On Failure of 0-1 Laws

8 years 4 days ago

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Let α ∈ (0, 1)R be irrational and Gn = Gn,1/nα be the random graph with edge probability 1/nα; we know that it satisﬁes the 0-1 law for ﬁrst order logic. We deal with the failure of the 0-1 law for stronger logics: L∞,k, k a large enough natural number and the inductive logic. Dedicated to Yuri Gurevich on the occasion of his 75th birthday Let Gn,p be the random graph with set of nodes [n] = {1, . . . , n}, each edge of probability p ∈ [0, 1]R, the edges being drawn independently, (see ⊞1 below). On 0-1 laws (and random graphs) see the book of Spencer [6] or Alon-Spencer [1], in particular on the behaviour of the random graph Gn,1/nα for α ∈ (0, 1)R irrational. On ﬁnite model theory see Flum-Ebbinghaus [2], e.g. on the logic L∞,k and on inductive logic, also called LFP logic (i.e. least ﬁx point logic). A characteristic example of what can be expressed in this logic is “in the graph G there is a path from the node x to the node y”, this is closed to what we...

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