Disambiguating "now" and "immediate"

Contents
  1. Now
    1. Absolute time
    2. The ability to stop time
  2. Immediate
  3. Conclusion

Now

The notion of "now" seems to be a good enough illusion for everyday perception of time1. Let us consider with some rigor what "now" means when used with respect to something physical, that is, the universe or its part, or something logical, e.g., a data object. Having the current state of a system X assumes

  • existence of absolute time, a.k.a. global time, which allows us to talk about snapshots of X;
  • the ability to stop time and explore a snapshot of X while X is not changing, as otherwise, the snapshot would not be the current one.

Absolute time

A physical system X can be thought as existing in a single timeline that represents absolute time iff the order of any two events a and b (the events are either simultaneous or are ordered one before the other) is the same for all observers, a.k.a. frames of reference.

A logical system X can be though as existing in a single timeline that represents absolute time iff there is a total order over all events in the system.

If there is absolute time, we can define what a snapshot is.

Snapshot—the state S(t) of X at a point t on its absolute timeline.

Note that the term "snapshot" can be defined without introducing the notion of absolute time, see "Distributed Snapshots: Determining Global States of Distributed Systems" by K. Mani Chandy and Leslie Lamport2. This is, however, not very relevant to our discussion about "now" because discussing this term requires discussing time.

According to the theory of special relativity, there is no absolute time in which the universe exists. For any event, i.e., a point in spacetime defined by spacial coordinates and a time coordinate, there are infinitely many events distanced from it with a space-like interval. Such events are seen as simultaneous by some observers and are not seen as simultaneous by other observers, which means the universe does not exist in absolute time.

What if we are talking about a logical system, may it have absolute time?—it definitely may, for example:

  • any sequential object, that is, a data object that is accessed only sequentially, may be thought of as having absolute logical time;
  • depending on the consistency model of a concurrent object, that is, a data object that may be accessed concurrently, we may be able to imagine it existing in absolute logical time. This may be done for an object as simple as a linearizable, a.k.a. atomic3, register, or as complex as a DB managed by a DBMS that guarantees that all transactions are serializable;
  • the work "Time, Clocks, and the Ordering of Events in a Distributed System" by Leslie Lamport describes the synchronization of logical clocks, allowing to totally order all events in a distributed system.

Do not get too excited, though—there are many logical concurrent systems, especially distributed ones, such that different parts of a system exist in different timelines, i.e., the events are ordered not totally, but partially. LongAdder from Java Platform, Standard Edition (Java SE) API is an example of a simple concurrent object with no absolute timeline and no snapshots, and the specification is explicit about this: "The returned value is NOT an atomic snapshot; invocation in the absence of concurrent updates returns an accurate result, but concurrent updates that occur while the sum is being calculated might not be incorporated."

The ability to stop time

At least on the macroscopic scale, we perceive time as flowing unstoppably and irreversibly. What is it exactly that allows us to perceive and measure the time? The answer to this question is—changes. We perceive the flow of time because we observe changes, and if all changes stopped, it would have been equivalent to time stopping. An observer within a part of the universe where there are no changes (this is a hypothetical situation which is not physically possible) or within a logical system where no changes are happening cannot perceive that the time is stopped, but an external observer can.

A snapshot S(tc) of X is the current snapshot iff

  1. all instants t on the absolute timeline of X are such that ttc;
  2. the above stays true while S(tc) is being explored, i.e., time is stopped for X but is not stopped for its external observer.

tc is called the current time or now.

It is impossible to stop the universe or its part from continuously changing, which means that even if there were absolute time in which the universe exists, there would be strictly speaking no "now". Nevertheless, it still may be useful in some situations to imagine a part of the universe as existing in absolute time and maybe even not changing, and imagine us as being external observers of that part. Within such a model "now" makes sense.

Unlike with the universe, preventing a logical system from changing its state and observing it externally is very much possible.

Immediate

Consider the following sentence: "I initiated the process of releasing a hotfix immediately after I finished fixing the problem." This sentence conveys that after fixing the problem I initiated a hotfix release instead of doing anything else. You probably also assumed that the time I spent between fixing the problem and initiating the release process is minuscule. Such temporal semantics may be fine when we use the words "immediately"/"immediate" in everyday life, but should not be applied when reasoning about concurrent systems.

When discussing concurrency, the word "immediately" should be used only to express ordering relations. The phrase "y immediately follows x" has the same meaning as the phrase "y is ordered immediately after x", which means that there is no element z different from both x and y such that x if ordered before z and z is ordered before y.

With this in mind, let us consider the following statement about linearizability: "Linearizability provides the illusion that each operation takes effect instantaneously at some point between its invocation and its response, implying that the meaning of a concurrent object's operations can still be given by pre- and post conditions."4 The authors clearly do not mean to imply that operations take effect instantaneously in a temporal sense—that is why they call it an illusion. But why exactly does such an illusion occur when a linearizable object is used? It is because the requirements imposed by the linearizable consistency model "allow us to describe acceptable concurrent behavior directly in terms of acceptable sequential behavior, an approach that simplifies both formal and informal reasoning about concurrent programs."4 If all operations on an object can be thought of as being sequential, then the duration of each operation in such a sequential history stops affecting the behaviour of the object, and therefore we may imagine that operations take effect instantaneously. I cannot speak for others, but to me, informal remarks in articles about concurrency / distributed systems that appeal to words with temporal semantics, introduce confusion instead of helping to clarify the concept being explained.

Conclusion

When using the word "now" with regard to a logical system, consider if it is actually applicable to the system, and even if it is applicable, do you actually need it? Use the word "immediately" only to express ordering relations because its temporal meaning is vague. The idea to write this post came to me as a result of seeing the question "Can you confirm if this is correct—a volatile write or atomic dec/inc becomes visible immediately to other threads …?", which was about the semantics of volatile fields in Java, and apparently implied the temporal meaning of the word "immediately". I like the two following answers to this question:

  • first

    "No there is no requirement for writes to become visible "immediately". Only the happens-before ordering guarantees visibility—if you see a given write then you must also see writes that happen-before that first write. In practice writes don't take long to become visible on most architectures—there is typically a delay until a write buffer drains."

  • second

    "When someone talks about "visible immediately", we need to ask them what they mean by "immediately". In particular, how would a external observer know that something became visible "not immediately".

    You will quickly see that the external observer can only make judgements about the order of some events with respect to other events. If observing arrival of event A implies arrival of event B can be observed, i.e., A implies B, then you have B happens before A. This is pretty much all the external observer can figure out. But this is enough to reason about the state of the system."

See also "There is No Now" by Justin Sheehy.

  1. The concept of physical time is not trivial, and I am not qualified in this area, but I can recommend a few videos of qualified people talking about / discussing this topic:

  2. By the way, despite the paper does not mention the term "consistent cut", it is widely used by the community when talking about the snapshot algorithm described in the paper. 

  3. The word "atomic" is overloaded: it is used both as a synonym of being linearizable (by the way, a volatile field in Java is a linearizable register) and as the atomicity property in ACID, which is about either committing all actions bundled together in a transaction, or ensuring that none of them happened. 

  4. Linearizability: A Correctness Condition for Concurrent Objects by Maurice P. Herlihy, Jeannette M. Wing  2