from an interdisciplinary point of view. Its scope
includes proof theory, constructive mathematics and type theory,
univalent mathematics and point-free approaches to topology,
extraction of certified programs from proofs, automated proofs in
the automotive industry, as well as the philosophical and
historical background of proof theory. By filling the gap between
(under-)graduate level textbooks and advanced research papers, the
book gives a scholarly account of recent developments and emerging
branches of the aforementioned fields.
As society comes to rely increasingly on software for its welfare
and prosperity there is an urgent need to create systems in which
it can trust. Experience has shown that confidence can only come
from a more profound understanding of the issues, which in turn can
come only if it is based on logically sound foundations.
This volume contains contributions from leading researchers in the
critical disciplines of computing and information science,
mathematics, logic, and complexity. All contributions are
self-contained, aiming at comprehensibility as well as
comprehensiveness. The volume also contains introductory hints to
technical issues, concise surveys, introductions, and various fresh
results and new perspectives.
The latest work by the world's leading authorities on the use of
formal methods in computer science is presented in this volume,
based on the 1995 International Summer School in Marktoberdorf,
Germany. Logic is of special importance in computer science, since
it provides the basis for giving correct semantics of programs, for
specification and verification of software, and for program
synthesis. The lectures presented here provide the basic knowledge
a researcher in this area should have and give excellent starting
points for exploring the literature. Topics covered include
semantics and category theory, machine based theorem proving, logic
programming, bounded arithmetic, proof theory, algebraic
specifications and rewriting, algebraic algorithms, and type
theory.
Constructive mathematics – mathematics in which 'there exists'
always means 'we can construct' – is enjoying a renaissance.
fifty years on from Bishop's groundbreaking account of constructive
analysis, constructive mathematics has spread out to touch almost
all areas of mathematics and to have profound influence in
theoretical computer science. This handbook gives the most complete
overview of modern constructive mathematics, with contributions
from leading specialists surveying the subject's myriad aspects.
Major themes include: constructive algebra and geometry,
constructive analysis, constructive topology, constructive logic
and foundations of mathematics, and computational aspects of
constructive mathematics. A series of introductory chapters
provides graduate students and other newcomers to the subject with
foundations for the surveys that follow. Edited by four of the most
eminent experts in the field, this is an indispensable reference
for constructive mathematicians and a fascinating vista of modern
constructivism for the increasing number of researchers interested
in constructive approaches.
This book is for graduate students and researchers, introducing
modern foundational research in mathematics, computer science, and
philosophy from an interdisciplinary point of view. Its scope
includes Predicative Foundations, Constructive Mathematics and Type
Theory, Computation in Higher Types, Extraction of Programs from
Proofs, and Algorithmic Aspects in Financial Mathematics. By
filling the gap between (under-)graduate level textbooks and
advanced research papers, the book gives a scholarly account of
recent developments and emerging branches of the aforementioned
fields.
Driven by the question, 'What is the computational content of a
(formal) proof?', this book studies fundamental interactions
between proof theory and computability. It provides a unique
self-contained text for advanced students and researchers in
mathematical logic and computer science. Part I covers basic proof
theory, computability and Godel's theorems. Part II studies and
classifies provable recursion in classical systems, from fragments
of Peano arithmetic up to PI11-CA0. Ordinal analysis and the
(Schwichtenberg-Wainer) subrecursive hierarchies play a central
role and are used in proving the 'modified finite Ramsey' and
'extended Kruskal' independence results for PA and PI11-CA0. Part
III develops the theoretical underpinnings of the first author's
proof assistant MINLOG. Three chapters cover higher-type
computability via information systems, a constructive theory TCF of
computable functionals, realizability, Dialectica interpretation,
computationally significant quantifiers and connectives and
polytime complexity in a two-sorted, higher-type arithmetic with
linear logic.
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