一阶逻辑的零-一定律与几乎处处赋值
Zero-One Laws and Almost Sure Valuations of First-Order Logic in Semiring Semantics
摘要：
半环语义通过交换半环K中的值来评价逻辑语句。随机半环的解释，诱导的概率分布K，概括随机结构，我们在这里调查的问题是如何经典的结果一阶逻辑随机结构，最重要的是0-1法律Glebskii等人。和Fagin，概括半环语义。对于正半环，经典的0-1定律意味着每个一阶句子，渐近地，或者几乎必然通过随机半环解释评估为0，或者几乎必然只取不同于0的值。然而，通过更复杂的分析，基于适当的扩展性质和一阶公式的代数表示，我们可以证明更强的结果。对于许多半环K，一阶句子可以被划分为类F（j），对于K中的所有半环值j，使得F（j）中的每个句子在随机半环解释下几乎必然求值为j。此外，对于有限或无限格半环，这种划分实际上只坍缩为三个句子类F（0）、F（1）和F（e），它们分别几乎肯定值为0、1和最小的非零值e。有限格半环上一阶句子的几乎处处赋值的计算问题是PSPACE-完全的。一个重要的半环的分析是有些不同的是自然数的半环。在这里，加法和乘法都相对于自然半环序而增加，并且对于自然数j，类F（j）不再覆盖所有FO句子，而是必须通过几乎肯定评估为无限大的句子类来扩展。值。
英文摘要：
Semiring semantics evaluates logical statements by values in some commutative semiring K. Random semiring interpretations, induced by a probability distribution on K, generalise random structures, and we investigate here the question of how classical results on first-order logic on random structures, most importantly the 0-1 laws of Glebskii et al. and Fagin, generalise to semiring semantics. For positive semirings, the classical 0-1 law implies that every first-order sentence is, asymptotically, either almost surely evaluated to 0 by random semiring interpretations, or almost surely takes only values different from 0. However, by means of a more sophisticated analysis, based on appropriate extension properties and on algebraic representations of first-order formulae, we can prove much stronger results. For many semirings K, the first-order sentences can be partitioned into classes F(j) for all semiring values j in K, such that every sentence in F(j) evaluates almost surely to j under random semiring interpretations. Further, for finite or infinite lattice semirings, this partition actually collapses to just three classes F(0), F(1), and F(e), of sentences that, respectively, almost surely evaluate to 0, 1, and to the smallest non-zero value e. The problem of computing the almost sure valuation of a first-order sentence on finite lattice semirings is PSPACE-complete. An important semiring where the analysis is somewhat different is the semiring of natural numbers. Here, both addition and multiplication are increasing with respect to the natural semiring order and the classes F(j), for natural numbers j, no longer cover all FO-sentences, but have to be extended by the class of sentences that almost surely evaluate to unboundedly large values.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
