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Exercises in Software Printing ANSI/AIM Code 39 in Software Exercises




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5.6 Exercises using barcode generating for software control to generate, create uss code 39 image in software applications. Microsoft Office Development. Microsoft Office 2000/2003/2007/2010 3. Consider the Code 3 of 9 for None Kripke model M = (W, R, L) where W = {a, b, c, d, e}; R = {(a, c), (a, e), (b, a), (b, c), (d, e), (e, a)}; and L(a) = {p}, L(b) = {p, q}, L(c) = {p, q}, L(d) = {q} and L(e) = . (a) Draw a graph for M.

(b) Investigate which of the formulas in exercise 1(b) on page 350 have a world which satis es it. 4. (a) Think about what you have to do to decide whether p 23q is true in a model.

* (b) Find a model in which it is true and one in which it is false. 5. For each of the following pairs of formulas, can you nd a model and a world in it which distinguishes them, i.

e. makes one of them true and one false In that case, you are showing that they do not entail each other. If you cannot, it might mean that the formulas are equivalent.

Justify your answer. (a) 2p and 22p (b) 2 p and 3p (c) 2(p q) and 2p 2q * (d) 3(p q) and 3p 3q (e) 2(p q) and 2p 2q * (f) 3(p q) and 3p 3q (g) 2(p q) and 2p 2q (h) 3 and (i) 2 and (j) 3 and . 6.

Show that the following formulas of basic modal logic are valid: * (a) 2( ) (2 2 ) (b) 3( ) (3 3 ) * (c) 2 (d) 3 (e) 3 (2 3 ) 7. Inspect De nition 5.4.

We said that we de ned x by structural induction on . Is this really correct Note the implicit de nition of a second relation x . Why is this de nition still correct and in what sense does it still rely on structural induction .

Exercises 5.3 1. For which of Software barcode 3/9 the readings of 2 in Table 5.7 are the formulas below valid * (a) ( 2 ) ( 3 ) (b) (2 ( 22 3 )) ((2 ( 22 )) (3 23 )).

2. Dynamic logic: Let P range over the programs of our core language in 4. Consider a modal logic whose modal operators are P and [P ] for all such programs P .

Evaluate such formulas in stores l as in De nition 4.3 (page 264)..

5 Modal logics and agents The relation l P Software Code 39 Full ASCII holds i program P has some execution beginning in store l and terminating in a store satisfying . * (a) Given that P equals [P ], spell out the meaning of [P ]. (b) Say that is valid i it holds in all suitable stores l.

State the total correctness of a Hoare triple as a validity problem in this modal logic. 3. For all binary relations R below, determine which of the properties re exive through to total from page 320 apply to R where R(x, y) means that * (a) x is strictly less than y, where x and y range over all natural numbers n 1 (b) x divides y, where x and y range over integers e.

g. 5 divides 15, whereas 7 does not (c) x is a brother of y * (d) there exist positive real numbers a and b such that x equals a y + b, where x and y range over real numbers. * 4.

Prove the Fact 5.16. 5.

Prove the informal claim made in item 2 of Example 5.12 by structural induction on formulas in (5.1).

6. Prove Theorem 5.17.

Use mathematical induction on the length of the sequence of negations and modal operators. Note that this requires a case analysis over the topmost operator other than a negation, or a modality. 7.

Prove Theorem 5.14, but for the case in which R is re exive, or transitive. 8.

Find a Kripke model in which all worlds satisfy p q, but at least one world does not satisfy q p; i.e. show that the scheme is not satis ed.

9. Below you nd a list of sequents in propositional logic. Find out whether you can prove them without the use of the rules PBC, LEM and e.

If you cannot succeed, then try to construct a model M = (W, R, L) for intuitionistic propositional logic such that one of its worlds satis es all formulas in , but does not satisfy . Assuming soundness, this would guarantee that the sequent in question does not have a proof in intuitionistic propositional logic. * (a) (p q) (q r) (b) The proof rule MT: p q, q p (c) p q p q (d) p q p q (e) The proof rule e: p p * (f) The proof rule i: p p.

10. Prove that the natural deduction rules for propositional logic without the rules e, LEM and PBC are sound for the possible world semantics of intuitionistic propositional logic. Why does this show that the excluded rules cannot be implemented using the remaining ones 11.

Interpreting 2 as agent Q believes , explain the meaning of the following formula schemes: (a) 2 3 * (b) 2 2 (c) 2( ) 2 2 ..
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