# Free Agency and Determinism: Is There a Sensible Definition of Computational Sourcehood?

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## Abstract

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## 1. Introduction

## 2. No-Shortcut Approaches to Apparent Free Agency

#### 2.1. Computational Irreducibility

“And it is this, I believe, that is the ultimate origin of the apparent freedom of human will. For even though all the components of our brains presumably follow definite laws, I strongly suspect that their overall behavior corresponds to an irreducible computation whose outcome can never in effect be found by reasonable laws.”

“If someone’s will is apparently free, it hardly follows that that will is in fact free. Nowhere in ANKS [his book] does Wolfram even intimate that he maintains that our decisions are in fact free.”

#### 2.2. Lloyd’s Time Complexity Argument

“The unpredictability of the decision-making process does not arise from any lack of determinism—the Turing machines involved could be deterministic, or could possess a probabilistic guessing module, or could be quantum mechanical. In all cases, the unpredictability arises because of uncomputability.”

“In summary, applying the Hartmanis–Stearns diagonalization procedure shows that any general method for answering the question ‘Does decider d make a decision in time T, and what is that decision?’ must for some decisions take strictly longer than T to come up with an answer. That is, any general method for determining d’s decision must sometimes take longer than it takes d actually to make the decision.”

“Now we see why most people regard themselves as possessing free will. Even if the world and their decision-making process is completely mechanistic—even deterministic—no decider can know in general what her decision will be without going through a process at least as involved as the decider’s own decision-making process. In particular, the decider herself cannot know beforehand what her decision will be without effectively simulating the entire decision-making process. However, simulating the decision-making process takes at least as much effort as the decision-making process itself.”

## 3. John the Cook: A Thought Experiment

“See, John? You think that you and your emotions were irreducibly involved in the decision to prepare the Canadian omelette, but what happened in the safe was only determined by your (and your apartment’s) physical state yesterday night. Your thoughts and emotions this morning had no impact on the decision whatsoever!”

## 4. Computational Sourcehood

**Informal Conjecture.**

- Establishing a formal version of this conjecture would allow us to reason that John can be viewed in a specific sense as the “source of their own actions”, assuming that the Turing machine T he implements is contained in the set $\mathbf{T}$. However, regardless of the problem of free agency, establishing or disproving formal versions of the above might yield interesting insights into the nature of universal computation.

## 5. Towards a Rigorous Formulation of the Conjecture

#### 5.1. Textbook Universal Turing Machines Behave as Conjectured

**Observation 1.**

- In other words, Hennie’s universal TM works exactly as described in our Informal Conjecture; it simulates every TM T step by step, and reproduces exact images of the computational state of T at suitable time steps.

#### 5.2. A Simulation Preorder on TMs

**Definition 1**

**.**In the remainder of this paper, unless mentioned otherwise, a TM T is always assumed to conform to the following requirements. The TM has a set of internal states $Q=\{0,1,2,\dots ,k-1\}$ with $k\in \mathbb{N}$, where ${q}_{0}:=0$ is the initial state and ${q}_{f}:=k-1$ the final state. The TM has two bidirectional tapes (input and work tape), and one unidirectional tape (the output tape). The input tape is read-only, i.e., its content cannot be modified during the computation. The finite alphabet for all tapes is $\Sigma =\{0,1,\#\}$. The input is a finite binary string $x\in {\{0,1\}}^{*}$ that is initially written on the cells $0,1,\dots ,\ell \left(x\right)-1$ of the input tape. All other cells of all tapes are initially blank (#). All tape heads start in position zero. At each step of operation, the input and work tape heads can independently either move to the left or to the right. Furthermore, the machine may write a bit (0 or 1) at the current cell of the output tape and move its output tape head one position to the right (but not to the left), or it leaves the output tape as it is. If the machine halts, i.e., enters the distinguished internal state ${q}_{f}$, then the TM’s output, y, is the finite binary string that has so far been written onto the output tape. If this happens, then we write $T\left(x\right)=y$. This defines a partial function from the finite binary strings to the finite binary strings.

**Definition 2**

**.**A TM U isuniversalif it satisfies the following conditions. There exists a decidable prefix code ${\left\{{p}_{T}\right\}}_{T}$, where T labels the Turing machines, such that ${p}_{T}\in {\{0,1\}}^{*}$ is a computable description of the TM T; that is, there is an algorithm that extracts the set of internal states Q and the transition function δ from ${p}_{T}$. Furthermore,

- That is, a universal TM U takes the description of a TM T as input, and then imitates its output behavior on the rest of the input. Note that this definition is strictly stronger than the usual definition of a universal TM as used in algorithmic information theory [22,23]; there, it is only demanded that for every TM T there is some ${p}_{T}\in {\{0,1\}}^{*}$, such that Equation (3) holds, but it is not explicitly demanded that a description of T can be reconstructed from ${p}_{T}$. Decidability of the prefix code $\left\{{p}_{T}\right\}$ means that there is a computable function $f:{\{0,1\}}^{*}\to \{0,1\}$ with $f\left(s\right)=1$ if, and only if, $s\in \left\{{p}_{T}\right\}$—i.e., there exists an algorithm that decides for every given string whether that string is a valid encoding of a TM or not.

**Definition 3**

**.**Let T and ${T}^{\prime}$ be TMs. Suppose that, relative to our choice of simple functions $\mathcal{S}$, there is some $\phi \in \mathcal{S}$ such that for every input $x\in {\{0,1\}}^{*}$, the sequence of configurations

- Since the identity function is in $\mathcal{S}$, we have $T{\u2aaf}_{\mathcal{S}}T$ for all TMs T. Furthermore, $T{\u2aaf}_{\mathcal{S}}{T}^{\prime}$ and ${T}^{\prime}{\u2aaf}_{\mathcal{S}}{T}^{\prime \prime}$ implies $T{\u2aaf}_{\mathcal{S}}{T}^{\prime \prime}$, since simple functions can be composed. This implies that ${\u2aaf}_{\mathcal{S}}$ is a preorder on the TMs.

**Conjecture (1st attempt).**

**Conjecture (2nd attempt).**

**Lemma 1.**

- Next we will discuss how to concretely choose a suitable set of simple functions $\mathcal{S}$.

#### 5.3. How Not to Choose the Set of Simple Functions $\mathcal{S}$

**Lemma 2.**

- Note that the right-hand side is not the same as $T={T}^{\prime}$; for example, $T={T}_{i}$ and ${T}^{\prime}={T}_{j}$ for $i\ne j$ from the family of inefficient TMs under Equation (4) will also satisfy it. However, TMs T and ${T}^{\prime}$ that satisfy the right-hand side above are “identical for all practical purposes”.

- Extract ${p}_{T}$ and x from the input tape and t from the work tape.
- Simulate T on input x for t steps and return the configuration ${\mathcal{C}}_{T}(x,t)$.

- Consequently, we obtain $T{\u2aaf}_{\mathcal{S}}U({p}_{T}\u2022)$ for all TMs T. However, by construction, U never actually performs any step-by-step simulation of any TM T (since V does not). We have thus shown the following undesirable feature of the maximal choice of $\mathcal{S}$ as the set of all total computable functions; if there exist universal TMs that violate our Informal Conjecture, then some of them will still satisfy $T{\u2aaf}_{\mathcal{S}}U({p}_{T}\u2022)$. Thus, ${\prec}_{\mathcal{S}}$ is not a reliable formalization of the notion of step-by-step simulation that our Informal Observation refers to.

**Definition 4**

**.**A clock Turing machine C is a TM that ignores its input and counts integer time steps $t\in \mathbb{N}$ on its work tape indefinitely.

- We are not giving a formal construction of a clock TM, but it is not difficult to think of a concrete set of internal states and a transition function that implements the clock. For example, at each time step, the TM may simply write a fixed symbol (say, 1) in the currently active cell of the work tape and move the work tape head to the right. Since this can be completed in different ways (e.g., writing only zeros or ones, or alternating in ways that are determined by changes of the internal state), there are infinitely many clock TMs. Clock TMs C never halt, i.e., $C\left(x\right)$ is undefined for every $x\in {\{0,1\}}^{*}$.

**Lemma 3.**

- Most total computable functions are intuitively extremely complex, so the maximal choice of $\mathcal{S}$ is obviously a very bad formalization of a “set of simple functions”. However, the argument above rules out other, more intuitively sensible choices of $\mathcal{S}$. For example, we may consider the set $\mathcal{S}$ of functions that have at most linear time complexity. A running time at least linear in the input length is required to read the input, and as we would like the functions in $\mathcal{S}$ to be simple, it is natural to demand that they shall not take significantly more time than this minimum.

**Lemma 4.**

- Can stricter time bounds give us a better choice of $\mathcal{S}$? This seems unlikely, given that at least linear time is needed to even read the input configuration. So is there another way to define a set of simple functions $\mathcal{S}$ that gives us a non-trivial simulation preorder, but that leaves some chance for our conjecture (say, in its third formalization) to be true? Unfortunately, there is a strong counterargument to this hope, as we will now demonstrate.

#### 5.4. An Encryption Counterexample to “Simplicity”

- It reads the symbol in the currently active work tape cell i (assumed unencrypted) and the other currently active tape cells and applies T’s tabulated transition function $\delta $ to determine whether it has to move left or right on the tapes, and which symbol ${w}_{i}$ it has to write into the work tape cell (and similarly for the other tapes).
- Then it computes, with a fixed program independent of any other tape content, the bits ${a}_{i}$ and ${a}_{i+\sigma}$, where $\sigma =-1$ if it has to turn left on the work tape or $\sigma =+1$ if it has to turn right.
- It determines ${w}_{i}^{\prime}$, which is the blank symbol # if ${w}_{i}$ is blank, and which is ${w}_{i}\oplus {a}_{i}$ if ${w}_{i}$ is a bit (encryption). It writes ${w}_{i}^{\prime}$ into work tape cell i and reads ${w}_{i+\sigma}^{\prime}$ from work tape cell $i+\sigma $. It then determines ${w}_{i+\sigma}$, which is the blank symbol # if ${w}_{i+\sigma}^{\prime}$ is blank, and ${w}_{i+\sigma}^{\prime}\oplus {a}_{i+\sigma}$ otherwise (decryption). The writing onto the input and output tape is performed without encryption, as determined by T’s transition function $\delta $.
- ${U}^{\prime}$ erases all data that results from the computation of ${a}_{i}$ and ${a}_{i+\sigma}$ and of the sums of those with the work tape bits from other parts of its memory.

- We assume that the TM ${U}^{\prime}$ is constructed such that the only relevant difference after the encryption resp. decryption step to before is the value of the simulated work cell bit. In other words, ${U}^{\prime}$ is supposed to “erase all the garbage” that it produced while computing ${a}_{i}$, leaving only a simple encoding of the encrypted configuration of T. We also assume that ${U}^{\prime}$ does not contain an explicit counter of the number of time steps that have passed since the start of the computation.

**Assumption 1.**

**Assumption 2.**

**Assumption 3.**

**Observation 2.**

#### 5.5. From Simplicity to Preservation of Structure

**Conjecture (third and final attempt).**

- By structure-preserving maps, as sketched above, we mean functions $\phi $ with the following property. If we have close-by configurations c and ${c}^{\prime}$, and another configuration ${c}_{U}$ with $\phi \left({c}_{U}\right)=c$, then there is another configuration ${c}_{U}^{\prime}$close to ${c}_{U}$ with $\phi \left({c}_{U}^{\prime}\right)={c}^{\prime}$. These functions are not necessarily assumed to be “easy to implement”.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A universal computer (in the safe, on the right) reproduces the outputs of another process, i.e., its observable actions (John preparing breakfast, on the left). Computational sourcehood means that this prediction cannot typically be successfully performed without representing all relevant elements (here: thoughts, emotions) of that process in the simulation.

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**MDPI and ACS Style**

Krumm, M.; Müller, M.P.
Free Agency and Determinism: Is There a Sensible Definition of Computational Sourcehood? *Entropy* **2023**, *25*, 903.
https://doi.org/10.3390/e25060903

**AMA Style**

Krumm M, Müller MP.
Free Agency and Determinism: Is There a Sensible Definition of Computational Sourcehood? *Entropy*. 2023; 25(6):903.
https://doi.org/10.3390/e25060903

**Chicago/Turabian Style**

Krumm, Marius, and Markus P. Müller.
2023. "Free Agency and Determinism: Is There a Sensible Definition of Computational Sourcehood?" *Entropy* 25, no. 6: 903.
https://doi.org/10.3390/e25060903