1,667 research outputs found
Randomizations of models as metric structures
The notion of a randomization of a first order structure was introduced by
Keisler in the paper Randomizing a Model, Advances in Math. 1999. The idea was
to form a new structure whose elements are random elements of the original
first order structure. In this paper we treat randomizations as continuous
structures in the sense of Ben Yaacov and Usvyatsov. In this setting, the
earlier results show that the randomization of a complete first order theory is
a complete theory in continuous logic that admits elimination of quantifiers
and has a natural set of axioms. We show that the randomization operation
preserves the properties of being omega-categorical, omega-stable, and stable
Observing, reporting, and deciding in networks of sentences
In prior work we considered networks of agents who prove facts from their knowledge bases and report them to their neighbors in their common languages in order to help a decider verify a single sentence. In report complete networks, the signatures of the agents and the links between agents are rich enough to verify any decider\u27s sentence that can be proved from the combined knowledge base. This paper introduces a more general setting where new observations may be added to knowledge bases and the decider must choose a sentence from a set of alternatives. We consider the question of when it is possible to prepare in advance a finite plan to generate reports within the network. We obtain conditions under which such a plan exists and is guaranteed to produce the right choice under any new observations
Craig interpolation for networks of sentences
The Craig Interpolation Theorem can be viewed as saying that in first order logic, two agents who can only communicate in their common language can cooperate in building proofs. We obtain generalizations of the Craig Interpolation Theorem for finite sets of agents with the following properties. (1) The agents are vertices of a directed graph. (2) The agents have knowledge bases with overlapping signatures. (3) The agents can only communicate by sending to neighboring agents sentences that they know and that are in the common language of the two agents
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