we have the mixture of "true concurrency" and interleaving semantics. The only disadvantage is that gso-structures are more complex to conceptualize than so-structures.

Since the works of Janicki et al. [7,5] focus on the algebraic properties of gsostructures, the number of axioms are kept to minimal and some of the assumptions are made implicit. Furthermore, the theorems of gso-structure theory frequently involve quantifying over relations, which requires the use of higher-order language. Hence, to apply first-order ontology and model-theoretic techniques in the manner as in [4], we will first define a formal ontology for gso-structure in first-order logic and characterize all possible models of gso-structure theory up to isomorphism. After that we can proceed to investigate to what extend the theorems of gso-structure theory hold within the first-order setting of PSL by studying possible ontological mappings from gso-structure theory to PSL.

The organization of this paper is as follows. In Section 2, we will give a first-order axiomatization of the gso-structure theory and end the section will a result showing that our theory is consistent. In Section 3, we will classify all possible models of the gsostructure theory from Section 2 using more natural and intuitive concepts from graph theory. In Section 4, we study a semantic mapping from our theory to PSL-core theory.

Section 5 contains our concluding remarks.

## 2 First-Order Axiomatization Of Gso-Structure Theory

The following table provides a summary of the lexicon of so-structure theory. The relations ≺, ⊏ and <> in the papers of Janicki et al. [5,7] correspond to the relations earlier than, not later than and nonsimultaneousrespectively in this paper. We rename these relations to make the theory more intuitive and accessible.

| Lexicon                  | Informal Semantics                  |                               |
|--------------------------|-------------------------------------|-------------------------------|
| Universe                 | event(e)                            | e is an event                 |
| event occurrence(o)      | o is an event occurrence            |                               |
| observation(x)           | x is an observation                 |                               |
| occurrence(o,e)          | o is an event occurrence of event e |                               |
| Gso-structure            | earlier than(o1,o2)                 | o1 must occur earlier than o2 |
| not later than(o1,o2)    | o1 must occur not later than o2     |                               |
| nonsimultaneous(o1,o2)   | o1 and o2 must not occur simultaneously                                     |                               |
| Observations             | observed before(o1,o2,x)            | event occurrence o1 is observed before event occurrence o2 in observation x                               |
| observed simult(o1,o2,x) | event occurrences o1 and o2 are observed simultaneously in observation x                                     |                               |

2