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

• Main Authors: Grädel, Erich, Helal, Hayyan, Naaf, Matthias, Wilke, Richard
• Format: Journal Article
• Language: English
• Published: 07-03-2022
• Subjects: Computer Science - Logic in Computer Science
• DOI: 10.48550/arxiv.2203.03425

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.