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Questiоn 1. [10+10+5 = 25 pts]. Twо-level minimizаtiоn. Consider the incompletely specified function FON = w'xy'z' + xy'z + wxy FDC = wx'z' + w'x'y' + w'xyz Mаnuаlly apply the following steps of the ESPRESSO algorithm to this example. (a) Compute the off-set as FOFF = complement(FON ∪{"version":"1.1","math":"∪"} FDC), by using the unate recursive paradigm based algorithm to compute complements that we discussed in class. (b) The initial cover C0 is set to FON. Perform expand(C0, FOFF), i.e., expand each cube in C0 as much as possible while ensuring that the cubes do not expand into the off-set. Denote the cover resulting from this step as C1. (c) Perform irredundant(C1, FDC). This step eliminates cubes in C1 if possible, resulting in an irredundant cover C2. For this step, you may find it convenient to use the Karnaugh map representation of C1. However, you may also use any other method of your choice. Question 2. [20+20 = 40 pts]. Unate Recursive Paradigm. Consider the application of the unate recursive paradigm to verifying the equivalence of two Sum-of-products (SOP) expressions. For example, suppose you wish to verify that the input provided to a two-level minimizer like ESPRESSO and the output it generates are functionally equivalent. (a) Given two functions f1 and f2 specified in SOP form, describe an algorithm to check whether f1 = = f2 using the unate recursive paradigm. HINTS: Since you have two functions (f1 and f2) rather than one, consider performing a coordinated co-factoring where you pick the same variable at each step to co-factor f1 and f2. Here is a list of questions to guide your thought process: What is the basic property of this problem that allows you to adopt a divide-and-conquer solution? What operation will you perform at each step of the co-factoring tree? What are the leaf cases? How will you merge the results as you go up the tree from the leaves to the root? The description of the algorithm should be clear and concise. Think through the algorithm before you write it down. Please avoid verbose or vague descriptions. (b) Apply the algorithm to check if the following SOP expressions are equivalent: f1 = a' c' d' + a' b' c + a b c' + b' c' d' + a' b c d + a b' c d f2 = c' d' + a' c d + a b c' + b' c d + a' b' d' Question 3. [20+15 = 35 pts]. Multi-level synthesis - Algebraic factoring. Consider the following algebraic expressions: F = acf + ad'ef + bcf + bd + bd'ef + cd G = ae + acd + be + ef ' (a) Generate all kernels and co-kernels for F and G. You may use either the recursive algorithm or the cube-literal matrix based method. (b) Determine if there exists a common multi-cube factor for F and G. If one exists, substitute it wherever possible into F and G, write the new expressions for F and G, and calculate the savings in literals due to the substitution(s). Any multi-cube factor is acceptable (not limited to the one with highest literal savings). Congratulations, you are almost done with this exam. DO NOT end the Honorlock session until you have submitted your work to Gradescope. When you have answered all questions: Use your smartphone or scanner to scan your answer sheet, and save the scan as a PDF. Make sure your scan is clear and legible. Submit your PDF as follows: Email your PDF to yourself or save it to the cloud (Google Drive, etc.). Submit your exam: Midterm Exam (online students only) Return to this window and click the button below to agree to the honor statement. Click Submit Quiz to end the exam. End the Honorlock session.