Zero-One Laws and Almost Sure Valuations of First-Order Logic in Semiring Semantics

TL;DR

This paper extends classical 0-1 laws of first-order logic to semiring semantics, showing that under random interpretations, sentences almost surely evaluate to specific semiring values, with detailed classifications for various semirings.

Contribution

It generalizes 0-1 laws to semiring semantics, providing a detailed classification of first-order sentences based on their almost sure valuations in different semirings.

Findings

Classical 0-1 law extends to positive semirings, with sentences evaluating almost surely to 0 or non-zero values.

For lattice semirings, sentences fall into three classes: almost surely 0, 1, or minimal non-zero value.

In natural number semirings, some sentences evaluate almost surely to unboundedly large values.

Abstract

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.