By Burichenko V.P.

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**Example text**

Then: μfV = V (G). t. G. We sketch a proof of this theorem for the particular case of equation (3). Let us “unfold” the grammar G of (4) by augmenting the nonterminal X with a counter keeping track of the height of a derivation: 24 J. Esparza and M. Luttenberger X 1 X [1] X 2 X [2] X 3 X [3] →c →X 1 → aX 1 X 1 | bX 1 → X 2 | X [1] → aX 2 X 2 | aX [1] X → X 3 | X [2] .. 2 2 | aX X h → aX h−1 X h−1 | aX [h−2] X X [h] → X h | X [h−1] .. X [1] | bX h−1 | aX 2 h−1 X [h−2] | bX h−1 Let G[h] (G h ) be the grammar consisting of those “unfolded” rules whose left-hand side is given by one of the variables of X [h] = {X 0 , X [0] , .

We show that Kleene’s theorem, which not only proves the existence of the least solution, but also provides an algorithm for approximating it, corresponds to approximating G by grammars G[1] , G[1] , . . where G[h] generates the derivation trees of G of height h. We then introduce (Section 4) a faster approximation by grammars H [1] , H [1] , . . where H [h] generates the derivation trees of G of dimension h [EKL08a, EKL10]. We show that this approximation is a generalization of Newton’s method for approximating the zero of a differentiable function, and present a new result about its convergence speed when multiplication is commutative [Lut]2 .

Liveness counterexamples can be found by (coinductively) enumerating all possible states that can be reached via inﬁnite loops and then determining if any of these states constitute valid counterexamples. To demonstrate the power of coinductive logic programming, we show how an interpreter for linear temporal logic can be written very elegantly. In LTL, one checks if a temporal logic formula is true along a path. Temporal operators whose meaning is given in terms of LFP s are realized via tabled logic programming, while those whose meaning is given in terms of GFP s are realized using coinductive logic programming.

### 2-Cohomologies of the groups SL (n,q) by Burichenko V.P.

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