Question

Construct npda's that accept the following languages on Σ = {a, b, c}.

(a) L = {a"b²n: n>0}.
(b) L = {wcwR: wЄ {a, b}*}.
(c) L = {anbmcn+m : n ≥ 0, m ≥ 0}.
(d) L = {a¹bn+m¿m : n ≥ 0, m ≥ 1}.
(e) L = {a³bc": n>0}.
(f) L = {a"b":n≤m≤3n}.
(g) L= {w: n (w) = nь (w) + 1}.
(h) L= {w: na (w) = 2ng (w)}.
(i) L = {w: na (w) + nz (w) = n。 (w)}.
(j) L= {w: 2na (w) ≤ ng (w) ≤ 3nc (w)}.
(k) L= {w: na (w)

Answer 1

To construct an NPDA that accepts language (a) L = {a"b²n: n>0}, we can use a stack to keep track of the number of b's. The NPDA can start by pushing a marker symbol onto the stack and transitions based on the input. for hence, option (b) is the correct answer.

Step-by-step explanation:

To construct an NPDA that accepts language (a) L = {a"b²n: n>0}, we can use a stack to keep track of the number of b's. The NPDA can start by pushing a marker symbol onto the stack. Then, for each 'a' input, it pops nothing from the stack and transitions to the next state. For each 'b' input, it pops a single 'b' from the stack. If the stack is empty after processing all inputs, the NPDA accepts the input string.

In particle physics, reactions must adhere to conservation laws, such as the conservation of strangeness, which is violated in reaction (b) p+n→p+p+K¯. For antiparticle decays, a muon antiparticle (μ+) decays into a positron and an electron neutrino (μ+ → e+ + ve), which conserves lepton number.

The student's question involves particle physics reactions and the conservation laws that govern these reactions. Specifically, the conservation of strangeness is crucial in understanding which particle reactions can occur. The question asks which of the following reactions cannot occur because they violate the law of conservation of strangeness. The law of conservation of strangeness states that the strangeness of a system must remain constant in any process that occurs via the strong interaction. However, changes in strangeness can occur through weak interactions which proceed more slowly than strong interactions.

Given the reactions provided, we can tell that the reaction (b) p+n→p+p+K¯ would violate the law of conservation of strangeness because antikaons, K¯, have strangeness +1 while protons and neutrons have no strangeness. This means the final state has a different total strangeness than the initial state, which violates the law when considering that the process is a strong interaction.

Regarding decay processes for antiparticles, for a muon antiparticle (μ+), one possible decay is μ+ → e+ + ve. This decay process conserves lepton number, as a muon lepton changes into an electron lepton, and the corresponding neutrinos account for the rest of the lepton balance.

Answer 2

(a) For the language L = {aⁿb²ⁿ : n > 0}, the non-deterministic pushdown automaton (NPDA) requires states to track the number of 'a's and 'b's, ensuring that the number of 'a's matches twice the number of 'b's.

(b) The language L = {w cwᵀ : w ∈ {a, b}*} can be accepted by an NPDA by using states to simulate the process of matching the mirrored portion of the string w.

Step-by-step explanation:

(a) NPDA for L = {aⁿb²ⁿ : n > 0}

To construct an NPDA for this language, we need states to keep track of the count of 'a's and 'b's. The automaton starts in the initial state, reads 'a's and pushes them onto the stack. For each 'b' encountered, it pops two 'a's from the stack. If the input is exhausted and the stack is empty, the NPDA accepts. This ensures that for every 'b', there are two preceding 'a's, satisfying the condition of the language.

(b) NPDA for L = {w cwᵀ : w ∈ {a, b}*}

This language requires checking if the string is a palindrome. The NPDA uses states to simulate the process of reading the input from left to right and then reading it again from right to left. It employs the stack to match corresponding symbols in the two halves of the string. The automaton accepts if the entire string is processed and the stack is empty, indicating a palindrome.

In summary, the NPDA construction for each language involves defining states to keep track of specific conditions based on the language's requirements. The transitions and stack operations are designed to ensure the automaton accepts strings that satisfy the given language criteria.