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**Extra info for 699-approximation algorithm for Max-Bisection**

**Sample text**

The remaining problem is then to prove the propositions that are given above the line. In particular, the condition {PI 1\ B}P{PI} amounts to saying that PI is an invariant of P: if it holds before the execution of P, it will also remain true after the termination of P (it may however be false during the execution of P). The problem is that invariants are not always detected easily so that the conditions above the line hold. For this reason, some investigations started to compute or approximate invariants.

However, this relationship has been considered relatively late. Thomas showed in 1988 [485] that the temporal logic that is obtained by using w-automata as temporal operators is strictly less expressive than the second order theory of two successors (525). In contrast to 515, where only one successor is available, the monadic second order logic of two successors, 525 has two successors. For this reason, its domains are not lined up, but form trees that may be finite or infinite. In particular, domains for 525 are binary trees, and for 5n5 trees of branching degree n.

Qn and corresponding variables q~, ... , q~ for the next state. This description of the structure can then be efficiently stored as a BOD, as can the sets of states of the structure. As the evaluation of the temporal logic formula simply consists of some fixpoint iterations over such sets of states, the entire model checking procedure can be implemented with BODs. Note that a structure with n state variables may have 2n reachable states. Hence, small propositional formulas can encode large structures.