This paper investigates whether the postmodern claim of unrestricted semantic indeterminacy, and its foundational Saussurean axiom of the arbitrary sign, are compatible with the structural architecture of Classical Arabic. We develop a formal mathematical model of Arabic non concatenative morphology in which lexical meaning is determined by the interaction between an invariant root and a morphosyntactic pattern. Within this framework, we establish a Morphological Correspondence Theorem, demonstrating that every lexical item is uniquely generated by a root pattern pair, and a Semantic Localization Theorem, proving that lexical meaning is determined at the derivational level prior to surface realization. To address Saussurean weaker notion of relative arbitrariness, we formalize it via conditional Kolmogorov complexity, defining arbitrariness algorithmically as the no rule property. We prove that general relative arbitrariness is formally undecidable, while Arabic relative arbitrariness is decidable and provably less than 1 for its motivated signifiers (Levels W and M), establishing a strict system complexity asymmetry over Indo-European languages.
Solomonoff Induction, or SolInd, provides an ideal unbounded model of a priori sequence prediction but cannot naturally describe extrapolation from a given training dataset, as performed by Large Language Models. We apply de Finetti's theorem on exchangeable distributions to SolInd to produce what we call Hierarchical Solomonoff Induction, or HSI, which maintains a hyperprior over all Solomonoff priors that can be conditioned on previously observed sequences. We extend Wood et al.'s proof that universal mixtures of semimeasures are equivalent to SolInd to show that universal mixtures of these mixtures are also equivalent, proving that HSI=SolInd. We also prove that HSI's excess error on any distribution, compared to its true generator, is bounded by that generator's complexity in the hyperprior. This result is directly comparable to SolInd's prediction error being bounded by the Kolmogorov complexity of the sequence being predicted, and forces HSI's average excess error to converge to 0 as a dataset grows, leading to optimal prediction in the limit. We claim that HSI is an ideal unbounded model of sequence prediction given a dataset in the same way that SolInd is ideal over individual sequences.
A randomized algorithm may terminate almost surely even though exceptional random tapes make it run forever. This paper studies the survival tail, the Kolmogorov complexity of one such tape, and the Hausdorff dimension of all of them. For each $s>0$ at which the powered repair matrices commute, the main theorem bounds $\sum_wP[w]^s$ over surviving prefixes $w$, uniformly over deterministic nonanticipating selectors. The case $s=1$ controls termination; the full family gives weak-source and dimension bounds. The source powers contain information absent even from the ordinary repair kernel and the complete stopping-time law. Under one common finite tape source, two overlapping disagreement-repair rules on a four-vertex path have the same ordinary kernels and the same stopping-time law for every selector, yet their nontermination dimensions can be arbitrarily close to zero and one. At one common source-power level, the same dominated tape source makes one rule run forever but gives the other an exponential stopping tail. The separation is caused by action labels that produce the same state transition and are therefore invisible at power one. For bounded-dependence $k$-SAT, conditional block min-entropy above the trace-growth threshold gives exponential termination, and the effective dimension of an individual infinite run is bounded by the trace growth induced by the clauses repaired infinitely often. Tree formulas asymptotically attain the maximum-degree dimension and global source bounds, while clique formulas attain the graph-specific one-step threshold in the stated regime. An exact backward likelihood identity complements these setwise results with tail and coding bounds for each run.
In this paper, we define the quantity of prompting complexity: for a fixed instruction-tuned language model, what is the shortest plausible prompt that makes deterministic decoding produce a target text? It is an LM-relative analogue of resource-bounded Kolmogorov complexity: the prompt is a program, the model interface is the interpreter, and information omitted from the prompt is supplied by the model's weights, training distribution, tokenizer, template, and decoding rule. Unlike classical Kolmogorov complexity, this measure is intentionally non-universal. In the finite-context setting it is computable by enumeration, but there is no model-independent invariance theorem; the same text may be cheap for one model and inaccessible or expensive for another. To keep the search space aligned with prompt engineering, we restrict programs to plausible human-readable texts rather than arbitrary token strings. We extend the exact definition to soft prompting complexity for approximate outputs, yielding a lossy notion of model-relative text compression and a formal target for prompt optimization. We also define prompting distance by comparing shortest generating prompts, and behavioral prompting complexity for reaching any output satisfying a specification. Based on these formulations, we define a research agenda for empirically studying which texts and behaviors are accessible from short plausible prompts under a fixed LM interface.
Pearl's causal hierarchy shows that observational, interventional, and counterfactual queries are qualitatively distinct. We ask a quantitative version of this question: how many additional bits are needed to specify higher-rung causal answers once lower-rung answers are known? We formalize this via query-class description length, the Kolmogorov complexity of the answer oracle induced by an SCM for a class of queries. Our main construction gives binary acyclic SCMs whose observational distribution has constant description length, while the single-variable interventional answer oracle has description length $Θ(n^2)$. A degree-sensitive upper bound shows that finite-gate-schema SCMs of indegree $d$ have observational-interventional gap at most $O(nd \log(en/d) + n \log n)$, making the quadratic construction order-optimal in the dense regime and a rooted-tree construction order-optimal for bounded indegree. The quadratic separation persists under $\varepsilon$-accurate total-variation descriptions for every fixed $\varepsilon < 1/4$. At the next rung, the full hard-do interventional oracle can still leave a $Θ(n)$ counterfactual description gap. A general ambiguity-to-bits theorem and Shannon analogue show that these gaps equal the logarithm of residual higher-rung ambiguity up to lower-order terms.