Partial Determinism: Theory and Applications

• Partial determinism is a concept where deterministic laws govern outcomes only within certain scales or contexts, allowing both predictable and unpredictable behavior.
• It is applied to model complex systems, from computer performance metrics to quantum phenomena, demonstrating how system complexity bounds forecastability.
• The framework unifies approaches like constraint-based modeling, projection-relative determinism, and engineered execution to control nondeterminism in practical systems.

Searching arXiv for the cited works to ground the synthesis. Searching arXiv for “Determinism, Complexity, and Predictability in Computer Performance”. Partial determinism is a family of positions in which deterministic laws, dynamics, or mechanisms coexist with bounded, context-dependent, or level-relative non-uniqueness in outcomes, predictability, or description. In the literature, the term appears in several non-identical but structurally related forms: deterministic dynamics whose temporal complexity bounds practical forecastability in computer performance (Garland et al., 2013); deterministic microphysics with emergent objective probabilities at the macro/experimental level (Vervoort, 2014); determinism of selected observables, subsystems, or chance assignments within a constraint-based theory (Adlam, 2021); determinism relative to a projection or level of description in formal theories (Halvorson et al., 7 Mar 2025); and per-step reliability $0<\delta<1$ in agentic environments, where long chains remain only partially predictable because success degrades as $\delta^k$ (Ding et al., 21 Jun 2026).

1. Conceptual forms and formal schemas

The term does not denote a single doctrine. In one prominent usage, partial determinism means that a system is deterministic “in principle,” but only some aspects of its behavior are effectively predictable or uniquely determined “in practice” (Garland et al., 2013). In another, it denotes a layered account in which microdynamics are deterministic while probabilities arise from frequency stabilization under repeatable initiating and probing conditions, coarse-graining, ignorance of hidden variables, or typical causal averaging (Vervoort, 2014). In formal metaphysics and philosophy of physics, it denotes determinism of some observables, subsystems, or event-types even when the total history is not uniquely fixed (Adlam, 2021).

The constraint-based framework of “holistic determinism” makes this selective structure explicit. If laws determine a solution set $S=\bigcap_{\ell\in L} C_\ell$ of admissible Humean mosaics, then an observable $O$ is determined by the laws iff $\forall m_1,m_2\in S,\; O(m_1)=O(m_2)$. A subsystem or spacetime region is likewise partially determined when the relevant restriction map is invariant across $S$. This permits weak holistic determinism—more than one admissible mosaic, but no objective probability distribution over them—together with lawlike fixation of selected features (Adlam, 2021).

A closely related model-theoretic formulation treats determinism as a property of theories rather than possible worlds. Let $\pi:S\to S'$ be a projection onto the variables or features of interest. Then partial determinism is determinism relative to $\pi$: weak when agreement extends at the $\pi$-level in the sense of Belot’s D1, and strong when the extension is unique in the sense of D3. On this view, qualitative determinism is already a form of coarse-grained or restricted determinism, because it fixes structure only up to isomorphism rather than up to name-preserving identity (Halvorson et al., 7 Mar 2025).

2. Deterministic dynamics with bounded predictability

Computer performance provides a concrete operationalization. The central claim is that computers, viewed at the level of their microarchitectural execution, are deterministic nonlinear dynamical systems; Takens’ embedding theorem then implies the existence of deterministic forecast rules for a scalar performance trace under smoothness and genericity assumptions. Yet deterministic forecast rules sometimes fail. The proposed explanation is that complexity can effectively overwhelm the predictive power of deterministic forecast models, producing a form of partial determinism: deterministic traces whose practical predictability is bounded by temporal complexity (Garland et al., 2013).

The paper studies L2 cache-miss rate and instructions per cycle (IPC) for two microkernels, row_major and col_major, and two SPEC CPU2006 applications, 403.gcc and 482.sphinx3. Forecasting uses delay-coordinate embedding with $\tau$ chosen as the first minimum of average mutual information, $\delta^k$0 chosen via the false-nearest-neighbor method with a 10% threshold, and the Lorenz method of analogues as a 1-nearest-neighbor predictor. The holdout protocol uses the last $\delta^k$1 samples. Prediction quality degrades monotonically with normalized permutation entropy: col_major cache has nRMSE $\delta^k$2 and row_major cache $\delta^k$3, while 403.gcc cache has $\delta^k$4 and 482.sphinx3 cache $\delta^k$5; for IPC the corresponding values are $\delta^k$6, $\delta^k$7, $\delta^k$8, and $\delta^k$9. The paper identifies two practical thresholds: persistent PE $S=\bigcap_{\ell\in L} C_\ell$0 marks regimes in which deterministic forecasts degrade markedly, whereas PE $S=\bigcap_{\ell\in L} C_\ell$1 marks high predictability (Garland et al., 2013).

The same theme appears in the family of non-chaotic $S=\bigcap_{\ell\in L} C_\ell$2 interval maps. For every finite horizon $S=\bigcap_{\ell\in L} C_\ell$3, determinism equals $S=\bigcap_{\ell\in L} C_\ell$4 for sufficiently small thresholds, so trajectories are perfectly predictable over finite horizons. At infinite horizon, however, determinism is strictly submaximal: the liminf lies in $S=\bigcap_{\ell\in L} C_\ell$5, for $S=\bigcap_{\ell\in L} C_\ell$6 the limsup equals $S=\bigcap_{\ell\in L} C_\ell$7, and for $S=\bigcap_{\ell\in L} C_\ell$8 the limsup is strictly less than $S=\bigcap_{\ell\in L} C_\ell$9. The paper therefore exhibits partial determinism without chaos in the usual sense: perfect finite-horizon predictability coexisting with only partial infinite-horizon predictability (Majerová, 2015).

3. Probability, quantum theory, and free choice

In the probabilistic and quantum-mechanical literature, partial determinism is a layered framework rather than a claim of unrestricted predictability. The central thesis is that microdynamics may be deterministic while probabilities at the macro/experimental level arise from frequency stabilization under repeatable experimental conditions. In Bell scenarios this allows deterministic hidden-variable models once one scrutinizes measurement independence, $O$0, rather than treating indeterminism as forced by experiment (Vervoort, 2014).

Within this framework, determinism means that individual outcomes are functions of hidden variables, $O$1 and $O$2, together with local factorizability $O$3. Supercorrelation provides a non-conspiratorial mechanism for violating measurement independence: $O$4 is associated with a local background medium or field that interacts locally with analyzers and particles, so $O$5 can arise through local analyzer–background interactions. The spin-lattice model gives a concrete example: a 10-spin square lattice with Ising Hamiltonian yields $O$6 while preserving locality in the Clauser–Horne sense and violating measurement independence (Vervoort, 2014).

The same paper links this to an objective frequency interpretation of probability. Probability attaches to repeatable experiments with specified initiating and probing conditions; conditional probabilities can be realized as automated experiments; and the law of large numbers and central limit theorem explain stable frequencies and Gaussian laws as emergent from deterministic causes. Partial determinism here means deterministic microphysics plus emergent, objective probabilities at the macro level (Vervoort, 2014).

A more recent deterministic model of free will pushes the same theme into decision theory and quantum foundations. It argues for determinism without Big-Bang predestination by replacing all-at-once initialization with just-in-time initialization of sub-Planck information and by introducing deterministic impact control through a map $O$7. The parameter $O$8 modulates the impact of low-level information on a classical decision bit. The model claims that violating Measurement Independence does not invalidate the free-will conclusion: $O$9 can hold through contextual rationality constraints rather than conspiratorial fixing of settings at the Big Bang (Palmer, 1 Apr 2025).

4. Restricted nondeterminism in computability, automata, and multirelations

In computability theory, partial determinism is formalized as the power of partially defined but deterministic gambling strategies. A partial martingale is a partial function whose domain is prefix-closed and satisfies the martingale fairness condition whenever both extensions are defined. Partial computable randomness is the property that no such partial computable martingale succeeds. The key separation result shows that there exists a sequence $\forall m_1,m_2\in S,\; O(m_1)=O(m_2)$0 that is partial computably random but not almost everywhere computably random; probabilistic computable strategies can therefore strictly outperform deterministic partial gamblers (Bienvenu et al., 2021).

Automata theory develops an explicit hierarchy between full determinism and unrestricted nondeterminism. An automaton is determinizable by pruning (DBP) if an equivalent deterministic automaton can be obtained by deleting transitions; it is history deterministic (HD) if nondeterministic choices can be resolved online as a function of the past; and it is semantically deterministic (SD) if all successor choices are language-equivalent. For automata on finite words, the three levels coincide. For Büchi, co-Büchi, and weak automata on infinite words, the hierarchy is strict, although for Büchi and weak acceptance the semantic hierarchy collapses to deterministic expressive power: $\forall m_1,m_2\in S,\; O(m_1)=O(m_2)$1 for $\forall m_1,m_2\in S,\; O(m_1)=O(m_2)$2 (Radi et al., 2022).

Binary multirelations provide a two-level algebraic version of the same idea. A multirelation $\forall m_1,m_2\in S,\; O(m_1)=O(m_2)$3 supports an outer, angelic choice of successor sets and an inner, demonic choice of elements within a chosen set. Partial determinism arises when one of these levels is functional and the other is not. The paper distinguishes inner deterministic, outer deterministic, inner univalent, and outer univalent classes, proves that deterministic classes form categories under Peleg composition, and introduces fusion and fission as determinisation maps that approximate general multirelations either by binary relations or by deterministic multirelations (Furusawa et al., 2023).

5. Space-time, asymmetry, and theory-relative determinism

General relativity exhibits partial determinism in a literal spacetime sense. Determinism holds on the domain of dependence $\forall m_1,m_2\in S,\; O(m_1)=O(m_2)$4 of a Cauchy surface $\forall m_1,m_2\in S,\; O(m_1)=O(m_2)$5, where the Einstein equations determine a unique maximal globally hyperbolic development, but it can fail at or beyond a Cauchy horizon. Taub–NUT spacetime is the standard example: the Taub region is globally hyperbolic and deterministic, whereas beyond the Cauchy horizon there are multiple smooth extensions and causality breaks down. Gödel spacetime displays a stronger failure, with closed timelike curves through every point and no nontrivial region in which experimental outcomes are predictably fixed by Cauchy data (Isenberg, 2016).

Recent work sharpens this by grading determinism relative to collections of spacetimes. For a class $\forall m_1,m_2\in S,\; O(m_1)=O(m_2)$6 of globally hyperbolic, time-oriented spacetimes, de dicto determinism requires that isometric initial segments imply global isometry; de re determinism additionally requires the global isometry to extend the given local one; and de re⋆ determinism requires uniqueness of that extension. In standard GR, rigidity collapses de re and de re⋆: any $\forall m_1,m_2\in S,\; O(m_1)=O(m_2)$7 is rigid, and any $\forall m_1,m_2\in S,\; O(m_1)=O(m_2)$8—the class of four-dimensional, inextendible, globally hyperbolic, vacuum solutions—is de re⋆ deterministic. Stronger asymmetry conditions, “giraffe” and “Heraclitus,” generate stronger forms de dicto⋆ and de dicto⋆⋆/de re⋆⋆, showing that determinism in GR is relative both to model class and to symmetry structure (Manchak et al., 7 Mar 2025).

This spacetime-relative picture connects back to the more abstract frameworks. In constraint-based language, weak holistic determinism allows multiple admissible mosaics with no objective chance over them, while delocalised holistic determinism forbids differences confined to a small spacetime subregion (Adlam, 2021). In model-theoretic language, the same phenomenon appears as uniqueness of extension relative to a chosen signature, projection, or morphism class (Halvorson et al., 7 Mar 2025).

6. Engineered computation and AI systems

In systems research, partial determinism is usually an engineering objective: enforce determinism at well-defined boundaries while allowing controlled nondeterminism elsewhere. “Deterministic execution” in Determinator means that a given program, run with the same inputs, yields exactly the same outputs every time, without internal event logging. The kernel provides only single-threaded, shared-nothing spaces with three synchronization primitives—Put, Get, and Ret—and only the root space can access nondeterministic I/O. The user-level runtime emulates Unix-like processes, a replicated file system, and shared-memory multithreading through deterministic consistency. For legacy pthreads, deterministic scheduling quantizes execution and orders synchronization operations, yielding a form of partial determinism: repeatable outcomes via deterministic boundaries rather than unconstrained shared-memory interleaving (Aviram et al., 2010).

The programming-languages survey generalizes this strategy. A parallel program is deterministic when its observable result does not depend on the schedule, $\forall m_1,m_2\in S,\; O(m_1)=O(m_2)$9. Partial determinism is the property that identifies the conditions under which execution yields a deterministic result even if the language or runtime permits nondeterminism elsewhere. The survey emphasizes Kahn process networks, synchronous languages, actors with FIFO mailboxes and blocking future reads, BSP supersteps, and type/effect systems of the form $S$0, where $S$1, as programming models that localize or control nondeterminism (Gonnord et al., 2022).

Current AI systems reintroduce the issue at scale. In code generation, non-determinism is operationalized as inconsistency in code candidates generated in different requests with identical prompts. Across 829 problems from CodeContests, APPS, and HumanEval, the ratio of tasks with zero equal test output across different requests is 75.76%, 51.00%, and 47.56%, respectively. Setting temperature to $S$2 reduces but does not eliminate variability; on CodeContests, OER mean rises from $S$3 at $S$4 to $S$5 at $S$6, yet 16.36% of tasks still have OER $S$7. The result is explicitly described as partial determinism: variability is attenuated compared to $S$8, but not zero (Ouyang et al., 2023).

At inference-system level, LLM-42 addresses the same problem by “verified speculation.” The fast path remains non-deterministic under dynamic batching, but a verifier replays candidate tokens under a fixed-shape reduction schedule, commits tokens guaranteed to be consistent across runs, overwrites the corresponding KV cache region with verifier KV, and rolls back mismatching suffixes. Because most kernels are shape-consistent, this yields selective determinism for requests that set is_deterministic=True, while overhead scales with the fraction of traffic requiring determinism (Gond et al., 25 Jan 2026).

Agentic AI introduces a distinct but related notion. Here partial determinism is the regime in which an environment has per-step success probability $S$9 rather than $\pi:S\to S'$0. Under independent steps and no retries, chain success is $\pi:S\to S'$1; with retries it becomes $\pi:S\to S'$2. The paper therefore treats environment determinism as a binding axis for long-chain execution and operationalizes it through a Supply Certainty Index and a Determinism Maturity Model (Ding et al., 21 Jun 2026).

7. Biological autonomy and recurring controversies

Biology supplies a version of partial determinism centered on constrained autonomy. The “twilight of determinism” argument holds that organisms and environments co-determine one another “in a nonlinear way,” and that controlled biophysical systems display “relative autonomy and flexibility in response which could not be predicted.” The genome still matters, but as a constraint rather than as a program: it provides “a set of constraints on the spectrum of regulatory modes, analogous to boundary conditions in physical dynamical systems.” Rapid adaptation in yeast, critical fluctuations and long memory in neuronal systems, and many-to-one and one-to-many relations among levels of organization are presented as evidence that outcomes are constrained yet not uniquely determined (Gilead, 2015).

Across fields, the recurrent controversy is whether partial determinism should be read as disguised indeterminism. The literature does not support a single answer. In some domains it is a limit on predictability rather than on causation: deterministic performance traces can be practically unpredictable because entropy, dimension, nonlinearity, nonstationarity, finite data, noise, and aggregation overwhelm simple predictors (Garland et al., 2013). In others it is a claim about layered description: deterministic microphysics with emergent macro-probabilities (Vervoort, 2014). In still others it is a formal weakening of uniqueness, as in D1-relative determinism, de dicto determinism, or determinism of only selected observables (Halvorson et al., 7 Mar 2025, Manchak et al., 7 Mar 2025, Adlam, 2021).

A persistent misconception is that partial determinism is merely a compromise between determinism and randomness. The cited work suggests a more precise picture. Partial determinism can mean deterministic state evolution but limited forecastability; deterministic hidden variables plus violated measurement independence; deterministic behavior enforced at synchronization points but not at every microstep; deterministic semantics relative to a projection, subsystem, or region; or constrained autonomy within a feasible repertoire. Its unifying feature is not vagueness, but the restriction of determinism to a specified scale, horizon, observable class, model class, or operational interface (Garland et al., 2013, Vervoort, 2014, Gonnord et al., 2022, Gilead, 2015).