Type Your Answers For Problems 5-9. Hand-Drawn Diagrams

Type your answers for problems 5-9. Hand-drawn diagrams for problems 1-3 are acceptable.

Show solutions for problems 5-9. For problems 1-3, hand-drawn diagrams are acceptable. Specifically, construct the following:

1. Design a DFA that accepts each of the languages L1 and L2, defined over alphabet {a, b}:
   • L1 contains strings with 3n+2 number of a’s for n ≥ 0.
   • L2 contains strings with 2n+1 number of b’s for n ≥ 0.
2. Create a DFA that accepts the intersection of the two languages from problem 1.
3. Design a PDA that accepts the language {a²ⁱ b^k c³ⁱ | i, k > 0} over alphabet {a, b, c}.

For the remaining problems:

• Identify whether the given languages are regular (R), context-free but not regular (C), or not context-free (N), by circling the appropriate letter for each.
• Use the pumping lemma to prove that the language {aⁱ b^k | 2i > k} over {a, b} is not regular, providing a clear proof outline with specified variables.
• Construct a set of 5 strings D that are pairwise distinguishable with respect to the language {a⁵ⁿ | n ≥ 0}, and for each pair, identify a string (distinguisher) that distinguishes them.
• Assuming an infinite set of pairwise distinguishable strings exists for the language A = {aⁿ bⁿ | n ≥ 0}, provide such an infinite set along with distinguishers, and show that A is not regular without using the pumping lemma.
• Work with the management accounting case of Genre company, which produces Basic and Advanced products, to perform cost calculations:
  • Calculate the traditional overhead rate based on direct labor hours.
  • Determine the unit costs for each product under traditional costing.
  • Set the selling prices based on the traditional costing approach.
  • Compute activity-based costs for each activity cost pool.
  • Calculate the product costs individually via Activity-Based Costing (ABC).
  • Evaluate whether products are overcosted or undercosted based on ABC results.
  • Discuss potential benefits of implementing ABC and its usefulness for the company’s costing systems.

Paper For Above instruction

In the diverse realm of automata theory and formal languages, understanding the computational models that recognize specific pattern-related sets of strings is critical. This paper addresses constructing deterministic finite automata (DFA) and push-down automata (PDA) for given languages, analyzing their properties, and employing formal methods such as the pumping lemma to determine language classification. Additionally, it integrates a practical case study in management accounting, illustrating the application of cost allocation techniques such as traditional costing and Activity-Based Costing (ABC) to a manufacturing scenario. The following sections elaborate on these themes with detailed analysis and reasoning.

Automata for Languages L1 and L2

Languages L1 and L2 are defined over the alphabet {a, b}, with constraints on the number of specific symbols. For L1, strings contain 3n + 2 a's, where n ≥ 0, indicating that the total count of a's is congruent to 2 modulo 3. Similarly, L2 contains strings with an odd number of b's (2n + 1). The construction of the DFA for each involves states tracking the count modulo their respective formulas.

For L1, the DFA has three states representing the remainder of the count of a's modulo 3, plus states to handle the initial count with 2 a's. Transition functions adjust the state based on reading an 'a' or 'b', where 'b' transitions do not alter the count for L1. For L2, the DFA tracks the parity of b's, creating two states—one for even b counts, another for odd. Accepting states are those where the total count adheres to the language specifications.

The intersection DFA merges the states of both automata, recognizing strings satisfying both conditions concurrently. The combined automaton maintains a state for each pair of states from the individual ones, with transitions reflecting the combined input, and accepting states where both criteria are met simultaneously.

Designing the PDA for the language {a²ⁱ b^k c³ⁱ} involves tracking the counts of 'a' and 'c' using the stack, pushing and popping symbols to correspond to the counts of a and c, while reading b's unrestrictedly. The PDA accepts when the counts satisfy the required relations, specifically, matching pairs of 2i a's with 3i c's, ensuring the string structure is maintained.

Language Classification and Pumping Lemma Application

The classification of languages involves assessing their regularity and context-freeness. For example, the language {aⁱ b^j c^k | i > j > k ≥ 0} is not regular due to its counting constraints and relative ordering, as demonstrated by the pumping lemma. Similarly, languages that involve mirrored strings—like {x x^R}—are context-free but not regular, with formal proof via pumping lemma or closure properties.

The language L = {aⁱ b^k | 2i > k} can be shown non-regular using the pumping lemma by selecting a string s = a^2p b^p+1 where p is the pumping length. Dividing s into xyz, with the constraints of the lemma, leads to a contradiction, indicating non-regularity.

Distinguishability in Formal Languages

To analyze the set D of five strings in {a⁵ⁿ} that are pairwise distinguishable, one can select strings with incremental exponents of 5, such as a⁰, a⁵, a¹⁰, a¹⁵, and a²⁰. The distinguisher d for each pair could be constructed as a string that reveals the difference in exponents when concatenated with the strings in the set.

For example, the string 'a' could serve as a distinguisher between different strings, as it can distinguish whether the total number of a's is divisible by 5, thereby confirming their pairwise distinguishability.

In the case of the language A = {aⁿ bⁿ | n ≥ 0}, an infinite set of strings with increasing n, such as aⁿ bⁿ for n = 0,1,2,..., can be distinguished using the string 'a^n'bⁿ' or other markers. Demonstrating that such sets are infinitely distinguishable supports the fact that A is not regular, since regular languages cannot contain infinitely many pairwise distinguishable strings.

Cost Analysis in Management Accounting

The case of Genre Company provides practical insights into costing procedures. Traditional costing computes overhead rates by dividing the total overhead budget by the total direct labor hours, resulting in a rate used to assign costs to products. These calculations reveal that the Basic product, produced in smaller batches, may have its costs over- or under-allocated.

Activity-Based Costing (ABC) refines this by allocating overhead based on activities, such as setup, engineering changes, machine hours, and floor space. For example, ABC assigns setup costs to production runs, engineering costs to engineering changes, and machine costs based on actual machine hours used. Calculating the activity rates involves dividing each activity's budgeted cost by its total driver level.

Using the activity rates, one computes the total activity cost for each product by multiplying the rates with the actual driver usage per product. With the derived activity costs, the unit cost per product can be recalculated, enabling the firm to identify whether each product was overcosted or undercosted under traditional approaches. Typically, ABC reveals that complex or high-usage products like the Advanced product might be undercosted, leading to pricing and strategic inaccuracies.

Implementing ABC offers advantages such as more accurate product costing, better cost management, and more informed decision-making. For the Genre Company, ABC can highlight profitability issues and help develop competitive pricing strategies, especially crucial amidst international competition offering lower prices.

Conclusion

In conclusion, the construction of automata tailored to specific languages provides insight into the boundaries between regular and non-regular languages, employing formal tools like the pumping lemma for proofs. Simultaneously, the practical application of costing systems like ABC demonstrates how nuanced cost information can lead to better managerial decision-making. Both domains exemplify the importance of formal analysis and precise calculation in understanding complex systems, whether in theoretical computer science or business management.

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