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 non-commutative Intuitionistic Linear Logic, including Lambek calculus. The quantale (Q, � , •) acts on an underlying Q-right 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 resource-sensitive structure on the epistemic actions. Keywords: Multi-agent communication, knowledge update, resource-sensitivity, quantale, Galois adjoints, dynamic epistemic logic, sequent calculus, Lambek calculus, Linear Logic.

Abstract. This article is a brief and subjective survey of quantum programming language research. 1 Quantum Computation Quantum computing is a relatively young subject. It has its beginnings in 1982, when Paul Benioff and Richard Feynman independently pointed out that a quantum mechanical system can be used to perform computations [11, p.12]. Feynman’s interest in quantum computation was motivated by the fact that it is computationally very expensive to simulate quantum physical systems on classical computers. This is due to the fact that such simulation involves the manipulation is extremely large matrices (whose dimension is exponential in the size of the quantum system being simulated). Feynman conceived of quantum computers as a means of simulating nature much more efficiently. The evidence to this day is that quantum computers can indeed perform certain tasks more efficiently than classical computers. Perhaps the best-known example is Shor’s factoring algorithm, by which a quantum computer can find

Domain theory is a branch of classical computer science. It has proven to be a rigourous mathematical structure to describe denotational semantics for programming languages and to study the computability of partial functions. In this paper, we study the extension of domain theory to the quantum setting. By defining a quantum domain we introduce a rigourous definition of quantum computability for quantum states and operators. Furthermore we show that the denotational semantics of quantum computation has the same structure as the denotational semantics of classical probabilistic computation introduced by Kozen [23]. Finally, we briefly review a recent result on the application of quantum domain theory to quantum information processing. 1

1. A finitary probability space (=all probability measures on a fixed finite support) can be faithfully represented by a partial order equipped with a measure of content (e.g. Shannon entropy). 2. This partial order can be obtained via a purely order-theoretic systematic procedure starting from an algebra of properties. This procedure applies to any poset envisioned as an algebra of properties. 1

We present an algebra and sequent calculus to reason about dynamic epistemic logic, a logic for information update in multi-agent systems. We contribute to it by equipping it with a logical account of resources, a semi-automatic way of reasoning through the algebra and sequent calculus, and finally by generalizing it to non-boolean settings. Dynamic Epistemic Logic (DEL) is a PDL-style logic [14] to reason about epistemic actions and updates in a multi-agent 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 e-commerce where agents need to have knowledge acquisition strategies over complex networks. The standard approach to information flow in a multi-agent system has been presented in [8] but it does not present a formal description of epistemic programs and their updates. The first attempts to

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We revisit the standard axioms of domain theory with emphasis on their relation to the concept of partiality, explain how this idea arises naturally in probability theory and quantum mechanics, and then search for a mathematical setting capable of providing a satisfactory unification of the two. 1

Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be

In this paper, we will investigate two partial orders on classical states ∆ n that induce a domain theoretic structure. In particular, we will look at the Bayesian order presented by Coecke and Martin (2002) and the implicative order presented by Martin (2004). For both of the domains that arise from these orderings on the classical states, Shannon entropy is a measurement in the domain theoretic sense. We will then investigate the behaviour of a simple learning algorithm for stochastic learning automata, which has previously been studied using domain theory (Edalat, 1995), in terms of how the states of the system change with respect to the domains induced by these partial orders. 1