The encoding and exploitation of semantics has been gaining popularity, as exemplified by the uptake of digital ontologies and knowledge graphs. However, the semantics of domain objects usually do not reflect how they evolved over time, i.e., which events their dynamic transitions are based on. While a number of methods have been proposed to trace events and their impacts on a domain, there is a paucity of approaches to efectively join them. Thus, we combine event calculus as an analytical approach for modeling causal relationships between events and efects with semantic drifts as an empirical approach for quantifying the impact of domain updates. We demonstrate how their respective weaknesses can be addressed and how their interaction can improve the representation of semantic transitions.
Semantics, often referred to as the study of meaning and truth, represent the foundation of
human cognitive abilities and thus most research fields [
Within a domain, objects are assigned meaning based on their interactions with each other.
Thus, knowledge graphs (KGs) structure knowledge based on ontological conceptualizations so
that the semantics of an object can be inferred from its graph neighborhood [
To incorporate semantic events within logical inferences, some works already exist, which can be divided into analytical and empirical approaches. Analytical approaches, such as event calculus [6], extend first-order logic (FOL) rules for temporal reasoning [ 7, 8]. Domain updates between two timestamps and are initiated by events with ∈ [ , ], and relationships between domain objects are regarded as fluent states that can be either true or false. Accordingly, efects are defined that take into account the semantics at time and the events that have taken place inbetween and to conclude the fluent states within the updated domain image. As prior domain knowledge about events is directly incorporated, temporal reasoning is a proactive approach that can also be applied to temporal knowledge graph extensions [9, 10].
Contrarily, empirical approaches assume two self-contained semantic representations of the domain knowledge. Besides knowledge graphs, external representation types, such as word or text annotations, are explicitly allowed as well. Based on the representations for both timestamps and , an attempt is made to identify so-called semantic drifts [ 11], i.e., to identify domain objects whose semantics have drifted from to , and to quantify these drifts [ 12, 13]. In conclusion, semantic drifts represent a reactive approach to the subsequent identification of semantic transitions and thus also events and their efects as enablers of these transitions.
In the following, both approaches are discussed in more detail and summarized in a compact manner. Thereby, we focus on their advantages and in particular their disadvantages with respect to their real-world applications. Based on these findings, we elaborate to what extent the combination of event calculus and semantic drifts can counteract their respective drawbacks and thus faciliate the incorporation of semantic events and transitions in dynamic domains.
As an extension of FOL, event calculus is based on propositions that can be either true or false. A proposition ( 1, .., ) is composed of some predicate that asserts a logical relationship among non-logical objects 1, .., ∈ Ω, such as entities or concepts of a domain. Here, Ω represents the set of all available domain objects and ∈ ℕ denotes the arity of a property, i.e., the number of objects within its logical expression. Accordingly, KGs G = (V, E ) with vertices V = Ω and edges E can be interpreted as sets of binary propositions, i.e., ( 1, 2) = implies the directed edge ( 1, , 2) ∈ E . For example, ( 1, 2) is binary (i.e., = 2 ) and true if a person 1 is the president of a country 2. Moreover, domain rules and composite propositions can be constructed via the connectives ∧, ∨, =, ¬, ⇒, ⇐ and the quantifiers ∀, ∃.
Accordingly, event calculus is based on the consideration of propositions as fluents , namely conditions that can change over time. These fluents are reified, i.e., they are formalized as non-logical objects so that they can serve as inputs for functions with range T . For example,
( , ) = [ 17, 21] ⊆ T describes the presidency term of = Donald Trump in the = United States inbetween the timestamps 17 ∶= January 20th, 2017 and 21 ∶= January 20th, 2021. Even though diferent interpretations of event calculus can be found in the literature [14], most of them introduce the additional functional predicates ℎ , ℎ , , and . The predicate ℎ is used to determine whether a proposition holds at a timestamp ∈ T , e.g., ℎ ( ( , ), ) = holds for all ∈ [ 17, 21]. While ℎ indicates whether a semantic event takes place at time ∈ T , the predicates and define its efects , i.e., how fluent states are afected by this event [15]. Events are encoded as non-logical objects and thus build the foundation for controlling dynamics in event calculus. In our example, the change in ofice (cio) in the US at time 21 represents the event that efectively terminated Donald Trump’s presidency and initiated the tenure of = Joe Biden, i.e., the fluent states are updated due to ℎ ( , 21) ∧ ( , ( , )) ∧ ( , ( , )) .
These events can be implicit as above, but also explicit, i.e., an event can explicitly characterize changes in fluent states, e.g., by considering the inauguration of Joe Biden on January 20th, 2021 as a single event. Such explicit events are commonly referred to as actions as they actively afect the structure of the domain knowledge. Accordingly, event calculus analyzes which actions need to be performed as efects of semantic events. Each efect is thus to be interpreted as an action. An overview of the dynamic transitions in event calculus can be found in Figure 1.
Since event calculus requires prior knowledge about a domain, it represents an analytical approach for modeling semantic events and their efects on fluent states. During the definition phase of the respective calculus, future events and efects must already be considered, which constitutes its major drawback. For example, the impacts of events may vary over time or they might even be unknown at the time of definition. Thus, for a dynamic domain, we want to trace the consistency and completeness of a given calculus. Empirical approaches are required for measuring the impact of events so that actions can be verified and missing efects can be indicated. For this purpose, we adopt approaches for determining so-called semantic drifts.
For two (not necessarily consecutive) timestamps , ∈ T , semantic drifts are introduced to measure the impact of semantic updates inbetween both timestamps [16, 17, 18]. Analogous to Section 3, entities and concepts of a domain are considered as non-logical objects ∈ Ω with semantic representations () ∈ Π for some ∈ T . The representation set Π may include sets of propositions (cf. Section 3), but also external representations, such as textual annotations.
Accordingly, a semantic drift measure ≻ ∶ Ω ∩ Ω → ℝ≥0 is derived from some distance measure ∶ Π ≻ × Π ≻ → ℝ≥0 by means of ≻ () ∶= ( represents a shared representation space. Thus, for objects , ′ ∈ Ω (), ()) . Here, Π ≻
∩ Ω , the inequality ≻ ( ′) >
≻ () indicates a larger semantic drift in ′. However, it must be assumed that ≻ is a space with some well-defined distance measure , which is generally not the case.
To solve this problem, embedding mappings ∶ Π → Π ⋆ can be used to embed the given semantic representations within a numerical representation space Π⋆ equipped with well-defined distance measures. For example, KG embedding methods like TransE [19] or RDF2Vec [20] assign numerical representations to the nodes of a KG [21]. Similarly, Natural Language Processing (NLP) introduces language models like Word2Vec [22], BERT [23], or T5 [24] to convert text within a joint embedding space Π ⋆≻ , embedding alignments can be performed via into numerical embeddings [25]. Typically, Π⋆ is chosen to be a real-valued embedding space, e.g., Π⋆ = ℝ with ∈ ℕ. To merge the numerical representations of both timestamps , ∈ T
∶ ℝ → ℝ ≻ and ∶ ℝ → ℝ ≻ with Π ≻ = ℝ ≻ and
≻ ∈ ℕ, to approximate diferent representations of identical objects.
Considering the alternative representations ⋆() ∶= ( ( ())) , these are adjusted through () ≈ () ⟺ ( ⋆() ) ≈ 0 ⋆(), for some predefined distance measure ∶ ℝ ≻ × ℝ ≻ → ℝ≥0 like the cosine or the euclidean distance, so that outliers are defined as semantic drifts. For the sake of completeness, it should be mentioned that only one or even no alignment may be performed. For example, embeddings can be aligned in an existing embedding space ℝ or ℝ via = or = . For some dynamic to quantify the semantic drift of an object ∈ Ω ∩ Ω via ≻ () ∶= ( embedding methods, such as [26, 27], it is even possible to a priori assume ( ()) ≈ for () ≈ () and thus also = = . In conclusion, distance measures can be applied (
()) ∗(), ∗()) .
In contrast to embedding-based methods for determining semantic drifts, other approaches exist which omit the prior embedding of the domain objects and instead consider representationbased distance measures ∶ Π ≻ × Π ≻ → ℝ≥0, where Π = Π = Π ≻ is always assumed.
For example, Π ≻ could represent all valid graph neighborhoods of some node representation within a KG, or it could represent the set of all english text fragments. Such formalisms can be found in [28, 29], among others, which consider compositions of semantic representations and corresponding distance measures. These distance measures are typically derived from similarity measures ∶ Π ≻ × Π
≻ → [
. Accordingly, semantic drifts can be defined via
≻ () = (
(), () ) = 1 − (
(), () ) ∈ [
Since arbitrary ℝ-valued distances measures can be derived almost analogously, we restrict
ourselves to such [
For example, textual object annotations () and () for some ∈ Ω ∩ Ω can be compared via text comparison methods like the Monge-Elkan similarity [30]. Analogously, KG-based representations can be considered, e.g., the number of adjacent nodes or the sets of common edges can be determined for both timestamps and to define the semantic drift as their diference or by applying set similarity measures like the Jaccard index [ 31], respectively.
Representation-based distance measures are also applied in [32], where additional KG-based aspects like URIs, superclasses and subclasses, and equivalent classes are incorporated. Overall, such approaches are always based on heuristics that can be directly applied to some graph structure or external representation (e.g., text) to measure the semantic drifts of domain objects.
Compared to the analytical approach of analyzing semantic transitions in event calculus, semantic drifts represent an empirical method for quantifying the impact of semantic transitions on domain objects. Order statistics of the drift scores { ≻ () ∶ ∈ Ω ∩ Ω } can be defined However, the selection of some well-defined threshold ∈ ℝ ≥0 with and compared, e.g., the < |Ω ∩ Ω
| domain objects can be determined that drifted the most. drifted significantly inbetween , ∈ T ∶ ⟺ ≻ () ≥ (1) is not trivial at all and solutions need to be elaborated for this problem.
In this chapter, we discuss how event calculus and semantic drifts can be combined for the modeling and analysis of semantic transitions in dynamic domains. According to Section 3, transitions between two timestamps , ∈ T are always based on events with ∈ [ , ] that implicitly or explicitly update the domain knowledge. In the following, we first reveal to what extent prior knowledge from an existing event calculus can improve the quality of semantic drifts. Subsequently, we show to what extent semantic drift measures can be applied to counteract the drawbacks of formalisms based on event calculus.
As mentioned in Section 4.3, one major drawback of semantic drifts is that conclusions regarding the significance of a semantic drift
≻ () can only be drawn in relation to those of other objects in Ω
⧵ { }. While the determination of a globally valid threshold ∈ ℝ ≥0 from Equation 1 seems impossible, an event calculus can be applied to derive a transition-specific threshold
≻ that only examines the updates between , ∈ T . Considering the example from Section 3, the inauguration of a person could be defined as a significant event with respect to his or her semantics (if not otherwise impacted). Thus, by specifying = 17 and = 21, drifted significantly inbetween , ∈ T ∶ ⟺ ≻ () ≥ ≻ ∶= ≻ ( ) can be defined so that all domain objects that drifted at least as much as Joe Biden are determined as objects whose semantics changed significantly. For this, knowledge is required about the semantic events underlying the dynamic transitions, which is made possible by event calculus.
In Section 3.1, two major drawbacks of formalisms based on event calculus are identified.
I Events are descriptive, i.e., their semantic impacts can be encoded in their efects, but their significance can not be measured in a quantitative manner.
II Events and their efects are defined prior to deployment. Thus, the calculus could be incomplete and might need to be updated in the future.
As the outcome of an event is encoded within its efects, i.e., their actions on a collection of domain properties (e.g., edges in a KG), we can identify objects whose semantics were updated as the cause of an event. Thus, the introduction of semantic drifts provides a possible solution to the first drawback since the drifts of afected objects can be measured and aggregated so that the impacts of diferent events can be compared, as exemplified in the following. Example. We assume the example from Section 3 regarding the change in ofice in the United States on ∶= 21 = January 20th 2021 that is defined as the semantic event . Further, we assume that ∶= January 19th 2021 and are consecutive. To analyze the impact of this event in more detail, we split up into two subevents that represent Donald Trump’s resignation and Joe Biden’s inauguration, respectively. We achieve this by means of , =̂ Donald Trump’s resignation and , =̂ Joe Biden’s inauguration.
Finally, we define an additional timestamp with < < , so that , happens at and , happens at . Since is directly afected by both subevents, we want to quantify their impacts on it. Semantic drift scores ≻ ( ) and ≻ ( ) can be utilized to answer this question.
In this context, it is important to note that, in the case of embedding-based semantic drifts, (⋅) = , to ensure comparability of the drift scores. alignments should be performed within the shared embedding space Π ⋆, i.e., by considering
Similarly, semantic drifts can counteract the second drawback mentioned above. Since they are based on empirical observations of semantic representations within the domain, we can identify how much domain objects have drifted semantically, to determine whether the calculus is capable of describing the dynamic transitions and the corresponding domain updates. Example. Analogous to the previous example, we assume the event and the timestamps , . Further, we consider some semantic drift measure ≻ ∶ Ω quality of an event calculus, so that it can be subsequently adjusted if necessary. △ △
In this work, we reviewed and analyzed event calculus and semantic drifts as approaches for identifying and understanding semantic transitions in dynamic domains. While event calculus is a proactive approach to the modeling of causal relationships between events and their efects, semantic drifts assume two self-contained images of a dynamic domain to reactively identify domain objects whose semantics have changed as an efect of a transition.
In this context, drawbacks are pointed out that impair or prevent their real-world applications. To counteract these drawbacks, we propose to conduct both approaches in a complementary manner, to combine the benefits of analytical modeling and empirical observations. On the one hand, semantic drifts are enriched by prior knowledge about the events underlying the transitions, so that it is made possible to assess whether drifts need to be regarded as significant or not. On the other hand, quantitative statements about impacts of events and their efects on domain objects enable the qualitative verification of temporal reasoning formalisms like event calculus. The proposed approach to combine both methods is exemplified and serves as a basis for future works to facilitate the modeling of dynamic knowledge based on semantic events.
• This work is part of the TEAMING.AI project which receives funding in the European
Commission’s Horizon 2020 Research Programme under Grant Agreement Number 957402. • We acknowledge the financial support from the Federal Ministry for Economic Afairs and
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