4,908 research outputs found

    Successive Local and Successive Global Omniscience

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    This paper considers two generalizations of the cooperative data exchange problem, referred to as the successive local omniscience (SLO) and the successive global omniscience (SGO). The users are divided into \ell nested sub-groups. Each user initially knows a subset of packets in a ground set XX of size kk, and all users wish to learn all packets in XX. The users exchange their packets by broadcasting coded or uncoded packets. In SLO or SGO, in the llth (1l1\leq l\leq \ell) round of transmissions, the llth smallest sub-group of users need to learn all packets they collectively hold or all packets in XX, respectively. The problem is to find the minimum sum-rate (i.e., the total transmission rate by all users) for each round, subject to minimizing the sum-rate for the previous round. To solve this problem, we use a linear-programming approach. For the cases in which the packets are randomly distributed among users, we construct a system of linear equations whose solution characterizes the minimum sum-rate for each round with high probability as kk tends to infinity. Moreover, for the special case of two nested groups, we derive closed-form expressions, which hold with high probability as kk tends to infinity, for the minimum sum-rate for each round.Comment: Accepted for publication in Proc. ISIT 201

    A Practical Approach for Successive Omniscience

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    The system that we study in this paper contains a set of users that observe a discrete memoryless multiple source and communicate via noise-free channels with the aim of attaining omniscience, the state that all users recover the entire multiple source. We adopt the concept of successive omniscience (SO), i.e., letting the local omniscience in some user subset be attained before the global omniscience in the entire system, and consider the problem of how to efficiently attain omniscience in a successive manner. Based on the existing results on SO, we propose a CompSetSO algorithm for determining a complimentary set, a user subset in which the local omniscience can be attained first without increasing the sum-rate, the total number of communications, for the global omniscience. We also derive a sufficient condition for a user subset to be complimentary so that running the CompSetSO algorithm only requires a lower bound, instead of the exact value, of the minimum sum-rate for attaining global omniscience. The CompSetSO algorithm returns a complimentary user subset in polynomial time. We show by example how to recursively apply the CompSetSO algorithm so that the global omniscience can be attained by multi-stages of SO

    Cooperative Data Exchange based on MDS Codes

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    The cooperative data exchange problem is studied for the fully connected network. In this problem, each node initially only possesses a subset of the KK packets making up the file. Nodes make broadcast transmissions that are received by all other nodes. The goal is for each node to recover the full file. In this paper, we present a polynomial-time deterministic algorithm to compute the optimal (i.e., minimal) number of required broadcast transmissions and to determine the precise transmissions to be made by the nodes. A particular feature of our approach is that {\it each} of the KdK-d transmissions is a linear combination of {\it exactly} d+1d+1 packets, and we show how to optimally choose the value of d.d. We also show how the coefficients of these linear combinations can be chosen by leveraging a connection to Maximum Distance Separable (MDS) codes. Moreover, we show that our method can be used to solve cooperative data exchange problems with weighted cost as well as the so-called successive local omniscience problem.Comment: 21 pages, 1 figur

    On the Optimality of Secret Key Agreement via Omniscience

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    For the multiterminal secret key agreement problem under a private source model, it is known that the maximum key rate, i.e., the secrecy capacity, can be achieved through communication for omniscience, but the omniscience strategy can be strictly suboptimal in terms of minimizing the public discussion rate. While a single-letter characterization is not known for the minimum discussion rate needed for achieving the secrecy capacity, we derive single-letter lower and upper bounds that yield some simple conditions for omniscience to be discussion-rate optimal. These conditions turn out to be enough to deduce the optimality of omniscience for a large class of sources including the hypergraphical sources. Through conjectures and examples, we explore other source models to which our methods do not easily extend

    Logical Omnipotence and Two notions of Implicit Belief

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    The most widespread models of rational reasoners (the model based on modal epistemic logic and the model based on probability theory) exhibit the problem of logical omniscience. The most common strategy for avoiding this problem is to interpret the models as describing the explicit beliefs of an ideal reasoner, but only the implicit beliefs of a real reasoner. I argue that this strategy faces serious normative issues. In this paper, I present the more fundamental problem of logical omnipotence, which highlights the normative content of the problem of logical omniscience. I introduce two developments of the notion of implicit belief (accessible and stable belief ) and use them in two versions of the most common strategy applied to the problem of logical omnipotence

    Omniscience, the Incarnation, and Knowledge de se

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    A knowledge argument is offered that presents unique difficulties for Christians who wish to assert that God is essentially omniscient. The difficulties arise from the doctrine of the incarnation. Assuming that God the Son did not necessarily have to become incarnate, then God cannot necessarily have knowledge de se of the content of a non-divine mind. If this is right, then God’s epistemic powers are not fixed across possible worlds and God is not essentially omniscient. Some options for Christian theists are discussed, including rejecting traditional theism in favour of some version of pantheism or panentheism

    Can Human Beings Truly Be Considered Free?

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    There exists a complex relationship between human freedom and God\u27s divine foreknowledge; the questions surrounding this topic abound and are difficult to answer. The question arises, if God knows all that we think and do, now and in the future, and has Providence over all of our actions, are we truly free; do we have free will? I assert that we do. The arguments that would pit God\u27s foreknowledge and human freedom against each other as incompatible have a faulty foundation surrounding the nature of God\u27s knowledge, His Being, and our purpose in the world. Essentially, the nature of God\u27s knowledge is not deterministic, and His knowing does not necessitate that actions happen as they will; they do not happen because He knows, He knows because they happen. Additionally, God created us with a particular nature, and that nature includes a free will as human beings. God can neither impede nor prevent this nature, as He would actually, in effect, be contradicting His own nature as well as ours. We as beings are made to fulfill our natures, and thus we were made to act freely. God\u27s providence exists in that He made us, and made us to fulfill our particular natures, and as such, we are naturally inclined towards the good and towards our divine end and Creator. The divine end and purpose exists for us all, but it is our free will which allows us to choose to fulfill or not fulfill this nature. Once the true nature and metaphysics of God\u27s knowledge and being, our nature as humans, and our purpose as humans are understood, we can better attempt to reconcile the idea of God\u27s foreknowledge and providence with our freedom and free will as human beings
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