The term trust is used in this weeks devotion and I want us to consider it as we begin this course on financial accounting.  Most of us rely on some type of financial reporting whether that is a pay check, a bank statement, or some other type of payment.  You may also have retirement funds, savings accounts, or personal investments or other types of interests in companies that your are relying on to earn a return, build up value in a stock investment, or to reinvest their earnings in developing safer products for us as consumers. Over the years we have seen many examples of companies who have replaced working towards developing that trust with enhancing their own personal interests. Many of these situations have come up in the area of misstating financial statements to reflect a better picture of the companys position than may truly exist.  Cases like Enron, WorldCom and Madoff Investments have destroyed some of the trust that many have had in the reporting process of corporate America.  Please post your thoughts on why being able to trust corporate financial reports is important to our economy.  What happens in the overall economy when people lose faith in financial reports?  Why might this be more important in an economy based on capitalism as compared to a country using a communistic or socialistic economic model?  Provide an example of a financial report that you trust” and why.

(a) Let n > 0 be an integer and Ln be the language of all linear equations a1X1 + a2X2 + · · · + anXn + an+1 = 0 in n unknowns X1, X2, . . . , Xn and over integer

coefficients a1, a2, . . . , an, an+1, which have a solution in the integers.

(i) Show that Ln is semi-decidable by describing, in general mathematical terms, an algorithm that takes as input a linear equation a1X1 + a2X2 + · · · + anXn + an+1 =

0 with a1, a2, . . . , an, an+1 in the integers, and halts exactly when this equation has a solution in the integers.

(ii) We now use the fact (stated in the lectures) that Ln is actually decidable. Describe an algorithm, using a decider for Ln as a subroutine, which for any linear

equation a1X1 + a2X2 + · · · + anXn + an+1 = 0, with a1, a2, . . . , an, an+1 in the integers, decides whether or not it has an integer solution, and if it does, finds

at least one such solution.

(b) Argue that the following languages over the alphabet {a, b, c} belong to the complexity class P. It is enough to give an implementation level description of the

relevant Turing machines, and explain why their complexity is polynomial.

(i) {w ? {a, b, c} * | |w|a = |w|b = |w|c}

(ii) {w ? {a, b, c} * | it is not the case that |w|a = |w|b = |w|c} Note that we use the notation |w|a to denote the number of occurrences of the letter a in w, and

similarly for |w|b and |w|c.

Sample Solution

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