By Deo S.
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Extra resources for Algebraic topology: a primer
Note that ∼ is the coarsest bisimulation on D. Intuitively, bisimilar states can stepwise simulate each other, meaning that they can be merged while preserving important properties. Deﬁnition 4 (Bisimulation quotient). For a DTMC D = (S, P, sinit , AP, L), the bisimulation quotient is deﬁned as D/∼ = (S/∼, P , [sinit ]∼ , AP, L ), where P ([s]∼ , [t]∼ ) = P(s, [t]∼ ) and L ([s]∼ ) = L(s). Note that D/∼ is well-deﬁned. The preservation theorem of Aziz et al.  states that strong bisimulation is sound and complete with respect to PCTL*: Proposition 1.
D. dissertation, Universit´e Scientifique et M´edicale de Grenoble, Grenoble, France (1978) 7. : Constructive design of a hierarchy of semantics of a transition system by abstract interpretation. Theoretical Computer Science 277(1-2), 47–103 (2002) 8. : Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: Proc. 4th ACM POPL, pp. 238– 252 (1977) 9. : Systematic design of program analysis frameworks. In: Proc. 6th ACM POPL, pp.
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. Actually, it turns out that completeness for (| x := x ± 1 |) completely characterizes constant propagation CP, in the sense that the complete shell of A0 for both (| x := x + 1 |) and (| x := x − 1 |) is precisely CP.
Algebraic topology: a primer by Deo S.