DeepAI

# Elementary equivalence versus isomorphism in semiring semantics

We study the first-order axiomatisability of finite semiring interpretations or, equivalently, the question whether elementary equivalence and isomorphism coincide for valuations of atomic facts over a finite universe into a commutative semiring. Contrary to the classical case of Boolean semantics, where every finite structure can obviously be axiomatised up to isomorphism by a first-order sentence, the situation in semiring semantics is rather different, and strongly depends on the underlying semiring. We prove that for a number of important semirings, including min-max semirings, and the semirings of positive Boolean expressions, there exist finite semiring interpretations that are elementarily equivalent but not isomorphic. The same is true for the polynomial semirings that are universal for the classes of absorptive, idempotent, and fully idempotent semirings, respectively. On the other side, we prove that for other, practically relevant, semirings such as the Viterby semiring, the tropical semiring, the natural semiring and the universal polynomial semiring N[X], all finite semiring interpretations are first-order axiomatisable (and thus elementary equivalence implies isomorphism), although some of the axiomatisations that we exhibit use an infinite set of axioms.

• 13 publications
• 1 publication
03/07/2022

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

Semiring semantics evaluates logical statements by values in some commut...
03/22/2023

### Locality Theorems in Semiring Semantics

Semiring semantics of first-order logic generalises classical Boolean se...
06/15/2021

### On Doctrines and Cartesian Bicategories

We study the relationship between cartesian bicategories and a specialis...
09/08/2022

### Quasi-Random Influences of Boolean Functions

We examine a hierarchy of equivalence classes of quasi-random properties...
07/12/2017

### Revisiting Elementary Denotational Semantics

Operational semantics have been enormously successful, in large part due...
10/21/2019

### Parametrized Complexity of Expansion Height

Deciding whether two simplicial complexes are homotopy equivalent is a f...
01/26/2023

### Quantitative Safety and Liveness

Safety and liveness are elementary concepts of computation, and the foun...