arXiv subjects

David Wolpert

Publications and source records attributed to David Wolpert.

At least 19 recordsLinked to original sources

arXiv2026

Entropy production bounds for systems running computer programs

Mismatch cost (MMC) is a universally applicable lower bound on the entropy production (EP) of any fixed physical process across a given time interval. In the first part of the paper, we establish results concerning MMC to prove that it scales at least linearly with the total heat flow in the worst case over initial distributions. We also prove that the MMC lower bound over a given time interval never decreases if the time interval is subdivided into a sequence of sub-intervals, and that the bound often increases. In the second part of the paper, we introduce a general framework for computing the minimal EP (i.e., the MMC) associated with running a computer program on any physical system that implements a modern digital computer. We apply this general framework to compare MMC of running two canonical sorting algorithms, bubble sort and bucket sort. The framework enables us to investigate how thermodynamic cost depends on features like input size and structure (e.g., with or without repeated entries). Finally, we extend the framework to programs that call subroutines.

cond-mat.stat-mech↗

arXiv2026

Stochastic Thermodynamics of Associative Memory

Dense Associative Memory networks (DenseAMs) unify several popular paradigms in Artificial Intelligence (AI), such as Hopfield Networks, transformers, and diffusion models, while casting their computational properties into the language of dynamical systems and energy landscapes. This formulation provides a natural setting for studying thermodynamics and computation in neural systems, because DenseAMs are simultaneously simple enough to admit analytic treatment and rich enough to implement nontrivial computational function. Aspects of these networks have been studied at equilibrium and at zero temperature, but the thermodynamic costs associated with their operation out of equilibrium are largely unexplored. Here, we define the thermodynamic entropy production associated with the operation of such networks, and study polynomial DenseAMs at intermediate memory load. At large system sizes and intermediate and low load, we use dynamical mean field theory to characterize out-of-equilibrium properties, work requirements, and memory transition times when driving the system with corrupted memories. We characterize a failure mode of higher order networks not observed at zero temperature. Further, we develop a method for calculating work and power costs in the mean field limit. Finally, we find tradeoffs between entropy production, memory retrieval accuracy, and operation speed.

cond-mat.stat-mech↗

arXiv2026

Unveiling hidden features of social evolution by inferring Langevin dynamics from data

Are there hidden dynamical common patterns in the evolution of social and cultural history? While the growing availability of digitized social data invites us to answer this question, prevailing quantitative methods often rely on deterministic snapshots or average effects. Such approaches overlook the continuous and inherently uncertain nature of historical trajectories. In this paper, we propose a framework for modeling historical dynamics as stochastic processes described by stochastic differential equations (SDEs). By viewing historical change through the lens of continuous-time dynamics, this framework provides a natural language to describe how structural trends and inherent random fluctuations interact to shape societal evolution. This approach allows us to handle the uncertainty in fragmentary historical records, moving beyond the dichotomy of structural determinism versus pure chance. We demonstrate that adopting this stochastic perspective unlocks a rich suite of analytical capabilities unavailable to static models. Specifically, we introduce methods to: (1) quantify the irreversibility; (2) detect exogenous perturbations; (3) perform multiple imputation for missing historical records. This framework offers a unified methodology for dissecting the stability, contingency, and dynamics of historical change.

cs.SI↗

arXiv2026

Disentangling Boltzmann brains, the time-asymmetry of memory, and the second law

Are your perceptions, memories and observational data, a statistical fluctuation out of the thermal equilibrium of the universe, having no correlation with the actual past state of the universe? Arguments are given in the literature for and against this "Boltzmann brain" hypothesis. Complicating these arguments have been the many subtle -- and very often implicit -- joint dependencies among these arguments and others that have been given for the past hypothesis, the second law, and even for Bayesian inference of the reliability of experimental data. These dependencies can easily lead to circular reasoning. To avoid this problem, since all of these arguments involve the stochastic properties of the dynamics of the universe's entropy, we begin by formalizing that dynamics as a time-symmetric, time-translation invariant Markov process, which we call the entropy conjecture. Crucially, like all stochastic processes, the entropy conjecture does not specify any time(s) which it should be conditioned on in order to infer the stochastic dynamics of our universe's entropy. Any such choice of conditioning times and associated entropy values must be introduced as an independent assumption. This observation allows us to disentangle the standard Boltzmann brain hypothesis, its "1000CE" variant, the past hypothesis, the second law, and the reliability of our experimental data, all in a fully formal manner. In particular, we show that these all make an arbitrary assumption that the dynamics of the universe's entropy should be conditioned on a single event at a single moment in time, differing only in the details of their assumptions. In this aspect, the Boltzmann brain hypothesis and the second law are equally legitimate (or not).

physics.hist-ph↗

arXiv2025

Non-coercive extortion in game theory

Commitments play a crucial role in game theory, shaping strategic interactions by either altering a player's own payoffs or influencing the incentives of others through outcome-contingent payments. While most research has focused on using commitments to achieve efficient equilibria, their potential applications beyond this goal remain largely unexplored. In this study, we introduce a non-coercive extortion mechanism that leverages commitments to outcome-contingent payments, demonstrating how a player or external agent can extract profit by offering rewards rather than threatening punishment. At the core of the mechanism is the introduction of sequentiality into a simultaneous-move game, fundamentally reshaping the strategic interaction. We derive the conditions under which extortion is successful, identify the class of games susceptible to this scheme, and determine both the maximum extractable profit and the minimum required payment. To illustrate the extortion mechanism, we apply it to 2x2 games, highlighting how even simple strategic settings can be vulnerable to this form of manipulation. Our results reveal strategic vulnerabilities in competitive settings, with significant implications for economic markets, diplomatic relations, and multi-agent systems operating in blockchain environments. This work broadens our understanding of commitments in game theory and raises critical questions about how to safeguard strategic interactions from exploitation through non-coercive extortion.

cs.GT↗

arXiv2025

Minimal thermodynamic cost of communication

Thermodynamic cost of communication is a major factor in the thermodynamic cost of real-world computers, both biological and digital. Despite its importance, the fundamental principles underlying this cost remain poorly understood. This paper makes two major contributions to addressing this gap. First, we derive a universal relationship between information transmission rate and minimal entropy production (EP) by focusing on the mismatch cost (MMC) component of thermodynamic cost. The resulting relationship holds independently of the underlying physical dynamics, making it broadly applicable. We discuss the implications of the derived minimal communication cost for work extraction in measurement-and-feedback protocols, and through examples involving binary channels, we show that the relationship between transmission rate and minimal thermodynamic cost can exhibit diminishing returns in certain scenarios. Second, we extend this thermodynamic analysis to the computational front and back ends critical to communication-namely, encoding and decoding to reduce errors in noisy transmission. Using the framework of periodic machines, we establish strictly positive minimal costs for implementing linear error-correcting codes. We compare these costs with end-to-end error rates, highlighting trade-offs between thermodynamic cost and decoding accuracy.

cond-mat.stat-mech↗

arXiv2025

A strictly positive lower bound on the thermodynamic cost of running a Boolean circuit

All digital computers implement input-output functions using logic gates connected into circuits. Different circuits computing the same function may nevertheless incur different resource costs, and circuit complexity theory studies these costs through measures such as the number of gates and the length of the longest path from input to output. Energetic cost is another important resource, however, that is typically not included among these measures. To address this, we use mismatch cost (MMC): a nonnegative contribution to the entropy production of a process that can be characterized largely independently of the detailed physical implementation of that process, providing a natural way to analyze the thermodynamic cost of Boolean circuits at an abstract, computational level. We derive an expression for the MMC of circuits composed of Boolean gates, relate it to standard complexity measures such as circuit size and depth, and use it to define mismatch cost complexity as a measure of thermodynamic resource cost. For Boolean circuits computing a non-constant Boolean function, this expression also implies a strictly positive MMC for every input distribution, and therefore a strictly positive lower bound on total entropy production. We characterize when MMC scales linearly with circuit size and when it does not, and compare the MMC of different circuit families that compute the same Boolean function. Together, these results lay the foundation for treating mismatch cost as a resource within circuit complexity theory.

cond-mat.stat-mech↗

arXiv2025

Maximizing free energy gain

Maximizing the amount of work harvested from an environment is important for a wide variety of biological and technological processes, from energy-harvesting processes such as photosynthesisto energy storage systems such as fuels and batteries. Here we consider the maximization of free energy -- and by extension, the maximum extractable work -- that can be gained by a classical or quantum system that undergoes driving by its environment. We consider how the free energy gain depends on the initial state of the system, while also accounting for the cost of preparing the system. We provide simple necessary and sufficient conditions for increasing the gain of free energy by varying the initial state. We also derive simple formulae that relate the free energy gained using the optimal initial state rather than another suboptimal initial state. Finally, we demonstrate that the problem of finding the optimal initial state may have two distinct regimes, one easy and one difficult, depending on the temperatures used for preparation and work extraction. We illustrate our results on a simple model of an information engine.

cond-mat.stat-mech↗

arXiv2024

Work extraction with feedback control using limited resources

Many physical, biological, and even social systems are faced with the problem of how to efficiently harvest free energy from an environment that can have many possible states, yet only have a limited number of harvesting protocols to choose among. We investigate this scenario by extending earlier work on using feedback control to extract work from nonequilibirum systems. Specifically, in contrast to that previous work on the thermodynamics of feedback control, we analyze the combined and separate effects of noisy measurements, memory limitations, and limitations on the number of possible work extraction protocols. Our analysis provides a general recipe to construct repertoires of allowed harvesting protocols that minimize the expected thermodynamic losses during free energy harvesting, i.e., that minimize expected entropy production. In particular, our results highlight that the benefits of feedback control over uninformed (random) actions extend beyond just the associated information gain, often by many orders of magnitude. Our results also uncover the effects of limitations on the number of possible harvesting protocols when there is uncertainty about the distribution over states of the environment.

cond-mat.stat-mech↗

arXiv2024

Stochastic Process Turing Machines

Computer science theory provides many different measures of complexity of a system including Kolmogorov complexity, logical depth, computational depth, and Levin complexity. However, these measures are all defined only for deterministic Turing machines, i.e., deterministic dynamics of the underlying generative process whose output we are interested in. Therefore, by construction they cannot capture complexity of the output of stochastic processes - like those in the real world. Motivated by this observation, we combine probabilistic Turing machines with a prior over the inputs to the Turing machine to define a complete stochastic process of Turing machines. We call this a stochastic process Turing machine. We use stochastic process Turing machines to define a set of new generative complexity measures based on Turing machines, which we call stochastic depth. As we discuss, stochastic depth is related to other such measures including Kolmogorov complexity and Levin complexity. However, as we elaborate, it has many desirable properties that those others measures lack. In addition, stochastic depth is closely related to various thermodynamic properties of computational systems. Stochastic process Turing machines and stochastic depth allow us to study complex, stochastic systems like the human brain, societies, and evolution all from within the framework of formal computation.

cs.CC↗

arXiv2024

Is stochastic thermodynamics the key to understanding the energy costs of computation?

The relationship between the thermodynamic and computational characteristics of dynamical physical systems has been a major theoretical interest since at least the 19th century, and has been of increasing practical importance as the energetic cost of digital devices has exploded over the last half century. One of the most important thermodynamic features of real-world computers is that they operate very far from thermal equilibrium, in finite time, with many quickly (co-)evolving degrees of freedom. Such computers also must almost always obey multiple physical constraints on how they work. For example, all modern digital computers are periodic processes, governed by a global clock. Another example is that many computers are modular, hierarchical systems, with strong restrictions on the connectivity of their subsystems. This properties hold both for naturally occurring computers, like brains or Eukaryotic cells, as well as digital systems. These features of real-world computers are absent in 20th century analyses of the thermodynamics of computational processes, which focused on quasi-statically slow processes. However, the field of stochastic thermodynamics has been developed in the last few decades - and it provides the formal tools for analyzing systems that have exactly these features of real-world computers. We argue here that these tools, together with other tools currently being developed in stochastic thermodynamics, may help us understand at a far deeper level just how the fundamental physical properties of dynamic systems are related to the computation that they perform.

cond-mat.stat-mech↗

arXiv2024

Boltzmann Bridges

It is often stated that the second law of thermodynamics follows from the condition that at some given time in the past the entropy was lower than it is now. Formally, this condition is the statement that $E[S(t)|S(t_0)]$, the expected entropy of the universe at the current time $t$ conditioned on its value $S(t_0)$ at a time $t_0$ in the past, is an increasing function of $t $. We point out that in general this is incorrect. The epistemic axioms underlying probability theory say that we should condition expectations on all that we know, and on nothing that we do not know. Arguably, we know the value of the universe's entropy at the present time $t$ at least as well as its value at a time in the past, $t_0$. However, as we show here, conditioning expected entropy on its value at two times rather than one radically changes its dynamics, resulting in a unexpected, very rich structure. For example, the expectation value conditioned on two times can have a maximum at an intermediate time between $t_0$ and $t$, i.e., in our past. Moreover, it can have a negative rather than positive time derivative at the present. In such "Boltzmann bridge" situations, the second law would not hold at the present time. We illustrate and investigate these phenomena for a random walk model and an idealized gas model, and briefly discuss the role of Boltzmann bridges in our universe.

cond-mat.stat-mech↗

arXiv2024

Why you do not need to worry about the standard argument that you are a Boltzmann brain

Are you, with your perceptions, memories and observational data, a Boltzmann brain, namely a fleeting statistical fluctuation out of the thermal equilibrium of the universe? Arguments are given in the literature claiming that this bizarre hypothesis needs to be considered seriously, that all of our data about the past is actually a mirage. We point to a difficulty in these arguments. They are based on the dynamical laws and on statistical arguments, but they disregard the fact that we infer the dynamical laws presupposing the reliability of our data records about the past. Hence the reasoning in favor of the Boltzmann brain hypothesis contradicts itself, relying on the reliability of our data about the past to conclude that that data is wrong. More broadly, it is based on incomplete evidence. Incomplete evidence notoriously leads to false conclusions.

physics.hist-ph↗

arXiv2024

Thermodynamics of computations with absolute irreversibility, unidirectional transitions, and stochastic computation times

Developing a thermodynamic theory of computation is a challenging task at the interface of non-equilibrium thermodynamics and computer science. In particular, this task requires dealing with difficulties such as stochastic halting times, unidirectional (possibly deterministic) transitions, and restricted initial conditions, features common in real-world computers. Here, we present a framework which tackles all such difficulties by extending the martingale theory of non-equilibrium thermodynamics to generic non-stationary Markovian processes, including those with broken detailed balance and/or absolute irreversibility. We derive several universal fluctuation relations and second-law-like inequalities that provide both lower and upper bounds on the intrinsic dissipation (mismatch cost) associated with any periodic process -- in particular the periodic processes underlying all current digital computation. Crucially, these bounds apply even if the process has stochastic stopping times, as it does in many computational machines. We illustrate our results with exhaustive numerical simulations of deterministic finite automata (DFA) processing bit strings, one of the fundamental models of computation from theoretical computer science. We also provide universal equalities and inequalities for the acceptance probability of words of a given length by a deterministic finite automaton in terms of thermodynamic quantities, and outline connections between computer science and stochastic resetting. Our results, while motivated from the computational context, are applicable far more broadly.

cond-mat.stat-mech↗

arXiv2023

Game Manipulators -- the Strategic Implications of Binding Contracts

Commitment devices are powerful tools that can influence and incentivise certain behaviours by linking them to rewards or punishments. These devices are particularly useful in decision-making, as they can steer individuals towards specific choices. In the field of game theory, commitment devices can alter a player's payoff matrix, ultimately changing the game's Nash equilibria. Interestingly, agents, whom we term game manipulators and who can be external to the original game, can leverage such devices to extract fees from players by making them contingent offers that modify the payoffs of their actions. This can result in a different Nash equilibrium with potentially lower payoffs for the players compared to the original game. For this scheme to work, it is required that all commitments be binding, meaning that once an offer is made, it cannot be revoked. Consequently, we analyse binding contracts as the commitment mechanism that enables game manipulation scenarios. The main focus of this study is to formulate the logic of this setting, expand its scope to encompass more intricate schemes, and analyse the behaviour of regret-minimizing agents in scenarios involving game manipulation.

cs.GT↗

arXiv2022

Inclusive Thermodynamics of Computational Machines

We introduce a framework designed to analyze the thermodynamics of an abstractly defined logical computer like a deterministic finite automaton (DFA) or a Turing machine, without specifying any extraneous parameters (like rate matrices, Hamiltonians, etc.) of a physical process that implements the computer. Earlier investigations of how to do this were based on the continuous-time Markov chain (CTMC) formulation of stochastic thermodynamics. These investigations either assumed that there was exactly zero irreversible entropy production (EP) generated by the physical system implementing the computation, or allowed the EP to be nonzero but only considered the mismatch cost component of the EP. In addition, they only applied to a single type of computer. Our framework neither requires that EP equal zero nor restricts attention to the mismatch cost component of EP, and is designed to apply to all types of computational machines. In contrast to earlier investigations using the CTMC-based formulation, our framework is based on the inclusive Hamiltonian formulation, in which the combination of the system of interest and the baths evolve in a Hamiltonian (or unitary) dynamics. Here, we use our framework to derive an integral fluctuation theorem for computers, in which the expectation value is strictly less than 1. We also derive an exchange fluctuation theorem, and a mismatch cost formula involving first-passage times. We analyze the EP generated by a DFA, a Markov information source, and a noisy communication channel. In particular, we use the Myhill-Nerode theorem of computer science to prove that out of all DFAs which recognize the same language, the minimal complexity DFA is the one with minimal EP for all dynamics and at all iterations.

cond-mat.stat-mech↗

arXiv2019

Thermodynamic Computing

The hardware and software foundations laid in the first half of the 20th Century enabled the computing technologies that have transformed the world, but these foundations are now under siege. The current computing paradigm, which is the foundation of much of the current standards of living that we now enjoy, faces fundamental limitations that are evident from several perspectives. In terms of hardware, devices have become so small that we are struggling to eliminate the effects of thermodynamic fluctuations, which are unavoidable at the nanometer scale. In terms of software, our ability to imagine and program effective computational abstractions and implementations are clearly challenged in complex domains. In terms of systems, currently five percent of the power generated in the US is used to run computing systems - this astonishing figure is neither ecologically sustainable nor economically scalable. Economically, the cost of building next-generation semiconductor fabrication plants has soared past $10 billion. All of these difficulties - device scaling, software complexity, adaptability, energy consumption, and fabrication economics - indicate that the current computing paradigm has matured and that continued improvements along this path will be limited. If technological progress is to continue and corresponding social and economic benefits are to continue to accrue, computing must become much more capable, energy efficient, and affordable. We propose that progress in computing can continue under a united, physically grounded, computational paradigm centered on thermodynamics. Herein we propose a research agenda to extend these thermodynamic foundations into complex, non-equilibrium, self-organizing systems and apply them holistically to future computing systems that will harness nature's innate computational capacity. We call this type of computing "Thermodynamic Computing" or TC.

cs.CY↗

arXiv2019

Deep Reinforcement Learning for Event-Driven Multi-Agent Decision Processes

The incorporation of macro-actions (temporally extended actions) into multi-agent decision problems has the potential to address the curse of dimensionality associated with such decision problems. Since macro-actions last for stochastic durations, multiple agents executing decentralized policies in cooperative environments must act asynchronously. We present an algorithm that modifies generalized advantage estimation for temporally extended actions, allowing a state-of-the-art policy optimization algorithm to optimize policies in Dec-POMDPs in which agents act asynchronously. We show that our algorithm is capable of learning optimal policies in two cooperative domains, one involving real-time bus holding control and one involving wildfire fighting with unmanned aircraft. Our algorithm works by framing problems as "event-driven decision processes," which are scenarios in which the sequence and timing of actions and events are random and governed by an underlying stochastic process. In addition to optimizing policies with continuous state and action spaces, our algorithm also facilitates the use of event-driven simulators, which do not require time to be discretized into time-steps. We demonstrate the benefit of using event-driven simulation in the context of multiple agents taking asynchronous actions. We show that fixed time-step simulation risks obfuscating the sequence in which closely separated events occur, adversely affecting the policies learned. In addition, we show that arbitrarily shrinking the time-step scales poorly with the number of agents.

cs.AI↗