By Davide Sangiorgi
Induction is a pervasive device in desktop technology and arithmetic for outlining gadgets and reasoning on them. Coinduction is the twin of induction and as such it brings in particularly diverse instruments. at the present time, it really is customary in computing device technology, but additionally in different fields, together with synthetic intelligence, cognitive technology, arithmetic, modal logics, philosophy and physics. the easiest recognized example of coinduction is bisimulation, frequently hired to outline and turn out equalities between probably countless items: tactics, streams, non-well-founded units, and so on. This ebook offers bisimulation and coinduction: the elemental techniques and strategies and the duality with induction. each one bankruptcy includes routines and chosen ideas, allowing scholars to attach thought with perform. a different emphasis is put on bisimulation as a behavioural equivalence for tactics. therefore the booklet serves as an creation to types for expressing techniques (such as method calculi) and to the linked recommendations of operational and algebraic research.
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Extra resources for Introduction to Bisimulation and Coinduction
Example text
The closure is ‘backward’ because the rules are used in the direction from the conclusion to the premises: we require that each element in the closure be the conclusion of a rule whose premises must also belong to the closure. Here, too, we will prove later that these different formulations coincide. We can, however, already see that the set resulting from the iterative construction is closed backward, and that the closure is lost by adding more processes to the set. The formulation with the closure gives us a proof principle: to prove that each process in a set T has an ω-trace under μ it suffices to show that T is closed backward under the rule above; this is the coinduction proof principle, for ω-traces.
Something to start from, and then, in the inductive part, one builds on top of what one has obtained so far. Indeed, the above definition of ∼, and its proof technique, are not inductive, but coinductive. It is good to stop for a while, to get a grasp of the meaning of coinduction, and a feeling of the duality between induction and coinduction. This will be useful for relating the idea of bisimilarity to other concepts, and it will also allow us to derive a few results for bisimilarity. We do this in Chapter 2.
Suppose, however, we did not notice this, and tried to prove that R is a bisimulation. 2 on each pair in R. As an example, we consider clause (1) on the pair a → P2 ; this is matched by Q1 via transition (P1 , Q1 ). The only transition from P1 is P1 − a Q1 − → Q2 , for (P2 , Q2 ) ∈ R as required. However, the checks on (P2 , Q2 ) fail, since, b for instance, the transition P2 − → P1 cannot be matched by Q2 , whose only transition is b Q2 − → Q3 and the pair (P1 , Q3 ) does not appear in R. ) We realise that we have to add the pair (P1 , Q3 ) to R.