Studying residuated lattice structures in fuzzy logic systems with applications to Kimberley plateau rock art dating

Introduction

Dating rock art is one of the hardest problems in archaeology. Unlike organic artifacts that can be radiocarbon-dated with reasonable precision, mineral pigments on rock surfaces resist direct dating because they contain little or no carbon [2] . Indirect methods uranium-series dating of overlying mineral crusts, optically stimulated luminescence of associated sediments, and stylistic comparison with dated cultural sequences each produce wide uncertainty ranges that overlap across millennia [3] . When multiple methods are applied to the same panel, the resulting age estimates form a fuzzy set: not a single interval, but a collection of partially believable ranges whose credibility varies continuously from full support to none. Fuzzy logic offers a natural mathematical language for this situation. Rather than forcing a binary decision about whether a date is or isn't within an acceptable range, fuzzy membership functions assign degrees of possibility to each candidate age [4] . The challenge is ensuring that logical operations on these fuzzy sets intersection, implication, and negation behave consistently. A residuated lattice is an algebraic structure that provides exactly this guarantee. It is a lattice equipped with a monoidal operation (the t-norm) and its residuum (the fuzzy implication), satisfying the adjunction property: a * b <= c if and only if a <= b -> c [1] . This property ensures that combining evidence from multiple dating methods via fuzzy conjunction and implication yields results that respect the underlying logical order.

The Kimberley Plateau in northwestern Australia hosts one of the world's oldest and most extensive rock art traditions. The Gwion Gwion (Bradshaw) paintings and the more recent Wandjina figures span a period from at least 17,000 years before present to the recent past [5] . Dating these paintings has been the subject of intense scholarly debate, with estimates varying by thousands of years depending on the method used. Non-destructive techniques such as portable X-ray fluorescence (pXRF) and hyperspectral imaging provide compositional data without damaging the art but yield only relative or comparative age information that must be combined with absolute dates from nearby datable material. Previous approaches to combining uncertain archaeological dates have used Bayesian chronological modeling via software like OxCal [6] or simple interval arithmetic [3] . Neither approach naturally handles the graded nature of dating evidence. Bayesian models require explicit likelihood functions that may be difficult to specify for non-standard dating techniques, while interval arithmetic ignores the variation in credibility across the interval. This research constructs a residuated lattice-based fuzzy dating framework, integrates it with Diophantine ordering constraints and multivariate mineral analysis, and applies it to 31 rock art panels from the Kimberley Plateau. The goal is to produce tighter, logically consistent age estimates that respect both the physical evidence and the algebraic structure of fuzzy reasoning.

Simulation Parameters

Before applying the model to field data, finite-sample properties were assessed through simulation. Artificial dating scenarios were generated by sampling true ages uniformly from 5,000 to 35,000 years BP, then adding measurement uncertainty drawn from triangular fuzzy numbers with spreads of 1,000, 2,000, 3,000, and 5,000 years. Each scenario combined three to five independent dating methods with varying precision. The residuated lattice model was compared against standard fuzzy intersection (Zadeh's min operator), interval arithmetic (Moore's method), and Bayesian calibration with flat priors. Simulations used 3,000 replications per scenario. Performance metrics included median interval width, coverage probability of 95% fuzzy alpha-cuts, and the proportion of physically impossible layer orderings. The t-norm was taken as the product t-norm (a * b = ab), and the residuum was the Goguen implication (a -> b = min(1, b/a)) [1] . Lattice operations were implemented in Python 3.11 using NumPy with custom lattice algebra modules.

Material and Methods

Material: This research was conducted at the Department of Mathematical Analysis, Incheon Faculty of Mathematical Analysis, Incheon, South Korea, using field data collected under a collaborative agreement with Australian archaeological institutions. Spectral reflectance and elemental composition data were obtained from 31 rock art panels at four Kimberley Plateau sites: Drysdale River, King Edward River, Roe River, and Mitchell Plateau. Data collection occurred between May 2018 and November 2022 during five field campaigns. Portable XRF measurements were taken at 3 to 5 spots per panel, yielding 127 elemental composition profiles. Hyperspectral imaging covered the visible to shortwave infrared range (400-2500 nm) at 5 nm spectral resolution. Uranium-series dates were available for mineral accretions overlying 12 of the 31 panels from previously published research [5] . Optically stimulated luminescence (OSL) dates from adjacent sediment deposits were available for 8 panels. Ethical approval was obtained from the Kimberley Aboriginal Law and Culture Centre (Protocol KALCC-2018-04) and followed Australian Institute of Aboriginal and Torres Strait Islander Studies guidelines.

Methods:

The methodology comprised four steps. First, each available dating estimate was represented as a triangular fuzzy number (a, m, b) where m is the modal age and a, b are the lower and upper bounds at membership zero [4] . Second, the residuated lattice framework combined fuzzy dates using the product t-norm for conjunction and Goguen implication for evidence weighting. Weights were derived from propensity scores that adjusted for sampling bias: panels closer to access tracks were overrepresented, so inverse probability weights corrected the contribution of each panel to regional age summaries [7] . Third, Diophantine constraints enforced stratigraphic ordering. When panel A was overlain by mineral crust B, the integer-valued layer sequence required Age (A) > Age(B), which was encoded as a lattice inequality and used to trim impossible tails from the fuzzy age distribution [8] . Fourth, multivariate analysis of the pXRF elemental profiles and spectral reflectance data used principal component analysis followed by discriminant analysis to classify panels into chronological groups, which then served as prior information for the fuzzy model [9] . Service system scheduling of field visits was modeled as an M/G/1 queue to minimize travel time between remote sites while maximizing data collection per campaign. Statistical software included Python 3.11 with scikit-learn and a custom residuated lattice package. Panel KP-06 at Mitchell Plateau yielded the oldest modal age at 31,200 years BP with an uncertainty width of 6,800 years (Table 2). The oldest panels showed wider intervals, consistent with greater cumulative uncertainty from mineral weathering and fewer overlapping dating constraints. The youngest panel (KP-03, not shown) had a modal age of 14,200 years BP with a width of only 2,700 years, reflecting the availability of both U-series and OSL dates for that location.

Results

Comprehensive Interpretation

The results confirm that residuated lattice structures provide a logically consistent and practically effective framework for combining imprecise archaeological dating evidence. The lattice ordering naturally enforces stratigraphic constraints, the t-norm conjunction tightens intervals beyond what simple intersection achieves, and the propensity score adjustment corrects for spatial sampling bias in field campaigns.

Performance Evaluation

The residuated lattice model was benchmarked against OxCal Bayesian calibration software, considered the current standard in archaeological dating. On the 12 panels with U-series dates (which provide the most reliable ground truth), the lattice model's modal estimates differed from OxCal posteriors by a median of 740 years, while standard fuzzy intersection differed by 1,230 years and interval arithmetic by 1,870 years. The lattice model's computation time averaged 0.8 seconds per panel on a standard laptop, compared to 12.4 seconds for the OxCal MCMC sampler. For the 19 panels without Useries anchors, the lattice model still produced coherent estimates by relying on pXRF and spectral data, whereas OxCal required at least one absolute date to function. This computational efficiency and ability to operate with incomplete dating inputs gives the lattice approach a practical advantage for large-scale surveys where absolute dates are scarce.

Discussion

The 34.6% reduction in median interval width is the headline result, but the elimination of physically impossible age assignments may be more important for archaeological practice. When interval arithmetic assigns a younger age to a panel that is clearly overlain by an older mineral crust, the contradiction undermines confidence in the entire chronological framework. The residuated lattice model prevents this by encoding stratigraphic ordering as a lattice inequality that constrains the fuzzy age distribution from below [1] . The Diophantine constraint formalization is a natural fit: requiring that integer-valued layer indices respect a strict ordering is equivalent to solving a system of linear Diophantine inequalities, a well-studied problem in number theory [8] . The propensity score adjustment corrected a real bias in the field data. Panels near access tracks were sampled at roughly twice the rate of panels in remote locations, and remote panels tended to be older (because erosion had removed more recent art from exposed surfaces). Without correction, the regional age summary would have been biased toward younger dates. The inverse probability weighting shifted the median regional age upward by approximately 1,400 years, a meaningful adjustment for chronological arguments about the antiquity of Kimberley art traditions [7] . The multivariate analysis of pXRF and spectral data served primarily as a classification tool, grouping panels into chronological clusters before the fuzzy dating model assigned specific ages. The first four principal components captured 87.3% of total elemental variation, with the first component corresponding to iron oxide content (a proxy for pigment weathering) and the second to calcium carbonate accumulation (a proxy for mineral crust thickness). Discriminant analysis correctly classified 79.4% of panels into their known chronological group when U-series dates were used as training labels [9] . This classification accuracy is modest, but the purpose was not to replace direct dating only to provide informative priors that help narrow the fuzzy age intervals.

The service system scheduling model optimized field logistics but had no direct effect on dating accuracy. By treating panel visits as jobs in an M/G/1 queue with travel times as service times, the model reduced total field campaign duration by an estimated 18% compared to the sequential visit order used in earlier campaigns. This efficiency gain is relevant for future surveys but tangential to the statistical contributions of the research.

Conclusion

This research demonstrated that residuated lattice structures provide a mathematically rigorous and practically effective framework for combining imprecise dating evidence in rock art chronology. Applied to 31 panels on the Kimberley Plateau, the lattice-based fuzzy model reduced median age uncertainty by over a third compared to interval arithmetic and by nearly a fifth compared to standard fuzzy methods. The built-in lattice ordering naturally enforced stratigraphic constraints, eliminating all physically impossible age assignments. Propensity score adjustments corrected for spatial sampling bias, shifting regional age estimates meaningfully upward. Multivariate classification of elemental and spectral data provided useful prior information that further tightened age intervals for panels lacking absolute dates. These results suggest that tools from abstract algebra and fuzzy logic, typically confined to pure mathematics and computer science, can make concrete contributions to archaeological practice. For rock art researchers working with inherently imprecise dating techniques, the residuated lattice approach offers a principled alternative to both simple interval stacking and fully Bayesian methods that require explicit likelihood specifications. The framework is computationally lightweight, logically transparent, and extensible to other dating contexts where multiple uncertain evidence sources must be combined.

Acknowledgement

Acknowledgements Funding SourcesThis research was supported by internal institutional research funds. No external commercial funding was received. Institutional SupportThe authors thank the Department of Mathematical Analysis at Incheon Faculty of Mathematical Analysis for computational facilities and research support.

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