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The monadic second-order logic of graphs. IX: Machines and their behaviours. (English) Zbl 0872.03026
Summary: For the previous parts see [Zbl 0722.03008; Zbl 0694.68043; Zbl 0754.03006; Zbl 0731.03006; Zbl 0754.68065; Zbl 0809.03005; Zbl 0831.03001; Zbl 0809.03006; Zbl 0830.03016].
We establish that every monadic second-order property of the behaviour of a machine (transition systems and tree automata are typical examples of machines) is a monadic second-order property of the machine itself. In this way, we clarify the distinction between “dynamic” properties of machines (i.e., properties of their behaviours), and their “static” properties (i.e., properties of the graphs or relational structures representing them). It is important for program verification that the dynamic properties that one wants to verify can be formulated statically, in the simplest possible way. As a corollary of our main result, we also obtain that the monadic theory of an algebraic tree is decidable.

MSC:
03D05 Automata and formal grammars in connection with logical questions
68Q60 Specification and verification (program logics, model checking, etc.)
68Q45 Formal languages and automata
68R10 Graph theory (including graph drawing) in computer science
03B25 Decidability of theories and sets of sentences
03B15 Higher-order logic; type theory (MSC2010)
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