有序分枝MDP的定性多目标可达性
Qualitative Multi-Objective Reachability for Ordered Branching MDPs
摘要：
我们研究定性的多目标可达性问题的有序分支马尔可夫决策过程（OBMDPs），或等价的上下文无关的MDPs，建立在以前的结果上的单目标可达性分支马尔可夫决策过程（BMDPs）。我们为OBMDP的“几乎确定”和“限制确定”多目标可达性提供了两种单独的算法。具体地，给定OBMDP，$\mathcal{A}$，给定起始非终端，并且给定大小为$k = |K|$的目标非终端$K$的集合，我们的第一算法决定生成包含集合$K$中的每个目标非终端的树的上确界概率是否为$1$。我们的第二个算法决定是否有一个策略，玩家几乎肯定（概率$1$）生成一棵树，其中包含集合$K$中的每个目标非终端。这两个单独的算法是必要的：我们表明，事实上，在这种情况下，“几乎肯定”$\not=$“有限肯定”的多目标可达性，这意味着有OBMDPs的球员可能没有任何策略，以实现概率恰好$1$达到所有目标的集合$K$在同一生成的树，但可能有一个序列的战略，实现概率任意接近$1$。两种算法都在时间上运行$2^{O(k)} \cdot |\mathcal{A}|^{O(1)}$，其中$|\mathcal{A}|$是给定OBMDP的总比特编码长度，$\mathcal{A}$。因此，当$k$固定时，它们在多项式时间内运行，并且相对于$k$是固定参数可处理的。此外，我们表明，即使是定性的几乎肯定（和有限肯定）的多目标可达性决策问题一般是NP-难的，当目标非终端的集合$K$的大小$k$是不固定的。
英文摘要：
We study qualitative multi-objective reachability problems for Ordered Branching Markov Decision Processes (OBMDPs), or equivalently context-free MDPs, building on prior results for single-target reachability on Branching Markov Decision Processes (BMDPs). We provide two separate algorithms for "almost-sure" and "limit-sure" multi-target reachability for OBMDPs. Specifically, given an OBMDP, $\mathcal{A}$, given a starting non-terminal, and given a set of target non-terminals $K$ of size $k = |K|$, our first algorithm decides whether the supremum probability, of generating a tree that contains every target non-terminal in set $K$, is $1$. Our second algorithm decides whether there is a strategy for the player to almost-surely (with probability $1$) generate a tree that contains every target non-terminal in set $K$. The two separate algorithms are needed: we show that indeed, in this context, "almost-sure" $\not=$ "limit-sure" for multi-target reachability, meaning that there are OBMDPs for which the player may not have any strategy to achieve probability exactly $1$ of reaching all targets in set $K$ in the same generated tree, but may have a sequence of strategies that achieve probability arbitrarily close to $1$. Both algorithms run in time $2^{O(k)} \cdot |\mathcal{A}|^{O(1)}$, where $|\mathcal{A}|$ is the total bit encoding length of the given OBMDP, $\mathcal{A}$. Hence they run in polynomial time when $k$ is fixed, and are fixed-parameter tractable with respect to $k$. Moreover, we show that even the qualitative almost-sure (and limit-sure) multi-target reachability decision problem is in general NP-hard, when the size $k$ of the set $K$ of target non-terminals is not fixed.
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