Let a and b be events in distributed systems. a and b happen currently if and only if:

To understand the statement "Let a and b be events in distributed systems. a and b happen currently if and only if:", we need to clarify what is typically meant by events in this context and what conditions would allow both events to occur simultaneously in a distributed systems environment.

In distributed systems, events can be thought of as actions or occurrences that can impact the system's state. For two events $a$ and $b$ to occur concurrently, several conditions might need to be satisfied depending on the system architecture, synchronization mechanisms, or communication protocols in place.

Here’s a breakdown of what might be involved:

1. Causality: Two events may have a causal relationship. For $a$ and $b$ to happen concurrently, neither event should causally depend on the other. This means that the occurrence of one should not influence the beginning or conclusion of the other.

2. Synchronization: In distributed systems, time and synchronization are vital. Both events need to be correctly synchronized using logical clocks (Lamport timestamps) or physical timestamps to ensure that they can perceive each other as happening at the same moment.

3. Resource Sharing: If both events require shared resources, such as processing power or access to a database, they must either handle their access in a manner that prevents deadlock or they must operate in separate pathways that do not interfere with each other.

4. Event Ordering: In certain cases, the system might use message passing to order events. The order of messages could affect how events are perceived; therefore, if both events rely on the same set of messages, they might not logically occur concurrently.

By considering these components, the conclusion would suggest that events $a$ and $b$ occur concurrently if they satisfy the necessary conditions of non-causality, appropriate synchronization, resource accessibility, and event ordering in the context of their distributed environment.

This statement could be formally expressed in logical terms as: $a \land b \text{ occurs if and only if } \neg (a \to b) \land \neg (b \to a) \land \text{synchronized time} \land \text{shared resources accessible}$

In conclusion, understanding how events can occur concurrently in distributed systems involves analyzing synchronization, causality, and resource management aspects of the events.