By Peter Abramenko

This booklet is addressed to mathematicians and complex scholars drawn to structures, teams and their interaction. Its first half introduces - presupposing sturdy wisdom of standard structures - the idea of dual structures, discusses its group-theoretic history (twin BN-pairs), investigates geometric features of dual constructions and applies them to figure out finiteness homes of yes S-arithmetic teams. This software depends upon topological houses of a few subcomplexes of round constructions. The historical past of this challenge, a few examples and the whole answer for all "sufficiently huge" classical structures are coated intimately within the moment a part of the booklet.

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**Extra resources for Twin Buildings and Applications to S-Arithmetic Groups**

**Example text**

3 Constant Propagation Let us consider the standard abstraction CP used in constant propagation analysis [16]: Z · · · −2 −1 0 ∅ 1 2 ··· Complete Abstractions Everywhere 21 CP is more precise than the basic abstraction A0 , hence it makes sense to ask whether CP can be viewed as a complete shell refinement of A0 for some transfer functions. From the viewpoint of its completeness properties, one may observe that CP is clearly complete for the transfer functions (| x := x + k |), for any k ∈ Z. Moreover, we also notice that if an abstraction A is complete for (| x := x + 1 |) and (| x := x − 1 |) then, since completeness is preserved by composing functions, A is also complete for all the transfer functions (| x := x + k |), for all k ∈ Z, and this allows us to focus on (| x := x ± 1 |) only.

As shown in [20], this allows us to design an efficient algorithm for computing the stuttering simulation preorder as an exact shell abstraction refinement for the union ∪ and the stuttering operator pos. 5 Probabilistic Bisimulation and Simulation The main behavioral relations between concurrent systems, like simulation and bisimulation, have been studied in probabilistic extensions of reactive systems like Markov chains and probabilistic automata. As recently shown in [11], we mention that bisimulation and simulation relations on probabilistic transition systems can still be characterized as exactness properties in abstract interpretation and as a byproduct this allows to design efficient algorithms that compute these behavioral relations as exact shell refinements.

However, its dependence on computationally expensive numerical operations makes it particularly susceptible to the state-space explosion problem. Among other abstraction techniques, bisimulation minimisation has proven to shorten computation times signiﬁcantly, but, usually, the full state space needs to be built prior to minimisation. We present a novel approach that leverages satisﬁability solvers to extract the minimised system from a high-level description directly. A prototypical implementation in the framework of the probabilistic model checker Prism provides encouraging experimental results.