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6 Binary decision diagrams using barcode development for software control to generate, create barcode 3 of 9 image in software applications. WinForms 7. (a) How would you de barcode 3 of 9 for None ne the notion of semantic entailment for the relational mu-calculus (b) De ne formally when two formulas of the relational mu-calculus are semantically equivalent..

Exercises 6.15 1. Using the model of F Software Code 3 of 9 igure 6.24 (page 384), determine whether f EX (x1 x2 ) holds, where is (a) (x1 , x2 ) (1, 0) (b) (x1 , x2 ) (0, 1) (c) (x1 , x2 ) (0, 0).

2. Let S be {s0 , s1 }, with s0 s0 , s0 s1 and s1 s0 as possible transitions and L(s0 ) = {x1 } and L(s1 ) = . Compute the boolean function f EX (EX x1 ) .

3. Equations (6.17) (page 395), (6.

19) and (6.20) de ne f EF , f AF and f EG . Write down a similar equation to de ne f AG .

4. De ne a direct coding f AU by modifying (6.18) appropriately.

5. Mimic the example checks on page 396 for the connective AU: consider the model of Figure 6.24 (page 384).

Since [[E[(x1 x2 ) U ( x1 x2 )]]] equals the entire state set {s0 , s1 , s2 }, your coding of f E[x1 x2 U x1 x2 ] is correct if it computes 1 for all bit vectors di erent from (1, 1). (a) Verify that your coding is indeed correct. (b) Find a boolean formula without xed points which is semantically equivalent to f E[(x1 x2 )U( x1 x2 )] .

6. (a) Use (6.20) on page 395 to compute f EG x1 for the model in Figure 6.

24. (b) Show that f EG x1 faithfully models the set of all states which satisfy EG x1 . 7.

In the grammar (6.10) for the relational mu-calculus on page 390, it was stated that, in the formulas Z.f and Z.

f , any occurrence of Z in f is required to fall within an even number of complementation symbols . What happens if we drop this requirement (a) Consider the expression Z.Z.

We already saw that our relation is total in the sense that either f or f holds for all choices of valuations and relational mu-calculus formulas f . But formulas like Z.Z are not formally monotone.

Let be any valuation. Use mutual mathematical induction to show: i. m Z.

Z for all even numbers m 0 ii. m Z.Z for all odd numbers m 1 Infer from these two items that Z.

Z holds according to (6.12). (b) Consider any environment .

Use mathematical induction on m (and maybe an analysis on ) to show: If m Z.(x1 + x2 Z) for some m 0, then k Z.(x1 + x2 Z) for all k m.

. 6.5 Exercises (c) In general, if f is Software barcode 3 of 9 formally monotone in Z then m Z.f implies m+1 Z.f .

Can you state a similar property for the greatest xed-point operator 8. Given the CTL model for the circuit in Figure 6.29 (page 389): * (a) code the function f EX (x1 x2 ) (b) code the function f AG (AF x1 x2 ) * (c) nd a boolean formula without any xed points which is semantically equivalent to f AG (AF x1 x2 ) .

9. Consider the sequential synchronous circuit in Figure 6.33 (page 408).

Evaluate f EX x2 , where equals (a) (x1 , x2 , x3 ) (1, 0, 1) (b) (x1 , x2 , x3 ) (0, 1, 0). 10. Prove Theorem 6.

19 Given a coding for a nite CTL model, let be a CTL formula from an adequate fragment. Then [[ ]] corresponds to the set of valuations such that f . by structural induction on .

You may rst want to show that the evaluation of f depends only on the values (xi ), i.e. it does not matter what assigns to xi or Z.

Argue that Theorem 6.19 above remains valid for arbitrary CTL formulas as long as we translate formulas which are not in the adequate fragment into semantically equivalent formulas in that fragment and de ne f to be f . Derive the formula f AF ( x1 x2 ) for the model in Figure 6.

32(b) on page 407 and evaluate it for the valuation corresponding to state s2 to determine whether s2 AF ( x1 x2 ) holds. Repeat the last exercise with f E[x1 x2 Ux1 ] . Recall the way the two labelling algorithms operate in 3.

Does our symbolic coding mimic either or both of them, or neither . 13. 14..

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