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Epistemic actions as resources
 Journal of Logic and Computation
, 2007
"... We provide algebraic semantics together with a sound and complete sequent calculus for information update due to epistemic actions. This semantics is flexible enough to accommodate incomplete as well as wrong information e.g. deceit. We give a purely algebraic treatment of the muddy children puzzle, ..."
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Cited by 18 (13 self)
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We provide algebraic semantics together with a sound and complete sequent calculus for information update due to epistemic actions. This semantics is flexible enough to accommodate incomplete as well as wrong information e.g. deceit. We give a purely algebraic treatment of the muddy children puzzle, which moreover extends to situations where the children are allowed to lie and cheat. Epistemic actions, that is, information exchanges between agents A, B,... ∈ A, are modeled as elements of a quantale, hence conceiving them as resources. Indeed, quantales are to locales what monoidal closed categories are to Cartesian closed categories, respectively providing semantics for Intuitionistic Logic, and for noncommutative Intuitionistic Linear Logic, including Lambek calculus. The quantale (Q, � , •) acts on an underlying Qright module (M, � ) of epistemic propositions and facts. The epistemic content is encoded by appearance maps, one pair f M A: M → M and f Q A: Q → Q of (lax) morphisms for each agent A ∈ A. By adjunction, they give rise to epistemic modalities [12], capturing the agents ’ knowledge on propositions and actions. The module action is epistemic update and gives rise to dynamic modalities [20] — cf. weakest preconditions. This model subsumes the crucial fragment of Baltag, Moss and Solecki’s [6] dynamic epistemic logic, abstracting it in a constructive fashion while introducing resourcesensitive structure on the epistemic actions. Keywords: Multiagent communication, knowledge update, resourcesensitivity, quantale, Galois adjoints, dynamic epistemic logic, sequent calculus, Lambek calculus, Linear Logic.
Algebra and Sequent Calculus for Epistemic Actions
 ENTCS PROCEEDINGS OF LOGIC AND COMMUNICATION IN MULTIAGENT SYSTEMS (LCMAS) WORKSHOP, ESSLLI 2004
, 2005
"... We introduce an algebraic approach to Dynamic Epistemic Logic. This approach has the advantage that: (i) its semantics is a transparent algebraic object with a minimal set of primitives from which most ingredients of Dynamic Epistemic Logic arise, (ii) it goes with the introduction of nondeterminis ..."
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Cited by 12 (3 self)
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We introduce an algebraic approach to Dynamic Epistemic Logic. This approach has the advantage that: (i) its semantics is a transparent algebraic object with a minimal set of primitives from which most ingredients of Dynamic Epistemic Logic arise, (ii) it goes with the introduction of nondeterminism, (iii) it naturally extends beyond boolean sets of propositions, up to intuitionistic and nondistributive situations, hence allowing to accommodate constructive computational, informationtheoretic as well as nonclassical physical settings, and (iv) introduces a structure on the actions, which now constitute a quantale. We also introduce a corresponding sequent calculus (which extends Lambek calculus), in which propositions, actions as well as agents appear as resources in a resourcesensitive dynamicepistemic logic.
Inverting weak dihomotopy equivalence using homotopy continuous flow
 Theory Appl. Categ
"... Abstract. A flow is homotopy continuous if it is indefinitely divisible up to Shomotopy. The full subcategory of cofibrant homotopy continuous flows has nice features. Not only it is big enough to contain all dihomotopy types, but also a morphism between them is a weak dihomotopy equivalence if and ..."
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Cited by 5 (3 self)
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Abstract. A flow is homotopy continuous if it is indefinitely divisible up to Shomotopy. The full subcategory of cofibrant homotopy continuous flows has nice features. Not only it is big enough to contain all dihomotopy types, but also a morphism between them is a weak dihomotopy equivalence if and only if it is invertible up to dihomotopy. Thus, the category of cofibrant homotopy continuous flows provides an implementation of Whitehead’s theorem for the full dihomotopy relation, and not only for Shomotopy as in previous works of the author. This fact is not the consequence of the existence of a model structure on the category of flows because it is known that there does not exist any model structure on it whose weak equivalences are exactly the weak dihomotopy equivalences. This fact is an application of a general result for the localization of a model category with respect to a weak factorization system. Contents
Towards `dynamic domains': totally continuous cocomplete Qcategories, Theoret
 Comput. Sci
, 2007
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SYMMETRY AND CAUCHY COMPLETION OF QUANTALOIDENRICHED CATEGORIES
"... Abstract. We formulate an elementary condition on an involutive quantaloid Q under which there is a distributive law from the Cauchy completion monad over the symmetrisation comonad on the category of Qenriched categories. For such quantaloids, which we call Cauchybilateral quantaloids, it follows ..."
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Cited by 2 (0 self)
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Abstract. We formulate an elementary condition on an involutive quantaloid Q under which there is a distributive law from the Cauchy completion monad over the symmetrisation comonad on the category of Qenriched categories. For such quantaloids, which we call Cauchybilateral quantaloids, it follows that the Cauchy completion of any symmetric Qenriched category is again symmetric. Examples include Lawvere’s quantale of nonnegative real numbers and Walters ’ small quantaloids of closed cribles.
Reasoning about Dynamic Epistemic Logic
"... We present an algebra and sequent calculus to reason about dynamic epistemic logic, a logic for information update in multiagent systems. We contribute to it by equipping it with a logical account of resources, a semiautomatic way of reasoning through the algebra and sequent calculus, and finally ..."
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Cited by 2 (0 self)
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We present an algebra and sequent calculus to reason about dynamic epistemic logic, a logic for information update in multiagent systems. We contribute to it by equipping it with a logical account of resources, a semiautomatic way of reasoning through the algebra and sequent calculus, and finally by generalizing it to nonboolean settings. Dynamic Epistemic Logic (DEL) is a PDLstyle logic [14] to reason about epistemic actions and updates in a multiagent system. It focuses in particular on epistemic programs, i.e. programs that update the information state of agents, and it has applications to modelling and reasoning about informationflow and information exchange between agents. This is a major problem in several fields such as secure communication where one has to deal with the privacy and authentication of communication protocols, software reliability for concurrent programs, Artificial Intelligence where agents are to be provided with reliable tools to reason about their environment and each other’s knowledge, and ecommerce where agents need to have knowledge acquisition strategies over complex networks. The standard approach to information flow in a multiagent system has been presented in [8] but it does not present a formal description of epistemic programs and their updates. The first attempts to
CATEGORIES ENRICHED OVER A QUANTALOID: ISBELL ADJUNCTIONS AND KAN ADJUNCTIONS
"... Abstract. Each distributor between categories enriched over a small quantaloid Q gives rise to two adjunctions between the categories of contravariant and covariant presheaves, and hence to two monads. These two adjunctions are respectively generalizations of Isbell adjunctions and Kan extensions in ..."
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Cited by 1 (0 self)
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Abstract. Each distributor between categories enriched over a small quantaloid Q gives rise to two adjunctions between the categories of contravariant and covariant presheaves, and hence to two monads. These two adjunctions are respectively generalizations of Isbell adjunctions and Kan extensions in category theory. It is proved that these two processes are functorial with infomorphisms playing as morphisms between distributors; and that the free cocompletion functor of Qcategories factors through both of these functors. 1.
Quantum triads: an algebraic approach
, 2008
"... frame, quantum triad. A concept of quantum triad and its solution is introduced. It represents a common framework for several situations where we have a quantale with a right module and a left module, provided with a bilinear inner product. Examples include Van den Bossche quantaloids, quantum frame ..."
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frame, quantum triad. A concept of quantum triad and its solution is introduced. It represents a common framework for several situations where we have a quantale with a right module and a left module, provided with a bilinear inner product. Examples include Van den Bossche quantaloids, quantum frames, simple and Galois quantales, operator algebras, or orthomodular lattices. 1
COVARIANT PRESHEAVES AND SUBALGEBRAS
"... Abstract. For small involutive and integral quantaloids Q it is shown that covariant presheaves on symmetric Qcategories are equivalent to certain subalgebras of a speci c monad on the category of symmetric Qcategories. This construction is related to a weakening of the subobject classi er axiom w ..."
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Abstract. For small involutive and integral quantaloids Q it is shown that covariant presheaves on symmetric Qcategories are equivalent to certain subalgebras of a speci c monad on the category of symmetric Qcategories. This construction is related to a weakening of the subobject classi er axiom which does not require the classi cation of all subalgebras, but only guarantees that classi able subalgebras are uniquely classi able. As an application the identi cation of closed left ideals of noncommutative C∗algebras with certain open subalgebras of freely generated algebras is given.
1.1. Guiding examples.
, 2007
"... (1) Ord: objects are (pre)ordered sets ( = sets with a reflexive and transitive relation, no antisymmetry condition), with monotone maps. Formally: (X, a) with 1. ⊤ � a(x, x), 2. a(x, y) ∧ a(y, z) � a(x, z), f: (X, a) − → (Y, b) with a(x, y) � b(f(x), f(y)). (2) Met: objects are (generalized) me ..."
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(1) Ord: objects are (pre)ordered sets ( = sets with a reflexive and transitive relation, no antisymmetry condition), with monotone maps. Formally: (X, a) with 1. ⊤ � a(x, x), 2. a(x, y) ∧ a(y, z) � a(x, z), f: (X, a) − → (Y, b) with a(x, y) � b(f(x), f(y)). (2) Met: objects are (generalized) metric spaces (=sets with a function a: X×X − → [0, ∞] that is 0 on the diagonal and satisfies the triangle inequality), with contractions (=nonexpansive maps). Formally: (X, a) with 1. 0 ≥ a(x, x), 2. a(x, y) + a(y, z) ≥ a(x, z), f: (X, a) − → (Y, b) with a(x, y) ≥ b(f(x), f(y)). (3) UMet: the full subcategory of Met containing all ultrametric spaces, for which (2) 2 is strengthened to 2 ′. max{a(x, y), a(y, z)} ≥ a(x, z).