# Allen’s Interval Algebra

#### T. S. Dye

#### 2025-09-22

ArchaeoPhases provides facilities to compute Allen’s interval algebra. This vignette provides an overview of the algebra and illustrates how it might be used analytically and empirically, given MCMC output from a Bayesian calibration.

    ## Load ArchaeoPhases
    library(ArchaeoPhases)

This vignette uses data available through the ArchaeoData package which is available in a separate repository. ArchaeoData provides MCMC outputs from ChronoModel, OxCal and BCal.

    ## Install data package
    install.packages("ArchaeoData", repos = "https://archaeostat.r-universe.dev")

# 1 Overview of Allen’s Interval Algebra

Allen’s interval algebra identifies 13 basic relations that are distinct, exhaustive, and qualitative (Allen 1983). The algebra is defined for definite intervals whose endpoints are single values. It can also be used with indefinite intervals whose endpoints are not single values.

It is conventional to express an Allen relation with a notation that indicates two intervals and a set of relations. For example, given definite intervals A and B, where A precedes B, their Allen relation can be represented as A(p)B. Given indefinite intervals C and D, where C starts before D starts and ends before D ends, an Allen relation that expresses this incomplete information is C(pmo)D. Note that the Allen set (pmo) in the relation of C and D is a superset of the Allen set (p) in the relation of A and B. In this circumstance, when two Allen sets are related as superset/subset, the relation indicated by the superset is weaker and the relation indicated by the subset is stronger. An Allen relation thus provides a precise vocabulary for characterizing the state of knowledge about the relation between two intervals, regardless of whether the intervals are definite or indefinite.

Relation |  |  | Converse
--- | --- | --- | ---
precedes | (p) | (P) | preceded by
meets | (m) | (M) | met by
overlaps | (o) | (O) | overlapped by
finished by | (F) | (f) | finishes
contains | (D) | (d) | during
starts | (s) | (S) | started by
equals | (e) |  |

It is often useful to visualize the basic Allen relations as a Nökel lattice. In a Nökel lattice, the immediate neighbors of a relation differ from it in the placement of a single interval end-point.

    ## Plot the basic Allen relations
    par(mar = c(1, 1, 4, 1))
    allen_illustrate("basic")

plot of chunk basic_relations

# 2 Analytic Inquiry with the Algebra

Interval algebra is carried out with a composition function that returns the relation of two intervals given their relations to a common third interval. As implemented here, the function expects the relation of one interval to the common interval and the relation of the common interval to the remaining interval. The function also accepts a title for the resulting Nokel lattice.

In this example, the composition function deduces that the interval represented by a context inferior to a reference context precedes the interval represented by a context superior to the reference context.

    ## Plot lattice for two contexts on the same line of a Harris matrix
    par(mar = c(1, 1, 4, 1))
    allen_analyze(
      x = "m",
      y = "m",
      main = "Composite relation of two contexts on the same line"
    )

plot of chunk analyze_stratigraphy

The same lattice can be created with a convenience function.

    ## Illustrate composite relations in a stratigraphic sequence
    par(mar = c(1, 1, 4, 1))
    allen_illustrate("sequence")

plot of chunk illustrate_stratigraphy

# 3 Empirical Inquiry with the Algebra

Empirical inquiry with Allen’s interval algebra compares intervals estimated by Bayesian calibration of age determinations from Anglo-Saxon female burials.

    ## Load the Anglo Saxon burials dataset
    path <- system.file("oxcal", "burials.csv", package = "ArchaeoData")
    burials <- read.table(path, header = TRUE, sep = ",", dec = ".",
                          check.names = FALSE)

    ## Coerce to event
    burials <- as_events(burials, calendar = CE())

First, we check for the expected result of the identity relation by comparing an interval with itself.

    ## Identify the burials with bead BE1-Dghnt
    be1_dghnt <- c(
      "UB-4503 (Lec148)", "UB-4506 (Lec172/2)", "UB-6038 (CasD183)",
      "UB-4512 (EH091)", "UB-4501 (Lec014)", "UB-4507 (Lec187)",
      "UB-4502 (Lec138)", "UB-4042 (But1674)", "SUERC-39100 (ERL G266)"
    )

    ## Build phases
    chains <- list("BE1-Dghnt" = be1_dghnt)
    be1_dghnt_phase <- phases(burials, groups = chains)

    ## Plot
    par(mar = c(1, 1, 4, 1))
    allen_observe(be1_dghnt_phase)

plot of chunk identity_relation

Next we compare the depositional histories of two bead types recovered from well-dated Anglo Saxon female burials in England. Note that the Allen relation and its converse are calculated.

    ## Identify the burials with bead BE1-CylRound
    be1_cylround <- c(
      "UB-4965 (ApD117)", "UB-4735 (Ber022)", "UB-4739 (Ber134/1)",
      "UB-6473 (BuD250)", "UB-6476 (BuD339)", "UB-4729 (MH068)",
      "UB-4835 (ApD134)", "UB-4708 (EH083)", "UB-4733 (MH095)",
      "UB-4888 (MelSG089)", "UB-4963 (SPTip208)", "UB-4890 (MelSG075)",
      "UB-4732 (MH094)", "SUERC-51539 (ERL G353)", "SUERC-51551 (ERL G193)"
    )

    ## Build phases
    chains <- list("BE1-Dghnt" = be1_dghnt, "BE1-CylRound" = be1_cylround)
    be1_phase <- phases(burials, groups = chains)

    ## Plot phase
    par(mar = c(4, 6, 1, 1))
    plot(be1_phase)

plot of chunk phases

    ## Plot relations
    par(mar = c(1, 1, 4, 1))
    allen_observe(be1_phase, converse = TRUE)

plot of chunk relation_and_converse

It can be useful to observe the frequency with which two intervals are related according to the relations in an Allen set. For instance, we might be interested in which of our two beads might be the trunk from which the other branched. This can be done by observing the frequency with which each bead type is related as row(oFD)col to the other bead. The following code shows that BE1-CylRound is the better trunk candidate.

    allen_observe_frequency(be1_phase, set = "oFD")
    #>              BE1-Dghnt BE1-CylRound
    #> BE1-Dghnt           NA            0
    #> BE1-CylRound    0.9962           NA

# References

Allen, James F. 1983. “Maintaining Knowledge about Temporal Intervals.” Communications of the ACM 26 (11): 832–43. https://doi.org/10.1145/182.358434.