PoPs: Probabilistic Bit-String Models

• PoPs is a framework that assigns probabilities to binary strings using various methods such as machine-word Bernoulli sampling, adaptive sequential estimation, and pattern-based laws.
• It integrates multiple regimes—from exact Bernoulli-word generation to substring frequency estimations—to model and analyze bit-string randomness and structure.
• Practical implementations, including Poisson-OR algorithms and FPGA-emulated p-bit architectures, demonstrate significant speedups and advancements in probabilistic computations.

Probabilities of Bit-String Probabilities (PoPs) denotes, in a broad research sense, the family of constructions in which binary strings are assigned probabilities, generated from per-bit probabilities, or compared through probability laws induced by sequential dynamics, pattern counts, or noisy channels. Across recent arXiv work, PoPs appear in at least four recurrent forms: exact machine-word Bernoulli sampling, where a whole word is drawn with i.i.d. Bernoulli$(p)$ bits; adaptive sequential models, where full-string probability is the chain-rule product of next-bit predictions; Bernoulli pattern laws, where a length-$k$ word $\omega$ has mass $p^{|\omega|}(1-p)^{k-|\omega|}$; and channel models, where a fixed or latent string induces a probability distribution on observed traces (Watanabe et al., 2018, Mattern, 2015, Kogan et al., 29 Sep 2025, Hartung et al., 2017).

1. Probability objects and semantic regimes

The most basic PoPs object is the probability of a fixed finite word under an i.i.d. Bernoulli source. In the machine-word formulation, one seeks a word

$X=(X_1,\dots,X_{N_{\mathrm{bit}}})$

such that

$$X_i \overset{\text{i.i.d.}}{\sim} \mathrm{Bernoulli}(p), \qquad \Pr(X=s)=p^m(1-p)^{N_{\mathrm{bit}}-m},$$

for any concrete bit string $s$ of Hamming weight $m$. The induced Hamming-weight law is binomial: $$\sum_{i=1}^{N_{\mathrm{bit}}} X_i \sim \mathrm{Binomial}(N_{\mathrm{bit}},p).$$ In this regime, PoPs is a word-level Bernoulli vector distribution, not merely a collection of unrelated single-bit events (Watanabe et al., 2018).

A second regime is sequential. In exponential-smoothing probability estimation, the model maintains a Bernoulli prediction for the next bit and updates it after each observation. For a binary sequence $x_{1:n}$, the induced whole-string probability is

$k$0

and the corresponding codelength is

$k$1

Here a bit-string probability is not specified directly; it is assembled online from conditional probabilities by the chain rule (Mattern, 2015).

A third regime concerns the distribution of probabilities across words of equal length. In an infinite Bernoulli$k$2 sequence, each length-$k$3 word $k$4 has probability

$k$5

Hence the probability of a word depends only on its Hamming weight, and there are only $k$6 distinct probability levels, with multiplicities $k$7. When $k$8, all words of length $k$9 are equiprobable; when $\omega$0, probabilities vary exponentially with weight (Kogan et al., 29 Sep 2025).

A fourth regime is sequential competition rather than static mass. In Penney’s ante, the central quantity is

$\omega$1

the probability that one target word appears before another in an iid Bernoulli$\omega$2 stream. This is not the same object as $\omega$3 or $\omega$4. A plausible implication is that PoPs should be treated as a family of probability semantics rather than a single scalar notion (Drexel et al., 2024).

2. Exact Bernoulli-word generation

A particularly concrete PoPs problem is exact word-level Bernoulli sampling: given $\omega$5 and $\omega$6, generate a machine word whose bits are mutually independent Bernoulli$\omega$7. The baseline is the obvious per-bit algorithm, which performs $\omega$8 random tests. The main alternatives developed for this problem are binomial-shuffle, Poisson-OR, and a finite-digit dyadic construction. Binomial-shuffle first samples

$\omega$9

then places exactly $p^{|\omega|}(1-p)^{k-|\omega|}$0 ones uniformly among the bit positions, with expected draw count approximately $p^{|\omega|}(1-p)^{k-|\omega|}$1. Poisson-OR samples

$p^{|\omega|}(1-p)^{k-|\omega|}$2

generates $p^{|\omega|}(1-p)^{k-|\omega|}$3 one-hot words, ORs them together, and yields exact mutual independence because

$p^{|\omega|}(1-p)^{k-|\omega|}$4

The finite-digit construction synthesizes a dyadic probability $p^{|\omega|}(1-p)^{k-|\omega|}$5 from raw fair-bit words using AND and OR recursions; for arbitrary real $p^{|\omega|}(1-p)^{k-|\omega|}$6, exactness is restored by a hybrid correction step using either

$p^{|\omega|}(1-p)^{k-|\omega|}$7

or

$p^{|\omega|}(1-p)^{k-|\omega|}$8

The paper’s exactness summary is sharp: the simple algorithm, binomial-shuffle, and Poisson-OR are exact; the finite-digit method alone is exact only for finite-binary $p^{|\omega|}(1-p)^{k-|\omega|}$9; the hybrid finite-digit-plus-correction method is exact for arbitrary $X=(X_1,\dots,X_{N_{\mathrm{bit}}})$0. In benchmark results on an Intel Xeon Gold 6184 with Intel C++ and $X=(X_1,\dots,X_{N_{\mathrm{bit}}})$1, the fastest method for $X=(X_1,\dots,X_{N_{\mathrm{bit}}})$2 was Poisson-OR with correction from $X=(X_1,\dots,X_{N_{\mathrm{bit}}})$3, giving about $X=(X_1,\dots,X_{N_{\mathrm{bit}}})$4 speedup for 32-bit words and $X=(X_1,\dots,X_{N_{\mathrm{bit}}})$5 for 64-bit words relative to the simple algorithm. The same generator accelerated multispin coding for one-dimensional bond-directed percolation by up to $X=(X_1,\dots,X_{N_{\mathrm{bit}}})$6 over optimized scalar code for cluster growth at criticality and about $X=(X_1,\dots,X_{N_{\mathrm{bit}}})$7 for relaxation from the fully active state (Watanabe et al., 2018).

3. Adaptive assignment and substring-based estimation

In adaptive sequential PoPs, the central object is an online Bernoulli predictor. The exponential-smoothing model $X=(X_1,\dots,X_{N_{\mathrm{bit}}})$8 updates

$X=(X_1,\dots,X_{N_{\mathrm{bit}}})$9

where $$X_i \overset{\text{i.i.d.}}{\sim} \mathrm{Bernoulli}(p), \qquad \Pr(X=s)=p^m(1-p)^{N_{\mathrm{bit}}-m},$$0. In closed form,

$$X_i \overset{\text{i.i.d.}}{\sim} \mathrm{Bernoulli}(p), \qquad \Pr(X=s)=p^m(1-p)^{N_{\mathrm{bit}}-m},$$1

so older bits are exponentially downweighted. The whole-string probability again follows by the chain rule, and the main redundancy guarantee is $$X_i \overset{\text{i.i.d.}}{\sim} \mathrm{Bernoulli}(p), \qquad \Pr(X=s)=p^m(1-p)^{N_{\mathrm{bit}}-m},$$2 against a piecewise stationary comparator with $$X_i \overset{\text{i.i.d.}}{\sim} \mathrm{Bernoulli}(p), \qquad \Pr(X=s)=p^m(1-p)^{N_{\mathrm{bit}}-m},$$3 segments, improving on $$X_i \overset{\text{i.i.d.}}{\sim} \mathrm{Bernoulli}(p), \qquad \Pr(X=s)=p^m(1-p)^{N_{\mathrm{bit}}-m},$$4 for previous approaches of similar time complexity. The estimator runs in $$X_i \overset{\text{i.i.d.}}{\sim} \mathrm{Bernoulli}(p), \qquad \Pr(X=s)=p^m(1-p)^{N_{\mathrm{bit}}-m},$$5 time per bit for the binary case and preserves strictly positive probabilities as long as $$X_i \overset{\text{i.i.d.}}{\sim} \mathrm{Bernoulli}(p), \qquad \Pr(X=s)=p^m(1-p)^{N_{\mathrm{bit}}-m},$$6 and $$X_i \overset{\text{i.i.d.}}{\sim} \mathrm{Bernoulli}(p), \qquad \Pr(X=s)=p^m(1-p)^{N_{\mathrm{bit}}-m},$$7 (Mattern, 2015).

A different, more empirical PoPs construction estimates probabilities from substring counts in a reference string. For a substring $$X_i \overset{\text{i.i.d.}}{\sim} \mathrm{Bernoulli}(p), \qquad \Pr(X=s)=p^m(1-p)^{N_{\mathrm{bit}}-m},$$8 and class string $$X_i \overset{\text{i.i.d.}}{\sim} \mathrm{Bernoulli}(p), \qquad \Pr(X=s)=p^m(1-p)^{N_{\mathrm{bit}}-m},$$9, the estimator is

$s$0

A test string $s$1 is then assigned the probability of its most probable segmentation: $s$2 equivalently

$s$3

The decisive modeling choice is the meaning of $s$4: overlapping counting counts all matching start positions, whereas non-overlapping counting extracts disjoint copies. The difference is especially pronounced for periodic patterns. In the reported classification study over 50 UCR datasets and 45,660 test strings, discordant outcomes were $s$5 in favor of overlapping and $s$6 in favor of non-overlapping, with McNemar $s$7, and the paper concludes that non-overlapping counting is better in that setting (Takamoto et al., 2022).

4. Pattern counts, Poisson laws, and first-occurrence probabilities

For pattern-count PoPs, the unbiased and biased Bernoulli cases diverge sharply. When $s$8, all length-$s$9 words are equiprobable with mass $m$0, and with $m$1 the number of occurrences of a random word converges to $m$2 for almost every sequence in the sense of Peres and Weiss. When $m$3, this global Poisson regime fails: for any choice of $m$4, words of the same length are exponentially nonuniform in probability, so most words are either too rare to appear or so common that their counts diverge. The paper proves a trichotomy: depending on the scale of $m$5 relative to $m$6, counts become trivial at zero, diverge in probability, or exhibit critical partial escape of mass with a nontrivial atom at $m$7 and all finite positive atoms vanishing. Poisson behavior is recovered only after restricting to a fixed Hamming-weight class

$m$8

so that all selected words have equal Bernoulli probability (Kogan et al., 29 Sep 2025).

Penney’s ante provides a complementary PoPs model in which overlap geometry, not merely raw Bernoulli mass, governs the outcome. For words $m$9 and $$\sum_{i=1}^{N_{\mathrm{bit}}} X_i \sim \mathrm{Binomial}(N_{\mathrm{bit}},p).$$0,

$$\sum_{i=1}^{N_{\mathrm{bit}}} X_i \sim \mathrm{Binomial}(N_{\mathrm{bit}},p).$$1

where $$\sum_{i=1}^{N_{\mathrm{bit}}} X_i \sim \mathrm{Binomial}(N_{\mathrm{bit}},p).$$2, $$\sum_{i=1}^{N_{\mathrm{bit}}} X_i \sim \mathrm{Binomial}(N_{\mathrm{bit}},p).$$3, $$\sum_{i=1}^{N_{\mathrm{bit}}} X_i \sim \mathrm{Binomial}(N_{\mathrm{bit}},p).$$4, and $$\sum_{i=1}^{N_{\mathrm{bit}}} X_i \sim \mathrm{Binomial}(N_{\mathrm{bit}},p).$$5 are correlation-polynomial values built from overlap sets. The same formalism gives

$$\sum_{i=1}^{N_{\mathrm{bit}}} X_i \sim \mathrm{Binomial}(N_{\mathrm{bit}},p).$$6

This immediately rules out a common misconception: a word with larger static mass need not be more likely to occur first. At $$\sum_{i=1}^{N_{\mathrm{bit}}} X_i \sim \mathrm{Binomial}(N_{\mathrm{bit}},p).$$7, if $$\sum_{i=1}^{N_{\mathrm{bit}}} X_i \sim \mathrm{Binomial}(N_{\mathrm{bit}},p).$$8, then

$$\sum_{i=1}^{N_{\mathrm{bit}}} X_i \sim \mathrm{Binomial}(N_{\mathrm{bit}},p).$$9

so a word longer by at least two bits can never be favorable. The paper also identifies systematic $x_{1:n}$0 symmetries and explicit bijections behind odd, even, and constant symmetry classes (Drexel et al., 2024).

5. Channel-induced string laws and latent probability strings

Deletion channels induce PoPs by mapping a fixed source string to a probability distribution on output traces. In the position-dependent deletion model, each source symbol $x_{1:n}$1 is retained with probability $x_{1:n}$2 and deleted with probability $x_{1:n}$3. For an output trace $x_{1:n}$4,

$x_{1:n}$5

The generating-function identity

$x_{1:n}$6

encodes output-coordinate expectations through

$x_{1:n}$7

Under weak monotonicity or 2-periodicity assumptions, the paper extends the $x_{1:n}$8 trace-reconstruction upper bound to varying deletion probabilities (Hartung et al., 2017).

A stronger generalization replaces the unknown deterministic source by a latent probability string

$x_{1:n}$9

Each trace is generated by first sampling

$k$00

independently, then deleting each sampled bit independently with probability $k$01. This contains classical trace reconstruction as the special case $k$02. The worst-case theorem states that for deletion probability at least order $k$03, no algorithm can approximate $k$04 to constant $k$05 distance or $k$06 distance $k$07 using fewer than $k$08 traces. By contrast, if the $k$09 are i.i.d. uniform on $k$10, then for any $k$11 and sufficiently small constant deletion probability, specifically $k$12, $k$13 can be recovered to $k$14 error at most $k$15 with high probability using $k$16 traces and computation time (Rivkin et al., 2024).

6. Computation, architectures, and alternative notions of randomness

When a bit-string probability factors into a product of local probabilities,

$k$17

PoPs becomes a streaming product problem. The approximation task $k$18 asks for a multiplicative $k$19-approximation to $k$20. A cited deterministic upper bound uses

$k$21

bits of state, while the paper proves matching lower bounds up to constants: $k$22 for both deterministic and randomized algorithms in the stated parameter regime. The threshold problem $k$23, deciding whether a product is below a threshold, is much harder: the paper proves randomized space lower bounds of $k$24 when $k$25. This establishes a sharp computational distinction between approximate scoring and exact thresholding of factored bit-string probabilities (Lohrey et al., 23 Apr 2025).

PoPs also appears as a hardware primitive. In p-bit computing, each stochastic binary unit obeys

$k$26

with $k$27 uniform on $k$28, so a p-bit is a 1-bit random number generator with controllable mean. Networks of p-bits can implement Bayesian networks, Ising/Boltzmann systems, and MCMC procedures. For Boltzmann-style sampling, the paper states that network states are generated with probabilities

$k$29

and local inputs satisfy

$k$30

The proposed architecture has an $k$31-bit RNG block, a deterministic kernel, and a deterministic data collector. The paper reports FPGA emulation at $k$32 MHz, systems with thousands of p-bits, and applications including Bayesian networks, optimization, Ising models, and quantum Monte Carlo (Kaiser et al., 2021).

A markedly different formulation treats randomness through witness complexity rather than probabilistic normalization. In the simple quantum model, a predicate $k$33 on an integer sequence $k$34 defines a bit string

$k$35

and a resource state

$k$36

contains witnesses required to certify the $k$37-positions. The paper’s compression criterion is cardinality-based: if $k$38, the string is compressible; if $k$39, it is incompressible and has maximal randomness. This is not a Shannon-entropy notion. It is an alternative PoPs perspective in which the randomness of a bit string is governed by the minimal resource needed to specify its positive instances (Ramos, 2015).

Taken together, these lines of work suggest that PoPs is best understood not as a single theory but as a technically connected research area spanning exact Bernoulli-word generation, adaptive sequence modeling, overlap-sensitive pattern laws, noisy-channel trace distributions, streaming computation of factored probabilities, and hardware or quantum formalisms for representing stochastic bit strings. The common invariant is that probabilities are attached to binary strings at the level of words, sequences, patterns, traces, or latent parameterizations, and that the operative mathematics depends on which of those objects is being treated as fundamental.