Compute the Distribution of LLM Decoding Stopping Time and Build a Strategy Against Adversaries

Problem: Model the Distribution of LLM Decoding Stopping Time and Implement a Minimax Strategy Against Adversaries In a simplified LLM decoding process, at each step the model may emit a special token <STOP> and terminate. You must do two things. Part A: Stopping-Time Distribution Given a maximum length N (generation runs for at most N tokens; if it does not stop early, it is forced to stop at N) and a per-step stop probability sequence: At step t (1-indexed), the probability of emitting <STOP> is p[t] (0 <= p[t] <= 1) If it does not stop at step t, it proceeds to step t+1 Compute: For all t = 1..N, the stopping-time probability P(T=t) The expected stopping time E[T] Note: If <STOP> is never emitted by step N, define T=N (forced stop). Part B: Strategy Against Adversaries There are multiple adversaries. Each adversary corresponds to a different mechanism that influences p[t] and/or the payoff. After observing p[1..N] (or its parametric form), your program must choose a decision (e.g., a cutoff step k, an early-stop policy, or an action a) to maximize the worst-case (minimax) expected reward. Implement a program that takes: N p[1..N] parameters for M adversaries (each defines a possible p[t] and/or reward function) a reward function reward(decision, T, adversary) Output: the chosen decision the expected reward under the worst adversary I/O Spec You must clearly specify: stdin input format stdout output format constraints (N, M) and complexity requirements Provide at least 5 test cases including edge cases. Example Input 3 0.5 0.0 0.0 Output P(T=1)=0.5 P(T=2)=0.0 P(T=3)=0.5 E[T]=2.0