This article reviews the main findings of the equally titled dissertation  with the main message: Causal loops are less problematic than initially thought. Causal structures can be represented by directed graphs, where the nodes represent events and the edges point from cause to effect . For instance, a computational circuit can be written in that way, where events are the intermediate states of computation, and the edges point forward in computation. Traditionally, causal structures come with two assumptions: 1) the underlying graph is acyclic, and 2) the graph is fixed. The first assumption forbids causal loops. The reason to forbid causal loops is that they might lead to logical problems. In a causal structure with loops, the state of an event does not only depend on the past events, but also on future events. The second assumption says that a process, e. g., some computation, does not alter the graph; the causal structure is not part of the computation, rather, computation takes place on top of the causal structure. Furthermore, both assumptions comply with our everyday experience.
What are the logical problems associated with such an endeavor? The most prominent one is called grandfather antinomy, where a causal loop leads to a logical contradiction. Assume some person travels to the past to kill his or her grandfather before this very same person has been born. By doing so, this person will not be born, and hence will not travel to the past, and hence will not kill his or her grandfather, and hence will be born, and hence, … – a logical contradiction. This problem has another face: the information antinomy. Suppose you wake up one morning and find, next to your bed, a novel you and no one else has ever seen before. A couple of years later you invent a time machine to travel to the past and place that book next to the bed of your younger self. What is the content of the book? Who wrote the book? This situation lacks any causal explanation for the information that becomes present at some time.1 A theory suffering from the information antinomy fails in certain cases to provide predictions. Both logical problems are schematically presented in Figure 1.
1.1 Motivations, results, and outline
The results stem from the interplay between computer science and physics: physics guides information processing, and limits on information processing restrict physics. We start by borrowing motivations from physics to relax the assumptions made on causality. General relativity theory is consistent with closed time-like curves , , . This means, there exist solutions to the equations of general relativity where particles bump into their younger selves; the causal structure is cyclic.
In quantum theory, physical properties are allowed to be in superposition: The state of a quantum-mechanical system can be in a superposition of being located at one and another point in space, which leads to a situation where the position of that system is not well defined. This superposition principle can be extended beyond the position degree of freedom to the temporal degree of freedom: We obtain a superposition of different causal structures, and by this, the causal structure is not well defined. Hence, the second assumption that a causal structure must be fixed is not satisfied. This phenomenon is illustrated in the following thought experiment, where quantum as well as relativistic effects take place . Take a large mass, e. g., a planet, and place it in superposition of being in two different locations. Due to general relativity, this planet transforms the space-time structure, resulting in a superposition of different space-times.
As a last motivation to mention, quantum theory comes with conceptual difficulties that might be overcome if we relax the assumptions made on causality. Most works that deal with such difficulties assume (often implicitly) a traditional notion of causality (see, e. g., References ,  for approaches that do not make such an assumption). One such conceptual difficulty is the causal understanding of non-local correlations . These correlations violate Reichenbach’s principle  that asks for correlations to be explained by a direct cause or a common cause.
The findings in the dissertation on causal loops mainly target the first assumption made on causal structures and are based on the “process matrix” formalism by Oreshkov, Costa, and Brukner . The authors show, by generalizing quantum theory, that the requirement for well-defined probabilities (without any assumption on causality) does not imply that the causal structure must be fixed and acyclic: Correlations arise that cannot be obtained from any fixed acyclic causal structure. We find that the same qualitative result also holds without having to invoke quantum theory, i. e., in classical theories (this is reported in the next section). Taking this as a starting point, we characterize these classical non-causal correlations, discuss time travel (see Section 3), and show that classical computational devices not limited to fixed acyclic causal structures cannot solve -hard problems efficiently2 (see Section 4).
The fact that logical consistency does not imply a fixed acyclic causal structure is shown via studying correlations among “parties.”
Definition 1 (Party, causal, and non-causal correlations).
A party is a tuple consisting of two random variables and a local operation . The random variable is the setting and the random variable the outcome.
For k parties, the correlations are called causal if and only if they can be simulated by arranging the parties on a fixed acyclic causal structure; otherwise, they are called non-causal.
Causal two-party correlations can hence be decomposed as where p is some probability: With probability p party acts before . Note that convex mixtures of fixed acyclic causal structures remain fixed and acyclic: The probability p might arise due to some “ignorance.”
Now, we describe how the correlations are obtained based on the local operations . Following Oreshkov, Costa, and Brukner , we do not assume an underlying fixed acyclic causal structure of the parties. Instead, we assume that the parties are isolated (A1), every party acts once (A2), the correlations are linear in the choice of local operations (A3), and logical consistency (A4). Condition (A1) means that the local operation of any party can be described independently of the other parties’ local operations. Then again, (A2) says that every local operation is applied once; this could be generalized, but as it stands, makes calculations easier and is sufficient for our claims. The requirement (A3) of linearity is natural: It translates to the requirement that convex combinations of local operations transform to convex combinations of correlations. Last but not least, (A4) is the core assumption in order to have a consistent theory.
Since we are interested in the case where the underlying theory is classical probability theory (as opposed to quantum theory), the most general form of the local operation is for some random variables : a stochastic channel that depends on the setting and produces an outcome . Hence, up to the logical consistency assumption that we formulate shortly, the most general correlations – without assuming an underlying fixed acyclic causal structure – are given by for some multi-linear function f. Note that f can be interpreted as a “supermap:” It maps operations to an operation . Logical consistency, finally, asks that for any choice of local operations, the distribution is well defined. This means that the parties are unrestricted in what local operation they want to perform.
Definition 2 (Logical consistency).
A supermap f is called logically consistent if and only if is a probability distribution of the form .
Given a supermap that is not logically consistent, a set of local operations of the parties exists such that the result is not a probability distribution (the “probabilities” are not normalized or negative).
Note that causal correlations satisfy all the assumptions (A1–A4). The question we are interested in now is: Do the four assumptions (A1–A4) imply causal correlations? If the underlying theory is quantum theory, then Oreshkov, Costa, and Brukner  answered this question negatively. In the same work the authors showed that if the underlying theory is classical, then for two parties the question is answered positively. This suggested that fixed acyclic causal structures might be the result from quantum-to-classical transitions (the quantum world may be causally exotic, yet the classical is not – supporting our everyday view on nature). In the dissertation we show that the latter is not true; classical theories allow for non-causal correlations if more than two parties are involved. Before we illustrate this result with an example, we present two theorems that help to understand logically consistent supermaps f.
Theorem 1 (Logically consistent supermaps (informal) ).
Logically consistent supermaps represent (stochastic) channels E from the variables to the variables (see Figure 2).
Following this theorem, the resulting distribution is calculated according to probability theory by taking the product over all local operations and the channel E, and by marginalizing over the variables. Note that some channels E do not lead to a logically consistent supermap, e. g., if we would plug in the identity channel, then a set of local operations exists such that the result is not a valid probability distribution as required.
Theorem 2 (Fixed-point characterization (informal) ).
A supermap f is logically consistent if and only if the average number of fixed points of concatenated with any choice of deterministic local operations is 1 (see Figure 2).
A corollary of the latter theorem is that a deterministic supermap is logically consistent if and only if for any choice of deterministic local operations, a unique fixed point exists. This result is tantamount to avoiding the grandfather as well as the information antinomy discussed in the introduction.
Example. Let be parties with local operations , where all random variables are binary and where the settings are uniformly distributed. To decide whether correlations are non-causal, we make use of causal inequalities , . If for a given distribution the inequality is violated, then this distribution must be non-causal. The reason for this is that for causal correlations (the parties are positioned on a fixed acyclic causal structure), at least one party has no other parties in her past and therefore no access to the settings of the other parties; hence, takes the correct value with probability at most 3/4.
Yet, in the framework presented, the supermap constructed from along with the local operations allow to violate the above inequality maximally: It is impossible to arrange the parties on a fixed acyclic causal structure in order to obtain the same correlations. By consulting the second theorem above, the reader can convince herself/himself that E indeed leads to a logically consistent supermap. The underlying causal structure is cyclic (see Figure 3).
3 Time travel
Scientific studies on time travel – even though controversial – obtained great interest after the development of general relativity. Already Einstein  was suspecting that his theory might allow for such a behavior, and approached  Carathéodory to resolve this issue – without any answer. To our knowledge, the first solution to the Einstein equations with closed time-like curves was discovered a decade later by Lanczos . We briefly discuss the more intuitive approach by Morris, Thorne, and Yurtsever  to time travel in general relativity. Their solution is based on wormholes. A wormhole has two mouths, and the proper times of both mouths are identified with each other, i. e., if a particle is dropped into one mouth at time τ, then it exists the other mouth at time τ. As the authors suggest, if we take one mouth, accelerate it to some high velocity and bring it back, we introduce some time dilation, which allows for time travel (see Figure 4).
In a series of texts, Thorne and coauthors ,  study the dynamics of time-travelling billiard balls. Motivated by the potential logical problems of time travel, they ask the following question: Are there some initial conditions of a billiard ball at a time before a closed time-like curve exists (e. g., at in Figure 4), such that the billiard ball travels to the past and collides with its younger self in such a way that it will not travel to the past? – Do some initial conditions lead to a logical contradiction à la grandfather antinomy? If yes, this would have rather strong consequences: It would mean that these initial conditions could not have been chosen even though at the time when we would have chosen them no time travel is possible – we would be restricted in todays actions if time travel were possible in the future. They answer: For any choice of initial conditions, consistent trajectories exist, even for time-travelling billiard balls! Yet, their result is surprisingly more extreme, and has been called the billiard-ball crisis : Not only consistent trajectories exist, but an infinity of such trajectories exist (information antinomy).
By abstracting away the general-relativistic settings, others ,  studied time travel on the basis of quantum theory; more concretely, on the basis of quantum circuits. The reason to avoid classical circuits is that classical circuits are prone to the grandfather antinomy (see Figure 1). If such a circuit is replaced by a quantum circuit, a self-consistent quantum state can always be found . These approaches, however, modify quantum theory in a way that it becomes non-linear, and hence unsatisfactory from a physical point of view.
The causal loops we study avoid and resolve the above raised issues. First, the logical consistency condition asks for consistent trajectories for all initial conditions and all intermediate transformations of the billiard ball. The second theorem above then states that the “trajectory” is uniquely defined – the billiard-ball crisis is avoided. Furthermore, we do not have to pursue a quantum-theoretic setup: Certain classical circuits avoid the mentioned logical problems and do not alter the underlying theory (whether it is classical or quantum) to a non-linear one. Moreover, we show that time travel in this fashion is reversible .
Since we cannot rule out causal loops on logical nor on physical grounds, what objections (apart from appealing to our everyday experiences where no exotic causal structures seem to pop up) against causal loops are left? Aaronson  suggests the -hardness assumption: “ -complete problems are intractable in the physical world.” This assumption says that if we build a computer based on some theory T and we find that this device can solve -hard problems efficiently, then we would suspect that T is not physical. This assumption might be debatable; yet, and what gives some weight to this assumption is that, all approaches to solve -hard problems efficiently by the use of exotic computational models implemented (e. g., with soap bubbles ), failed. Interestingly, the above mentioned circuit-based models ,  of time travel are ruled out by this assumption – -complete problems become solvable in polynomial time .
Motivated by Aaronson’s thesis, and out of general computer-science interest, we study the computational power of causal loops. Clearly, a computational model where causal loops are possible can efficiently solve all problems efficiently solvable without loops; but maybe more.
Before we discuss the results on this track, we introduce the computational model. The second theorem above states that a deterministic supermap is logically consistent if and only if for any choice of deterministic local operations, a unique fixed point exists. In a computational model, we usually do not think in terms of “parties,” but rather in terms of gates. So, the assumption that parties are unrestricted in their choice of local operations becomes obsolete. Instead, we say that a circuit with causal loops is logically consistent if and only if a unique fixed point on the looping wires exists. This is formulated in the following definition:
Definition 3 (Non-causal circuit ).
A circuit together with an integer is a non-causal circuit if and only if where ∃! is the uniqueness quantifier, and is some bit-string of length .
The “power” of such circuits with polynomial size equals an already known complexity class:
Theorem 3 (Polynomial-time logically consistent circuits (informal) ).
The set of languages decidable in polynomial time with a non-causal circuit equals .
Only few problems are known that are in and conjectured to be outside of : Factoring numbers, discrete logarithms, simple stochastic games, …. The computational-complexity class is conjectured to be strictly contained in as otherwise, the polynomial hierarchy would collapse. For the first two examples mentioned, efficient quantum algorithms exist that can solve them . This leaves open the question for simple stochastic games (or parity games, which are believed to be easier  than simple stochastic games). Unfortunately, this open question has remained unresolved. The above theorem states that allowing computers to make use of logically consistent causal loops may alter the computational power, but just not strong enough in order to violate the -hardness assumption.
Example. We sketch an efficient non-causal circuit for factoring some number N into its prime factors. The circuit takes as input a bit b, pairs , with and , where all wires are looped (). The circuit performs multiple tests. If at least one test fails, then the circuit outputs , otherwise, the circuit outputs . The tests are:
all are distinct,
all are prime (this can be done efficiently ),
the are ordered decreasingly,
the implication holds,
and finally, .
Causal loops have been ruled out mainly based on logical grounds. The most common objection against causal loops is the grandfather antinomy; with the liar paradox “All Cretans are liars,” as has been attributed to Epimenides from Crete , as a prime example. This motivated, e. g., Tarski  to rule out self-referential statements in language. In the dissertation, however, we show that such a restriction might be too strong: Some causal loops comply with the requirements of logics, computer science as well as physics. The motivations to study causal loops at all stem from taking our current physical theories (quantum theory and general relativity) seriously. The main message is that not only time travel now appears less implausible – which is anyways a controversial topic –, but that causal loops might be considered as theoretical building blocks for approaching problems in physics, computer science and possibly other fields.
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The dissertation of Dr. Ämin Baumeler has been awarded by the GI Dissertation Award 2017.
Deutsch  calls this creationism.
About the article
Dr. Ämin Baumeler
Dr. Ämin Baumeler studied computer science at ETH Zurich, Switzerland. His master’s project on quantum cryptography was conducted at the Institute for Quantum Computing in Waterloo, Canada, under the supervision of Prof. Anne Broadbent. After having obtained the master’s degree from ETH, he started his PhD studies in the research group of Prof. Stefan Wolf at the Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland. The research performed in his PhD studies are on causal structures, foundations of quantum theory, and complexity theory. After having obtained the PhD degree in 2016, he moved to Vienna to join Prof. Časlav Brukner’s group at the Institute for Quantum Optics and Quantum Information.
Published Online: 2019-07-03