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Synchronization and linearity: an algebra for discrete event systems
, 2001
"... The first edition of this book was published in 1992 by Wiley (ISBN 0 471 93609 X). Since this book is now out of print, and to answer the request of several colleagues, the authors have decided to make it available freely on the Web, while retaining the copyright, for the benefit of the scientific ..."
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Cited by 252 (10 self)
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The first edition of this book was published in 1992 by Wiley (ISBN 0 471 93609 X). Since this book is now out of print, and to answer the request of several colleagues, the authors have decided to make it available freely on the Web, while retaining the copyright, for the benefit of the scientific community. Copyright Statement This electronic document is in PDF format. One needs Acrobat Reader (available freely for most platforms from the Adobe web site) to benefit from the full interactive machinery: using the package hyperref by Sebastian Rahtz, the table of contents and all LATEX crossreferences are automatically converted into clickable hyperlinks, bookmarks are generated automatically, etc.. So, do not hesitate to click on references to equation or section numbers, on items of thetableofcontents and of the index, etc.. One may freely use and print this document for one’s own purpose or even distribute it freely, but not commercially, provided it is distributed in its entirety and without modifications, including this preface and copyright statement. Any use of thecontents should be acknowledged according to the standard scientific practice. The
Methods and Applications of (max,+) Linear Algebra
 STACS'97, NUMBER 1200 IN LNCS, LUBECK
, 1997
"... Exotic semirings such as the "(max, +) semiring" (R # {#},max,+), or the "tropical semiring" (N #{+#},min,+), have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete ev ..."
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Cited by 76 (26 self)
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Exotic semirings such as the "(max, +) semiring" (R # {#},max,+), or the "tropical semiring" (N #{+#},min,+), have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, HamiltonJacobi theory; asymptotic analysis (low temperature asymptotics in statistical physics, large deviations, WKB method); language theory (automata with multiplicities) . Despite this apparent profusion, there is a small set of common, nonnaive, basic results and problems, in general not known outside the (max, +) community, which seem to be useful in most applications. The aim of this short survey paper is to present what we believe to be the minimal core of (max, +) results, and to illustrate these results by typical applications, at the frontier of language theory, control, and operations research (performance evaluation of...
Multimodal Linguistic Inference
, 1995
"... In this paper we compare grammatical inference in the context of simple and of mixed Lambek systems. Simple Lambek systems are obtained by taking the logic of residuation for a family of multiplicative connectives =; ffl; n, together with a package of structural postulates characterizing the resourc ..."
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Cited by 40 (6 self)
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In this paper we compare grammatical inference in the context of simple and of mixed Lambek systems. Simple Lambek systems are obtained by taking the logic of residuation for a family of multiplicative connectives =; ffl; n, together with a package of structural postulates characterizing the resource management properties of the ffl connective. Different choices for Associativity and Commutativity yield the familiar logics NL, L, NLP, LP. Semantically, a simple Lambek system is a unimodal logic: the connectives get a Kripke style interpretation in terms of a single ternary accessibility relation modeling the notion of linguistic composition for each individual system. The simple systems each have their virtues in linguistic analysis. But none of them in isolation provides a basis for a full theory of grammar. In the second part of the paper, we consider two types of mixed Lambek systems. The first type is obtained by combining a number of unimodal systems into one multimodal logic. The...
PartialGaggles Applied to Logics with Restricted Structural Rules
 In Peter SchroederHeister and Kosta Dosen, editors, Substructural Logics
, 1991
"... Law of Residuation (in their jth place) when f and g are contrapositives (with respect to their jth place) and S(f; a 1 ; : : : ; a j ; : : : ; a n ; b) iff S(g; a 1 ; : : : ; b; : : : ; a n ; a j ). (5) Two operators f , g 2 OP are relatives when they satisfy the Abstract Law of Residuation in ..."
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Cited by 40 (1 self)
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Law of Residuation (in their jth place) when f and g are contrapositives (with respect to their jth place) and S(f; a 1 ; : : : ; a j ; : : : ; a n ; b) iff S(g; a 1 ; : : : ; b; : : : ; a n ; a j ). (5) Two operators f , g 2 OP are relatives when they satisfy the Abstract Law of Residuation in some position. (6) The family of operations OP is founded when there is a distinguished operator f 2 OP (the head) such that any other operator g 2 OP is a relative of f . Definition. A partialgaggle is a tonoid T = (X; ; OP), in which OP is a founded family. As examples, consider a p.o. residuated groupoid, with OP chosen to be any of the following families of operations (ffi is the head of the families of which it is a member): fffig, fffi; /g, fffi; !g, fffi; /;!g, f/g, f!g. Note that f!;/g does not formally fall under our definition since the trace of one is not directly the contrapositive of the trace of the other, even though the trace of each is a contrapositive of the trace of f...
A Primer On Galois Connections
 York Academy of Science
, 1992
"... : We provide the rudiments of the theory of Galois connections (or residuation theory, as it is sometimes called) together with many examples and applications. Galois connections occur in profusion and are wellknown to most mathematicians who deal with order theory; they seem to be less known to to ..."
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Cited by 29 (3 self)
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: We provide the rudiments of the theory of Galois connections (or residuation theory, as it is sometimes called) together with many examples and applications. Galois connections occur in profusion and are wellknown to most mathematicians who deal with order theory; they seem to be less known to topologists. However, because of their ubiquity and simplicity, they (like equivalence relations) can be used as an effective research tool throughout mathematics and related areas. If one recognizes that a Galois connection is involved in a phenomenon that may be relatively complex, then many aspects of that phenomenon immediately become clear; and thus, the whole situation typically becomes much easier to understand. KEY WORDS: Galois connection, closure operation, interior operation, polarity, axiality CLASSIFICATION: Primary: 06A15, 0601, 06A06 Secondary: 5401, 54B99, 54H99, 68F05 0. INTRODUCTION Mathematicians are familiar with the following situation: there are two "worlds" and t...
Peirce Algebras
, 1992
"... We present a twosorted algebra, called a Peirce algebra, of relations and sets interacting with each other. In a Peirce algebra, sets can combine with each other as in a Boolean algebra, relations can combine with each other as in a relation algebra, and in addition we have both a relationforming o ..."
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Cited by 25 (10 self)
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We present a twosorted algebra, called a Peirce algebra, of relations and sets interacting with each other. In a Peirce algebra, sets can combine with each other as in a Boolean algebra, relations can combine with each other as in a relation algebra, and in addition we have both a relationforming operator on sets (the Peirce product of Boolean modules) and a setforming operator on relations (a cylindrification operation). Two applications of Peirce algebras are given. The first points out that Peirce algebras provide a natural algebraic framework for modelling certain programming constructs. The second shows that the socalled terminological logics arising in knowledge representation have evolved a semantics best described as a calculus of relations interacting with sets.
Adjoining declassification and attack models by abstract interpretation
 In Proc. European Symp. on Programming, volume 3444 of LNCS
, 2005
"... Abstract. In this paper we prove that attack models and robust declassification in languagebased security can be viewed as adjoint transformations of abstract interpretations. This is achieved by interpreting the well known Joshi and Leino’s semantic approach to noninterference as a problem of mak ..."
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Cited by 17 (8 self)
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Abstract. In this paper we prove that attack models and robust declassification in languagebased security can be viewed as adjoint transformations of abstract interpretations. This is achieved by interpreting the well known Joshi and Leino’s semantic approach to noninterference as a problem of making an abstraction complete relatively to a program’s property on confidential data which flows, here called private observation, and the most concrete harmless attacker observing public data, here called public observable, both modeled as abstractions of the program’s semantics, are respectively the adjoint solutions of a completeness problem in standard abstract interpretation theory. In particular declassification corresponds to refining the given model of an attacker with the minimal amount of information in order to achieve completeness, which is noninterference, while the harmless attacker corresponds to remove this information. This proves an adjunction relation between two basic approaches to languagebased security: declassification and the construction of suitable attack models, and allows us to apply relevant techniques for abstract domain transformation in languagebased security.
Kernels, Images And Projections In Dioids
 PROCEEDINGS OF WODES’96
, 1996
"... We consider the projection problem for linear spaces and operators over dioids such as the (max, +) semiring. We give existence and uniqueness conditions for the projection onto the image of an operator, parallel to the kernel of another one, together with an explicit formula for the projector. Th ..."
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Cited by 15 (12 self)
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We consider the projection problem for linear spaces and operators over dioids such as the (max, +) semiring. We give existence and uniqueness conditions for the projection onto the image of an operator, parallel to the kernel of another one, together with an explicit formula for the projector. The theory is not limited to linear operators: the result holds more generally for residuated operators over complete dioids. Illustrative examples are provided.
Linear Projector in the maxplus Algebra
 IEEE MEDITERRANEEN CONFERENCE ON CONTROL, CHYPRE
, 1997
"... In general semimodules, we say that the image of a linear operator B and the kernel of a linear operator C are direct factors if every equivalence class modulo C crosses the image of B at a unique point. For linear maps represented by matrices over certain idempotent semifields such as the (max, +) ..."
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Cited by 14 (11 self)
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In general semimodules, we say that the image of a linear operator B and the kernel of a linear operator C are direct factors if every equivalence class modulo C crosses the image of B at a unique point. For linear maps represented by matrices over certain idempotent semifields such as the (max, +)semiring, we give necessary and sufficient conditions for an image and a kernel to be direct factors. We characterize the semimodules that admit a direct factor (or equivalently, the semimodules that are the image of a linear projector): their matrices have a ginverse. We give simple effective tests for all these properties, in terms of matrix residuation.