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ACM Transactions on Embedded Computing Systems

Determinism

Edward A. Lee

Published May 29, 2021• 28 citations
Publisher

Abstract

This article is about deterministic models, what they are, why they are useful, and what their limitations are. First, the article emphasizes that determinism is a property of models, not of physical systems. Whether a model is deterministic or not depends on how one defines the inputs and behavior of the model. To define behavior, one has to define an observer. The article compares and contrasts two classes of ways to define an observer, one based on the notion of “state” and another that more flexibly defines the observables. The notion of “state” is shown to be problematic and lead to nondeterminism that is avoided when the observables are defined differently. The article examines determinism in models of the physical world. In what may surprise many readers, it shows that Newtonian physics admits nondeterminism and that quantum physics may be interpreted as a deterministic model. Moreover, it shows that both relativity and quantum physics undermine the notion of “state” and therefore require more flexible ways of defining observables. Finally, the article reviews results showing that sufficiently rich sets of deterministic models are incomplete. Specifically, nondeterminism is inescapable in any system of models rich enough to encompass Newton’s laws.

DOI
10.1145/3453652

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This article is about deterministic models, what they are, why they are useful, and what their limitations are. First, the article emphasizes that determinism is a property of models, not of physical systems. Whether a model is deterministic or not depends on how one defines the inputs and behavior of the model. To define behavior, one has to define an observer. The article compares and contrasts two classes of ways to define an observer, one based on the notion of “state” and another that more flexibly defines the observables. The notion of “state” is shown to be problematic and lead to nondeterminism that is avoided when the observables are defined differently. The article examines determinism in models of the physical world. In what may surprise many readers, it shows that Newtonian physics admits nondeterminism and that quantum physics may be interpreted as a deterministic model. Moreover, it shows that both relativity and quantum physics undermine the notion of “state” and therefore require more flexible ways of defining observables. Finally, the article reviews results showing that sufficiently rich sets of deterministic models are incomplete. Specifically, nondeterminism is inescapable in any system of models rich enough to encompass Newton’s laws.
This article focuses on three themes concerning determinism and indeterminism. The first is observational equivalence between deterministic and indeterministic models. The article discusses several results about observational equivalence and presents an argument on how to choose between deterministic and indeterministic models involving indirect evidence. The second is whether Newtonian physics is indeterministic. The article argues that the answer depends on what one takes Newtonian mechanics to be and highlights how contemporary debates on this issue differ from those in the nineteenth century. The third theme is how the method of arbitrary functions can be used to make sense of deterministic probabilities. The article discusses various ways of interpreting the initial probability distributions and argues that they are best understood as physical, biological, and other quantities characterizing the particular situation. The fact that the method of arbitrary functions deserves more attention than it has received so far is also emphasized.
The purpose of this paper is to give a brief survey the implications of the theories of modern physics for the doctrine of determinism. The survey will reveal a curious feature of determinism: in some respects it is fragile, requiring a number of enabling assumptions to give it a fighting chance; but in other respects it is quite robust and very difficult to kill. The survey will also aim to show that, apart from its own intrinsic interest, determinism is an excellent device for probing the foundations of classical, relativistic, and quantum physics. The survey is conducted under three major presuppositions. First, I take a realistic attitude towards scientific theories in that I assume that to give an interpretation of a theory is, at a minimum, to specify what the world would have to be like in order for the theory to be true. But we will see that the demand for a deterministic interpretation of a theory can force us to abandon a naively realistic reading of the theory. Second, I reject the “no laws” view of science and assume that the field equations or laws of motion of the most fundamental theories of current physics represent science’s best guesses as to the form of the basic laws of nature. Third, I take determinism to be an ontological doctrine, a doctrine about the temporal evolution of the world. This ontological doctrine must not be confused with predictability, which is an epistemological doctrine, the failure of which need not entail a failure of determinism. From time to time I will comment on ways in which predictability can fail in a deterministic setting. Finally, my survey will concentrate on the Laplacian variety of determinism according to which the instantaneous state of the world at any time uniquely determines the state at any other time. The plan of the survey is as follows. Section 2 illustrates the fragility of determinism by means of a Zeno type example. Then sections 3 and 4 survey successively the fortunes of determinism in the Newtonian and the special relativistic settings..
Reliable verification techniques for quantum communication protocols are of paramount importance, given their high implementation cost and critical contexts of application. Extensions of process calculi have been proposed, together with various notions of behavioural equivalence. However, their standard probabilistic models turn out to introduce some non-deterministic capabilities not aligned with the observational properties of physical quantum systems, leading to bisimilarity notions that distinguish physically equivalent processes. Nonetheless, non-deterministic features are fundamental to account for inputs, environments and adversarial behaviour. To address this issue, we propose lqCCS, a process calculus that integrates concurrency, non-determinism and quantum capabilities. We introduce a novel semantics in terms of distributions, where explicit physically admissible schedulers constrain probabilistic composition and forbid ill-defined non-deterministic moves, while preserving the expressivity needed to model real-world protocols. We investigate a scheduled version of saturated bisimilarity, pairing two processes if no observer can tell them apart, and we verify its adequacy by lifting a known result from quantum mechanics to lqCCS: equivalent processes acting on indistinguishable mixtures of quantum states are correctly recognized as bisimilar. Finally, we give an alternative semantics and a labelled bisimilarity based on a quantum generalization of probability distributions. This characterizes our behavioural equivalence as a congruence with respect to the parallel operator, enabling compositional reasoning without the need to explicitly check all possible contexts. We describe a rich class of lqCCS processes for which equivalence is decidable using standard techniques, and we analyse real-world quantum communication protocols.
Defining the Observer: A General Physical Criterion, Its Relation to Existing Models, Impossibility of the final ascent, and the Impossibility of the Ideal Limit. No treatment of quantum gravity that takes observers seriously can avoid the question of what an observer physically is, and existing treatments - including the author's own (Materov 2026a,b,c) - each state a modelling assumption for the observer without deriving it. This paper's contribution is narrow and specific, not a claim to be the first physical account of an observer: a particular combination of requirements (D1–D4) on the trajectory of an embedded tensor factor, a separately-argued self-capacity criterion for individuation (D3′) tied to the no-cloning theorem, and a short, self-contained no-go argument (Proposition 4.1) for complete reconstruction of an observer's complement. We show that the observer-notions already in use in Materov (2026a,b,c) relate to this combination in three structurally different ways - as a more general object, as a concrete realization, and as a structurally parallel system not yet unified with it - and work out one concrete case, the photonic Friend of Local-Friendliness experiments, in detail. Since no observer can be ideal in this omniscient sense, we instead investigate uniqueness for a weaker, separately-defined notion (a self-dominant observer, whose own representational capacity exceeds its observing capacity); no dimension-counting argument we have found forces this either, and the question remains, honestly, open.