Figure 5.2. The parse tree for 23q r 2p. in Software Compose barcode 39 in Software Figure 5.2. The parse tree for 23q r 2p.

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Figure 5.2. The parse tree for 23q r 2p. generate, create 3 of 9 barcode none with software projects Bar Code Types 2 p r De nition 5.3 A model M of basic modal logic is speci ed by three things:. 1. A set W , Software Code 39 Extended whose elements are called worlds; 2. A relation R on W (R W W ), called the accessibility relation; 3.

A function L : W P(Atoms), called the labelling function.. We write R(x, y) to denote that (x, y) is in R. These models are often called Kripke models, in honour of S. Kripke who invented them and worked extensively in modal logic in the 1950s and 1960s.

Intuitively, w W stands for a possible world and R(w, w ) means that w is a world accessible from world w. The actual nature of that relationship depends on what we intend to model. Although the de nition of models looks quite complicated, we can use an easy graphical notation to depict nite models.

We illustrate the graphical notation by an example. Suppose W equals {x1 , x2 , x3 , x4 , x5 , x6 } and the relation R is given as follows:. r R(x , x ), 3 of 9 for None R(x , x ), R(x , x ), R(x , x ), R(x , x ), R(x , x ), R(x , x ), 1 2 1 3 2 2 2 3 3 2 4 5 5 4 R(x5 , x6 ); and no other pairs are related by R.. Suppose furth er that the labelling function behaves as follows: x1 x2 x3 x4 x5 x6 x L(x) {q} {p, q} {p} {q} {p}. 5 Modal logics and agents Then, the Kri pke model is illustrated in Figure 5.3. The set W is drawn as a set of circles, with arrows between them showing the relation R.

Within each circle is the value of the labelling function in that world. If you have read 3, then you might have noticed that Kripke structures are also the models for CTL, where W is S, the set of states; R is , the relation of state transitions; and L is the labelling function. De nition 5.

4 Let M = (W, R, L) be a model of basic modal logic. Suppose x W and is a formula of (5.1).

We will de ne when formula is true in the world x. This is done via a satisfaction relation x by structural induction on : p x x 2 x 3 x x x. x x x x i p L(x) i x i x and x i x , or x i x , whenever we have x i (x i x ) i , for each y W with R(x, y), we have y i there is a y W such that R(x, y) and y .. When x hold Software Code 39 Extended s, we say x satis es , or is true in world x. We write M, x if we want to stress that x holds in the model M. The rst two clauses just express the fact that is always true, while is always false.

Next, we see that L(x) is the set of all the atomic formulas that are true at x. The clauses for the boolean connectives ( , , , and ) should also be straightforward: they mean that we apply the usual truthtable semantics of these connectives in the current world x. The interesting cases are those for 2 and 3.

For 2 to be true at x, we require that be true in all the worlds accessible by R from x. For 3 , it is required that there is at least one accessible world in which is true. Thus, 2 and 3 are a bit like the quanti ers and of predicate logic, except that they do not take variables as arguments.

This fact makes them conceptually much simpler than quanti ers. The modal operators 2 and 3 are also rather like AX and EX in CTL see Section 3.4.

1. Note that the meaning of 1 2 coincides with that of ( 1 2 ) ( 2 1 ); we call it if and only if. De nition 5.

5 A model M = (W, R, L) of basic modal logic is said to satisfy a formula if every state in the model satis es it. Thus, we write M i , for each x W , x ..

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