PAST: Multi-Domain Formalizations & Applications

• PAST is a multifaceted concept defined differently across fields, including low-entropy conditions in physics, weak traces in quantum systems, and conditional lifetime distributions in reliability theory.
• Its formal framework varies from discrete, reversible expansion mechanisms in thermodynamics to modal operators and proof-theoretic structures in temporal logic.
• Recent applications in machine learning treat PAST as an optimization variable, enabling data corrections and separation of historical patterns to improve system performance.

Searching arXiv for the cited papers so the article can be grounded in current arXiv records. First, locating the thermodynamics/time-arrow paper and several core "past"-related works across physics, logic, and ML. Now checking additional papers on temporal logic, runtime monitoring, and applied ML systems using “PAST” as a formalism or model name. In the cited literature, “past” denotes several non-equivalent technical objects: a low-entropy boundary condition in statistical mechanics, a weakly measured history in pre- and postselected quantum systems, a conditional lifetime distribution in reliability theory, a family of temporal operators in logic, and an explicit informational resource in learning systems. Taken together, these usages show that the past is not treated merely as an informal temporal backdrop. It is formalized as a state space, an operator class, a conditional distribution, a proof-theoretic direction of reference, or an optimization variable, depending on the domain (Arrighi et al., 2023, Vaidman, 2013, S. et al., 2023, Ghari, 2018, Baptista et al., 2021).

1. Scope and terminology

The term appears in at least five stable research meanings. In physics, it is tied to the “past hypothesis” and to the problem of explaining the arrow of time. In quantum foundations, it concerns the location or determination of a system between preparation and detection. In reliability theory, it refers to the elapsed lifetime conditioned on failure by time $t$. In temporal logic, it denotes modal access to earlier states or events. In machine learning, it can mean historical payoffs, historical observations, or previously seen datasets.

Area	Technical meaning of “past”	Representative work
Statistical physics	Low-entropy past, past hypothesis, entropic arrow	(Arrighi et al., 2023)
Quantum foundations	Weak trace of pre- and postselected systems	(Vaidman, 2013)
Reliability theory	Past lifetime, past entropy, inactivity time	(S. et al., 2023)
Temporal logic	Past operators, past-present rules, bounded past	(Ghari, 2018)
Learning systems	Historical payoffs, corrected histories, past datasets	(Danku et al., 2019)

This taxonomy is not merely lexical. Each usage comes with a distinct formal apparatus: equivalence classes and entropy counts in physics, weak values in quantum theory, reversed hazard rates in reliability, modal operators such as $\mathsf{Y}$, $\mathsf{S}$, and $\mathsf{H}$ in logic, and gradient- or payoff-based update rules in learning systems.

2. Thermodynamic and cosmological past

A central use of “past” in physics is the “past hypothesis”: the claim that entropy increases toward the future because the universe began in a highly special low-entropy state. The toy-model analysis in "Time arrow without past hypothesis: a toy model explanation" argues that this assumption can, in principle, be replaced by reversible local dynamics with expansion (Arrighi et al., 2023). The paper starts from the standard tension between microscopic reversibility and macroscopic irreversibility and formulates entropy in Boltzmann form as $S(x)=\ln |[x]|$, where $[x]$ is the macro-equivalence class of microstate $x$. For finite reversible systems, periodicity implies that entropy cannot be strictly monotone for all times, and average entropy change over the whole state space vanishes. In that setting, the usual Boltzmann story needs the past hypothesis.

The contribution of the toy model is to show a different route. The system consists of finite circular graphs with particles moving and interacting under local reversible rules, together with local graph-expansion and graph-shrinking operations. Named graph machinery is introduced so that split and merge operations remain local and reversible. The principal result is that almost all states expand, and entropy always increases as a consequence of expansion. The paper therefore presents a local explanation of an entropic arrow of time without assuming a special low-entropy beginning (Arrighi et al., 2023).

The model is deliberately discrete. That allows full use of theoretical-computer-science proof techniques, and also supports numerical exploration of a time-symmetric variant, two inflationary variants, and a damping variant that slows down thermal death. The authors emphasize that these variants exhibit similar qualitative behavior, which suggests that local reversible expansion mechanisms may be a robust recipe for a time arrow without past hypothesis. The paper states this only in a qualitative sense, but it explicitly notes possible relevance at the cosmological level (Arrighi et al., 2023).

3. Quantum past, weak traces, and undetermined histories

In quantum foundations, “the past” is often the question of where a particle was between preparation and detection. Vaidman’s proposal is operational: a particle’s past is the set of regions where it leaves a nonzero weak trace, quantified by weak values of local projectors in the two-state vector formalism (TSVF). At intermediate time $t$, the system is described by $\langle \phi(t)|\,|\psi(t)\rangle$, and the weak value of an observable $A$ is

$\mathsf{Y}$0

For a local projector $\mathsf{Y}$1, a nonzero $\mathsf{Y}$2 is taken to indicate presence in region $\mathsf{Y}$3 (Vaidman, 2013).

This criterion leads to the familiar nested Mach–Zehnder result: weak traces can appear inside an interferometer while vanishing on the apparent routes in or out. The paper’s conclusion is that naive trajectory talk must be replaced by a description based on both forward- and backward-evolving states. Presence is tied to overlap of the two states, not to a single classical path (Vaidman, 2013).

Entanglement complicates the picture further. "Past of a particle in an entangled state" shows that single-particle weak traces are generally insufficient. In Hardy’s paradox, the single-particle overlapping-arm projectors have weak value $\mathsf{Y}$4 for each particle, while the joint overlapping-arm projector has weak value $\mathsf{Y}$5; in a two-box example, single-particle weak traces vanish although pair projectors have nonzero values such as $\mathsf{Y}$6, $\mathsf{Y}$7, and $\mathsf{Y}$8. The paper’s main claim is that multipartite traces carry information not recoverable from single-particle traces because the product rule for weak values fails in general. It therefore advocates a top-down account in which pair, triplet, and $\mathsf{Y}$9-particle traces are part of the system’s past, not optional corrections to a single-particle story (Paneru et al., 2017).

The controversy is interpretational as well as formal. A separate line, developed under “Convivial Solipsism,” rejects backward-in-time influence but argues that parts of an observer’s phenomenal past may become determined only when later measurements are made. On that view, there is no physical change in the past, but there may be no determinate truth value for some past facts until a later measurement fixes them within the observer’s phenomenal world (Zwirn, 2020). This does not replace the weak-trace program, but it shows that “the past of a quantum system” remains a contested object even when precise formal criteria are available.

4. Past lifetime, past entropy, and information-generating functions

In reliability theory, the past is formalized as a conditional lifetime distribution. If $\mathsf{S}$0 is a nonnegative absolutely continuous lifetime random variable, the past lifetime at time $\mathsf{S}$1 is

$\mathsf{S}$2

that is, the inactivity time given that failure has already occurred by time $\mathsf{S}$3 (S. et al., 2023). The associated conditional density is $\mathsf{S}$4 on $\mathsf{S}$5, where $\mathsf{S}$6 is the c.d.f. of $\mathsf{S}$7.

Past entropy, due to Di Crescenzo and Longobardi, is defined by

$\mathsf{S}$8

It is the Shannon entropy of the conditional distribution of $\mathsf{S}$9 given $\mathsf{H}$0, and it can also be written in terms of the reversed hazard rate $\mathsf{H}$1 (S. et al., 2023). The paper "Entropy generating function for past lifetime and its properties" introduces the past entropy-generating function

$\mathsf{H}$2

and proves that past entropy is recovered by the first derivative at $\mathsf{H}$3: $\mathsf{H}$4 This places past entropy inside a broader one-parameter family of information functionals (S. et al., 2023).

The paper establishes several structural properties. The PEGF behaves predictably under affine transformation, satisfies a differential relation with the reversed hazard rate,

$\mathsf{H}$5

and, under an increasing-in-$\mathsf{H}$6 assumption, uniquely determines the underlying distribution (S. et al., 2023). It then derives characterization theorems for uniform, power, exponential, and generalized power distributions via simple functional relations between $\mathsf{H}$7, $\mathsf{H}$8, and mean inactivity time. In this literature, “past” is not a metaphorical direction of inference but a conditional law with its own entropy, generating function, and identifiability theory.

5. Past operators in logic, programming, and verification

Temporal logic treats the past as a family of modalities with explicit proof theory and model theory. "Linear Temporal Justification Logics with Past Operators" combines Artemov-style justification logic with linear temporal logic over discrete time with finite past and infinite future. Its language includes $\mathsf{H}$9, $S(x)=\ln |[x]|$0, $S(x)=\ln |[x]|$1, and $S(x)=\ln |[x]|$2, together with explicit justification terms $S(x)=\ln |[x]|$3. The paper proves soundness and completeness and studies interaction principles between justifications and temporal operators, including past-oriented ones such as $S(x)=\ln |[x]|$4 and $S(x)=\ln |[x]|$5 (Ghari, 2018). In this setting, the past is a direction in which evidence can persist, be reconstructed, or fail to be learnable.

A more radical step is taken in "Intransitive Linear Temporal Logic, Knowledge from Past, Decidability, Admissible Rules," where time is past-directed and intransitive. The point is to model limited access to the past: bounded memory, partial archives, or local inspection windows. The paper studies both non-uniform and uniform intransitivity, proves decidability for the non-uniform logic, and proves admissibility decidability for the uniformly intransitive variant (Rybakov, 2015). Here the past is not merely earlier-than; it is earlier-than within an access relation that need not be transitive.

In logic programming, "Past-present temporal programs over finite traces" isolates a syntactic class in TEL$S(x)=\ln |[x]|$6 whose rule bodies refer to the past and heads to the present. This restriction guarantees that the past is independent of the future. The paper extends completion and loop formulas to this class and shows that temporal stable models of past-present temporal programs can be captured by an LTL$S(x)=\ln |[x]|$7 expression (Cabalar et al., 2023). The formal advantage is the same as in several other past-oriented systems: past formulas depend only on prefixes, which supports incremental reasoning.

Runtime verification pushes the same idea into distributed execution. "Predictive Semantics for Past-CTL Runtime Monitors" works with a past-only branching-time logic over event structures and enriches its semantics from Booleans to a six-valued lattice

$S(x)=\ln |[x]|$8

The refinement distinguishes currently true or false verdicts from locally or globally final verdicts, and the paper shows how these semantics translate compositionally into Field Calculus monitors for swarms of devices (Audrito et al., 2022). Finally, in real-time temporal logic, "One-Pass and Tree-Shaped Tableau Systems for TPTL and TPTLb+Past" emphasizes the cost of unrestricted past: adding past to TPTL makes satisfiability non-elementary, whereas the bounded variant TPTLb+Past recovers EXPSPACE-complete complexity and admits a one-pass, tree-shaped tableau (Geatti et al., 2018). Across these works, the past is a source of expressivity, but also a major determinant of computational behavior.

6. Historical data as an explicit resource in learning and adaptation

In several learning systems, the past is neither memory state nor temporal operator but a variable that enters the optimization problem directly. "Knowing the past improves cooperation in the future" modifies evolutionary game dynamics so that imitation depends on historical payoff rather than only current payoff. One variant uses an exponentially weighted moving average,

$S(x)=\ln |[x]|$9

and another intermittently substitutes a payoff from $[x]$0 rounds in the past with probability $[x]$1. On square lattices, random graphs, and small-world networks, both schemes promote cooperation by slowing fast invasions of defectors, while the details of how the past is incorporated are reported to be of second-order importance (Danku et al., 2019).

"Pastprop-RNN: improved predictions of the future by correcting the past" goes further by treating parts of the input history as trainable. In addition to updating LSTM parameters, it computes gradients with respect to input samples and applies data corrections $[x]$2 to past observations. The paper evaluates three variants—Epoch-wise, Instance-wise, and Selective Pastprop—on M4, M5, and NAB. It reports that Selective Pastprop with correction rate $[x]$3 reduces average MSE from $[x]$4 to $[x]$5 across M4 and M5, a $[x]$6 decrease relative to standard LSTM, while the method is especially useful when standard LSTM errors are high (Baptista et al., 2021).

A different operational problem appears in continual learning. "Fast Evaluation of DNN for Past Dataset in Incremental Learning" asks how to estimate the effect of an incremental update on accuracy over all earlier datasets without re-evaluating the full past test set. The proposed method precomputes gradient summaries on the past dataset and, after training, combines them with the parameter update $[x]$7 to obtain an effect score

$[x]$8

A per-class linear regression then maps this score to predicted accuracy change, yielding constant-time post-training evaluation with respect to the size of the past dataset (Sato, 2024). In these works, the past becomes an explicit optimization object: historical payoffs, mutable historical observations, or accumulated historical benchmarks.

7. PAST as an acronymic model family

The same word also appears as an acronym naming specific architectures. "PAST: Phonetic-Acoustic Speech Tokenizer" defines PAST as a speech tokenizer built on RVQ and EnCodec, augmented with a transformer encoder and supervised phonetic auxiliary heads. Its first codebook is trained to carry strong phonetic information through a CTC character head and a phoneme classification head, while the overall loss also enforces signal reconstruction. The paper reports that PAST surpasses evaluated baseline tokenizers on phonetic representation and speech reconstruction metrics and also serves as a superior speech representation for speech LLMs; it additionally introduces a streamable causal variant for real-time applications (Har-Tuv et al., 20 May 2025).

In traffic data imputation, "PAST: A Primary-Auxiliary Spatio-Temporal Network for Traffic Time Series Imputation" uses the acronym for a network that explicitly separates primary patterns, arising from internal relationships among data points, from auxiliary patterns, induced by timestamps and node attributes. Its graph-integrated module uses dynamic graphs, interval-aware dropout, and multi-order convolutions, while its cross-gated module extracts auxiliary patterns through bidirectional gating on embedded external features. On three datasets and 27 missing-data conditions, the paper reports that PAST outperforms seven baselines by up to $[x]$9 in RMSE and $x$0 in MAE (Hu et al., 17 Nov 2025).

These acronymic uses are conceptually distinct from the temporal or epistemic uses discussed above. They indicate that “PAST” has also become a naming convention for architectures that foreground historical structure—phonetic-acoustic histories in speech tokenization, or primary and auxiliary temporal patterns in traffic imputation—even when the central contribution is architectural rather than foundational.