两状态极大加比较是可判定的
Two-State Max-Plus Comparison Is Decidable
AI总结：
该研究解决了状态数2到552中两状态的极大加自动机比较问题，证明任意有限极大加自动机与至多两状态的极大加自动机的比较可判定，进而得到两状态极大加的比较、等价性和正性均可判定的结论。
AI中文摘要：
Daviaud、Guillon和Merlet证明，在状态数固定为553时，极大加自动机的比较是不可判定的，且明确留下了状态数2到552的范围未解决。我们解决了两状态这个端点问题。更重要的是，给定任意有限极大加自动机A和状态数至多为2的极大加自动机B，可判定对每个字w是否满足[[A]](w)≤[[B]](w)。其结构原因在于两状态动力学存在一维射影范式。在一个有效有界区域之外，转移具有三种尾行为之一：传播无界射影间隙且间隙独立的高度增量、遗忘间隙且间隙独立的高度增量、将间隙大小读入高度增量后遗忘。特别地，任何输出依赖于无界间隙的转移必然会破坏该间隙。这得到了B的精确单计数器实现，而上下文无关Parikh像的有效半线性性将比较问题归约为Presburger算术，因此两状态极大加的比较、等价性和正性都是可判定的。
英文摘要：
Daviaud, Guillon, and Merlet proved that comparison of max-plus automata is undecidable under a fixed state bound of 553 and explicitly left the range from 2 to 552 states open. We resolve the two-state endpoint. More strongly, given an arbitrary finite max-plus automaton $A$ and a max-plus automaton $B$ with at most two states, it is decidable whether $[\![A]\!](w)\leq [\![B]\!](w)$ for every word $w$. The structural reason is a one-dimensional projective normal form for two-state dynamics. Outside an effective bounded region, a transition has one of three tail behaviors: it propagates the unbounded projective gap with gap-independent height increment, forgets the gap with gap-independent height increment, or reads the gap magnitude into the height increment and then forgets it. In particular, any transition whose output depends on the unbounded gap necessarily destroys that gap. This yields an exact one-counter realization of $B$. Effective semilinearity of context-free Parikh images then reduces comparison to Presburger arithmetic. As a consequence, two-state max-plus comparison, equivalence, and positivity are decidable.
