A framework for consensus modelling is introduced using Kleene's three valued logic as a means to express vagueness in agents' beliefs. Explicitly borderline cases are inherent to propositions involving vague concepts where sentences of a propositional language may be absolutely true, absolutely false or borderline. By exploiting these intermediate truth values, we can allow agents to adopt a more vague interpretation of underlying concepts in order to weaken their beliefs and reduce the levels of inconsistency, so as to achieve consensus. We consider a consensus combination operation which results in agents adopting the borderline truth value as a shared viewpoint if they are in direct conflict. Simulation experiments are presented which show that applying this operator to agents chosen at random (subject to a consistency threshold) from a population, with initially diverse opinions, results in convergence to a smaller set of more precise shared beliefs. Furthermore, if the choice of agents for combination is dependent on the payoff of their beliefs, this acting as a proxy for performance or usefulness, then the system converges to beliefs which, on average, have higher payoff.
A fundamental difference between our approach and the above models is that we use Kleene's three valued logic to represent beliefs in a propositional logic setting, rather than identify opinions with real values or intervals. @cite_5 have shown that through use of a three-state model for networked consensus of complete graphs, nodes converge to a consensus much faster and with greater accuracy when compared to a restrictive binary model. In the sequel we extend this approach to a more general setting involving larger languages and incorporating a measure of payoff for beliefs.
@cite_5
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We consider the following distributed consensus problem: Each node in a complete communication network of size n initially holds an opinion, which is chosen arbitrarily from a finite set Σ. The system must converge toward a consensus state in which all, or almost all nodes, hold the same opinion. Moreover, this opinion should be valid, i.e., it should be one among those initially present in the system. This condition should be met even in the presence of a malicious adversary who can modify the opinions of a bounded subset of nodes, adaptively chosen in every round. We consider the 3-majority dynamics: At every round, every node pulls the opinion from three random neighbors and sets his new opinion to the majority one (ties are broken arbitrarily). Let k be the number of valid opinions. We show that, if k ≤ nα, where α is a suitable positive constant, the 3-majority dynamics converges in time polynomial in k and log n with high probability even in the presence of an adversary who can affect up to o([EQUATION]) nodes at each round. Previously, the convergence of the 3-majority protocol was known for |Σ| = 2 only, with an argument that is robust to adversarial errors. On the other hand, no anonymous, uniform-gossip protocol that is robust to adversarial errors was known for |Σ| > 2. The use of conventional classical logic is misleading for characterizing the behavior of logic programs because a logic program, when queried, will do one of three things: succeed with the query, fail with it, or not respond because it has fallen into infinite backtracking. In [7] Kleene proposed a three-valued logic for use in recursive function theory. The so-called third truth value was really undefined: truth value not determined. This logic is a useful tool in logic-program specification, and in particular, for describing models. (See [11].) Tarski showed that formal languages, like arithmetic, cannot contain their own truth predicate because one could then construct a paradoxical sentence that effectively asserts its own falsehood. Natural languages do allow the use of "is true", so by Tarski's argument a semantics for natural language must leave truth-value gaps: some sentences must fail to have a truth value. In [8] Kripke showed how a model having truth-value gaps, using Kleene's three-valued logic, could be specified. The mechanism he used is a famiUar one in program semantics: consider the least fixed point of a certain monotone operator. But that operator must be defined on a space involving three-valued logic, and for Kripke's application it will not be continuous. We apply techniques similar to Kripke's to logic programs. We associate with each program a monotone operator on a space of three-valued logic interpretations, or better partial interpretations. This space is not a complete lattice, and the operators are not, in general, continuous. But least and other fixed points do exist. These fixed points are shown to provide suitable three-valued program models. They relate closely to the least and greatest fixed points of the operators used in [1]. Because of the extra machinery involved, our treatment allows for a natural consideration of negation, and indeed, of the other prepositional connectives as well. And because of the elaborate structure of fixed points available, we are able to Our goal is to combine the rich multistep inference of symbolic logical reasoning with the generalization capabilities of neural networks. We are particularly interested in complex reasoning about entities and relations in text and large-scale knowledge bases (KBs). (2015) use RNNs to compose the distributed semantics of multi-hop paths in KBs; however for multiple reasons, the approach lacks accuracy and practicality. This paper proposes three significant modeling advances: (1) we learn to jointly reason about relations, entities, and entity-types; (2) we use neural attention modeling to incorporate multiple paths; (3) we learn to share strength in a single RNN that represents logical composition across all relations. On a largescale Freebase+ClueWeb prediction task, we achieve 25 error reduction, and a 53 error reduction on sparse relations due to shared strength. On chains of reasoning in WordNet we reduce error in mean quantile by 84 versus previous state-of-the-art. The code and data are available at this https URL We consider distributed plurality consensus in a complete graph of size @math with @math initial opinions. We design an efficient and simple protocol in the asynchronous communication model that ensures that all nodes eventually agree on the initially most frequent opinion. In this model, each node is equipped with a random Poisson clock with parameter @math . Whenever a node's clock ticks, it samples some neighbors, uniformly at random and with replacement, and adjusts its opinion according to the sample. A prominent example is the so-called two-choices algorithm in the synchronous model, where in each round, every node chooses two neighbors uniformly at random, and if the two sampled opinions coincide, then that opinion is adopted. This protocol is very efficient and well-studied when @math . If @math for some small @math , we show that it converges to the initial plurality opinion within @math rounds, w.h.p., as long as the initial difference between the largest and second largest opinion is @math . On the other side, we show that there are cases in which @math rounds are needed, w.h.p. One can beat this lower bound in the synchronous model by combining the two-choices protocol with randomized broadcasting. Our main contribution is a non-trivial adaptation of this approach to the asynchronous model. If the support of the most frequent opinion is at least @math times that of the second-most frequent one and @math , then our protocol achieves the best possible run time of @math , w.h.p. We relax full synchronicity by allowing @math nodes to be poorly synchronized, and the well synchronized nodes are only required to be within a certain time difference from one another. We enforce this synchronicity by introducing a novel gadget into the protocol.