The framework of Pearl's Causal Hierarchy (PCH) formalizes three types of reasoning: probabilistic (i.e. purely observational), interventional, and counterfactual, that reflect the progressive sophistication of human thought regarding causation. We investigate the computational complexity aspects of reasoning in this framework focusing mainly on satisfiability problems expressed in probabilistic and causal languages across the PCH. That is, given a system of formulas in the standard probabilistic and causal languages, does there exist a model satisfying the formulas? Our main contribution is to prove the exact computational complexities showing that languages allowing addition and marginalization (via the summation operator) yield NP^{PP}-, PSPACE-, and NEXP-complete satisfiability problems, depending on the level of the PCH. These are the first results to demonstrate a strictly increasing complexity across the PCH: from probabilistic to causal and counterfactual reasoning. On the other hand, in the case of full languages, i.e.~allowing addition, marginalization, and multiplication, we show that the satisfiability for the counterfactual level remains the same as for the probabilistic and causal levels, solving an open problem in the field.

Causal effect estimation from observational data is a fundamental task in empirical sciences. It becomes particularly challenging when unobserved confounders are involved in a system. This paper focuses on front-door adjustment – a classic technique which, using observed mediators allows to identify causal effects even in the presence of unobserved confounding. While the statistical properties of the front-door estimation are quite well understood, its algorithmic aspects remained unexplored for a long time. In 2022, Jeong, Tian, and Bareinboim presented the first polynomial-time algorithm for finding sets satisfying the front-door criterion in a given directed acyclic graph (DAG), with an O(n³(n+m)) run time, where n denotes the number of variables and m the number of edges of the causal graph. In our work, we give the first linear-time, i.e., O(n+m), algorithm for this task, which thus reaches the asymptotically optimal time complexity. This result implies an O(n(n+m)) delay enumeration algorithm of all front-door adjustment sets, again improving previous work by a factor of n³. Moreover, we provide the first linear-time algorithm for finding a minimal front-door adjustment set. We offer implementations of our algorithms in multiple programming languages to facilitate practical usage and empirically validate their feasibility, even for large graphs.

Learning the unknown causal parameters of a linear structural causal model is a fundamental task in causal analysis. The task, known as the problem of identification, asks to estimate the parameters of the model from a combination of assumptions on the graphical structure of the model and observational data, represented as a non-causal covariance matrix. In this paper, we give a new sound and complete algorithm for generic identification which runs in polynomial space. By a standard simulation result, namely $\mathsf{PSPACE} \subseteq \mathsf{EXP}$, this algorithm has exponential running time which vastly improves the state-of-the-art double exponential time method using a Gröbner basis approach. The paper also presents evidence that parameter identification is computationally hard in general. In particular, we prove, that the task asking whether, for a given feasible correlation matrix, there are exactly one or two or more parameter sets explaining the observed matrix, is hard for $\forall \mathbb{R}$, the co-class of the existential theory of the reals. In particular, this problem is $\mathsf{coNP}$-hard. To our best knowledge, this is the first hardness result for some notion of identifiability.

Enumerating the directed acyclic graphs (DAGs) of a Markov equivalence class (MEC) is an important primitive in causal analysis. The central resource from the perspective of computational complexity is the delay, that is, the time an algorithm that lists all members of the class requires between two consecutive outputs. Commonly used algorithms for this task utilize the rules proposed by Meek (1995) or the transformational characterization by Chickering (1995), both resulting in superlinear delay. In this paper, we present the first linear-time delay algorithm. On the theoretical side, we show that our algorithm can be generalized to enumerate DAGs represented by models that incorporate background knowledge, such as MPDAGs; on the practical side, we provide an efficient implementation and evaluate it in a series of experiments. Complementary to the linear-time delay algorithm, we also provide intriguing insights into Markov equivalence itself: All members of an MEC can be enumerated such that two successive DAGs have structural Hamming distance at most three.

Messengers have become an essential means of interpersonal interaction. Yet untraceable private communication remains an elusive goal, as most messengers hide content, but not communication patterns. The knowledge of communication patterns can by itself reveal too much, as happened, e. g., in the context of the Arab Spring. Subliminal channels in cryptographic systems enable untraceable private communication in plain sight. In this context, bulletin boards in the form of blockchains are a natural object for subliminal communication: accessing them is innocuous, as they rely on distributed access for verification and extension. At the same time, blockchain users generate hundreds of thousands of transactions per day that are individually signed and placed on the blockchain. Thus blockchains may serve as innocuous repository for publicly accessible cryptographic transactions where subliminal channels can be placed. In this paper, we propose a public-key subliminal channel using secret-recoverable splittable signature schemes on blockchains and prove that our construction is undetectable in the random oracle model under common cryptographic assumptions. Our approach is applicable to any secret-recoverable splittable signature scheme and introduces a constant overhead of a single signature per message. Such schemes are used by 98 of the top 100 cryptocurrencies. We also analyze the applicability of our approach to the Bitcoin, Monero, and RippleNet networks and present proof of concept implementations for Bitcoin and RippleNet.

We study formal languages which are capable of fully expressing quantitative probabilistic reasoning and do-calculus reasoning for causal effects, from a computational complexity perspective. We focus on satisfiability problems whose instance formulas allow expressing many tasks in probabilistic and causal inference. The main contribution of this work is establishing the exact computational complexity of these satisfiability problems. We introduce a new natural complexity class, named succ∃R, which can be viewed as a succinct variant of the well-studied class ∃R, and show that these problems are complete for succ∃R. Our results imply even stronger limitations on the use of algorithmic methods for reasoning about probabilities and causality than previous state-of-the-art results that rely only on the NP- or ∃R-completeness of the satisfiability problems for some restricted languages.

Counting and sampling directed acyclic graphs from a Markov equivalence class are fundamental tasks in graphical causal analysis. In this paper we show that these tasks can be performed in polynomial time, solving a long-standing open problem in this area. Our algorithms are effective and easily implementable. As we show in experiments, these breakthroughs make thought-to-be-infeasible strategies in active learning of causal structures and causal effect identification with regard to a Markov equivalence class practically applicable.

Linear structural equation models represent direct causal effects as directed edges and confounding factors as bidirected edges. An open problem is to identify the causal parameters from correlations between the nodes. We investigate models, whose directed component forms a tree, and show that there, besides classical instrumental variables, missing cycles of bidirected edges can be used to identify the model. They can yield systems of quadratic equations that we explicitly solve to obtain one or two solutions for the causal parameters of adjacent directed edges. We show how multiple missing cycles can be combined to obtain a unique solution. This results in an algorithm that can identify instances that previously required approaches based on Gröbner bases, which have doubly-exponential time complexity in the number of structural parameters.

Ancestral graphs are an important tool for encoding causal knowledge as they represent uncertainty about the presence of latent confounding and selection bias, and they can be inferred from data. As for other graphical models, several maximal ancestral graphs (MAGs) may encode the same statistical information in the form of conditional independencies. Such MAGs are said to be Markov equivalent. This work concerns graphical characterizations and computational aspects of Markov equivalence between MAGs. These issues have been studied in past years leading to several criteria and methods to test Markov equivalence. The state-of-the-art algorithm, provided by Hu and Evans [UAI 2020], runs in time $O(n^5)$ for instances with $n$ vertices. We propose a new constructive graphical criterion for the Markov equivalence of MAGs, which allows us to develop a practically effective equivalence test with worst-case runtime $O(n^3)$. Additionally, our criterion is expressed in terms of natural graphical concepts, which is of independent value.

Counting and uniform sampling of directed acyclic graphs (DAGs) from a Markov equivalence class are fundamental tasks in graphical causal analysis. In this paper, we show that these tasks can be performed in polynomial time, solving a long-standing open problem in this area. Our algorithms are effective and easily implementable. Experimental results show that the algorithms significantly outperform state-of-the-art methods.

Finding a consistent DAG extension for a given partially directed acyclic graph (PDAG) is a basic building block used in graphical causal analysis. In 1992, Dor and Tarsi proposed an algorithm with time complexity O(n^4), which has been widely used in causal theory and practice so far. It is a long-standing open question whether an extension can be computed faster and, in particular, it was conjectured that a linear-time method may exist. The main contributions of our work are two-fold: Firstly, we propose a new algorithm for the extension problem for PDAGs which runs in time O(n^3); secondly, we show that, under a computational intractability assumption, our cubic algorithm is optimal. Thus, our impossibility result disproves the conjecture that a linear-time method exists. Based on these results, we present a full complexity landscape for finding extensions in various causal graphical models. We extend the techniques to recognition problems and apply them to design an effective algorithm for closing a PDAG under the orientation rules of Meek.

Residuality plays an essential role for learning finite automata. While residual deterministic and non-deterministic automata have been understood quite well, fundamental questions concerning alternating automata (AFA) remain open. Recently, Angluin, Eisenstat, and Fisman (2015) have initiated a systematic study of residual AFAs and proposed an algorithm called AL* – an extension of the popular L* algorithm – to learn AFAs. Based on computer experiments they have conjectured that AL* produces residual AFAs, but have not been able to give a proof. In this paper we disprove this conjecture by constructing a counterexample. As our main positive result we design an efficient learning algorithm, named AL** and give a proof that it outputs residual AFAs only. In addition, we investigate the succinctness of these different FA types in more detail.

The goal of an algorithm substitution attack (ASA), also called a subversion attack (SA), is to replace an honest implementation of a cryptographic tool by a subverted one which allows to leak private information while generating output indistinguishable from the honest output. Bellare, Paterson, and Rogaway provided at CRYPTO '14 a formal security model to capture this kind of attacks and constructed practically implementable ASAs against a large class of symmetric encryption schemes. At CCS'15, Ateniese, Magri, and Venturi extended this model to allow the attackers to work in a fully-adaptive and continuous fashion and proposed subversion attacks against digital signature schemes. Both papers also showed the impossibility of ASAs in cases where the cryptographic tools are deterministic. Also at CCS'15, Bellare, Jaeger, and Kane strengthened the original model and proposed a universal ASA against sufficiently random encryption schemes. In this paper we analyze ASAs from the perspective of steganography - the well known concept of hiding the presence of secret messages in legal communications. While a close connection between ASAs and steganography is known, this lacks a rigorous treatment. We consider the common computational model for secret-key steganography and prove that successful ASAs correspond to secure stegosystems on certain channels and vice versa. This formal proof allows us to conclude that ASAs are stegosystems and to "rediscover" several results concerning ASAs known in the steganographic literature.

Spectral based heuristics belong to well-known commonly used methods which determines provably minimal graph bisection or outputs "fail" when the optimality cannot be certified. In this paper we focus on Boppana's algorithm which belongs to one of the most prominent methods of this type. It is well known that the algorithm works well in the random planted bisection model - the standard class of graphs for analysis minimum bisection and relevant problems. In 2001 Feige and Kilian posed the question if Boppana's algorithm works well in the semirandom model by Blum and Spencer. In our paper we answer this question affirmatively. We show also that the algorithm achieves similar performance on graph classes which extend the semirandom model. Since the behavior of Boppana's algorithm on the semirandom graphs remained unknown, Feige and Kilian proposed a new semidefinite programming (SDP) based approach and proved that it works on this model. The relationship between the performance of the SDP based algorithm and Boppana's approach was left as an open problem. In this paper we solve the problem in a complete way by proving that the bisection algorithm of Feige and Kilian provides exactly the same results as Boppana's algorithm. As a consequence we get that Boppana's algorithm achieves the optimal threshold for exact cluster recovery in the stochastic block model. On the other hand we prove some limitations of Boppana's approach: we show that if the density difference on the parameters of the planted bisection model is too small then the algorithm fails with high probability in the model.

In practice the vast majority of causal effect estimations from observational data are computed using adjustment sets which avoid confounding by adjusting for appropriate covariates. Recently several graphical criteria for selecting adjustment sets have been proposed. They handle causal directed acyclic graphs (DAGs) as well as more general types of graphs that represent Markov equivalence classes of DAGs, including completed partially directed acyclic graphs (CPDAGs). Though expressed in graphical language, it is not obvious how the criteria can be used to obtain effective algorithms for finding adjustment sets. In this paper we provide a new criterion which leads to an efficient algorithmic framework to find, test and enumerate covariate adjustments for chain graphs - mixed graphs representing in a compact way a broad range of Markov equivalence classes of DAGs.

Instrumental Variables are a popular way to identify the direct causal effect of a random variable X on a variable Y. Often no single instrumental variable exists, although it is still possible to find a set of generalized instrumental variables (GIVs) and identify the causal effect of all these variables at once. Till now it was not known how to find GIVs systematically or even test efficiently, if given variables satisfy GIV conditions. We provide fast algorithms for searching and testing restricted cases of GIVs. However, we prove that in the most general case it is NP-hard to verify if given variables fulfill the conditions of a general instrumental sets.

We present the first complexity-theoretic secure steganographic protocol which, for any communication channel, is provably secure, reliable, and has nearly optimal bandwidth. Our system is unconditionally secure, i.e. our proof does not rely on any unproven complexity-theoretic assumption, like e.g. the existence of one-way functions. This disproves the claim that the existence of one-way functions and access to a communication channel oracle are both necessary and sufficient conditions for the existence of secure steganography, in the sense that secure and reliable steganography exists independently of the existence of one-way functions.

This paper considers steganography - the concept of hiding the presence of secret messages in legal communications - in the computational setting and its relation to cryptography. Very recently the first (non-polynomial time) steganographic protocol has been shown which, for any communication channel, is provably secure, reliable, and has nearly optimal bandwidth. The security is unconditional, i.e. it does not rely on any unproven complexity-theoretic assumption. This disproves the claim that the existence of one-way functions and access to a communication channel oracle are both necessary and sufficient conditions for the existence of secure steganography in the sense that secure and reliable steganography exists independently of the existence of one-way functions. In this paper, we prove that this equivalence also does not hold in the more realistic setting, where the stegosystem is polynomial time bounded. We prove this by constructing (a) a channel for which secure steganography exists if and only if one-way functions exist and (b) another channel such that secure steganography implies that no one-way functions exist. We therefore show that security-preserving reductions between cryptography and steganography need to be treated very carefully.

Artificial immune systems (AIS) inspired by lymphocyte repertoires include negative and positive selection, clonal selection, and B~cell algorithms. Such AISs are used in computer science for machine learning and optimization, and in biology for modeling of fundamental immunological processes. In both cases, the necessary size of repertoire models can be huge. Here, we show that when lymphocyte repertoire models based on string patterns can be compactly represented as finite automata (FA), this allows to efficiently perform negative selection, positive selection, insertion into, deletion from, uniform sampling from, and counting the repertoire. Specifically, for r-contiguous pattern matching, all these tasks can be performed in polynomial time. But even in NP-hard cases like Hamming distance matching, the FA representation can still lead to practically important efficiency gains. We demonstrate the feasibility and flexibility of this approach by implementing T~cell positive selection simulations based on human genomic data using four different pattern rules. Hence, FA-based repertoire models generalize previous efficient negative selection algorithms to perform several related algorithmic tasks, are easy to implement and customize, and are applicable to real-world bioinformatic problems.

Negative selection algorithms are immune-inspired classifiers that are trained on negative examples only. Classification is performed by generating detectors that match none of the negative examples, and these detectors are then matched against the elements to be classified. This can be a performance bottleneck: A large number of detectors may be required for acceptable sensitivity, or finding detectors that match none of the negative examples may be difficult. In this paper, we show how negative selection can be implemented without generating detectors explicitly, which for many detector types leads to polynomial time algorithms whereas the common approach to sample detectors randomly takes exponential time in the worst case.

We study deterministic one-way communication complexity of functions with Hankel communication matrices. Some structural properties of such matrices are established and applied to the one-way two-party communication complexity of symmetric Boolean functions. It is shown that the number of required communication bits does not depend on the communication direction, provided that neither direction needs maximum complexity. Moreover, in order to obtain an optimal protocol, it is in any case sufficient to consider only the communication direction from the party with the shorter input to the other party. These facts do not hold for arbitrary Boolean functions in general. Next, gaps between one-way and two-way communication complexity for symmetric Boolean functions are discussed. Finally, we give some generalizations to the case of multiple parties.

This paper tries to fully characterize the properties and relationships of space classes defined by Turing machines (TMs) that use less than logarithmic space—be they deterministic, nondeterministic, or alternating (DTMs, NTMs, or ATMs). We provide several examples of specific languages and show that such machines are unable to accept these languages. The basic proof method is a nontrivial extension of the $1^n \mapsto 1^{n + n!} $ technique to alternating TMs. Let 1log denote the logarithmic function log iterated twice, and let $\Sigma _k {\textit{Space}}(S) $ and $\prod _k {\textit{Space}}(S)$ be the complexity classes defined by S-space-bounded ATMs that alternate at most $k - 1$ times and start in an existential (resp., universal) state. Our first result shows that for each $k > 1$, the sets \[ \begin{gathered} \hfill \Sigma _k {\textit{Space}}(1\log )\backslash \Pi _k Space(o(\log )) \quad {\text{and}} \\ \hfill \Pi _k {\textit{Space}}(1\log )\backslash \Sigma _k {\textit{Space}}(o(\log )) \\ \end{gathered} \] are both not empty. This implies that for each $S \in \Omega (1\log ) \cap o(\log )$, the classes \[ \begin{gathered} \Sigma _1 {\textit{Space}}(S) \subset \Sigma _2 {\textit{Space}}(S) \subset \Sigma _3 {\textit{Space}}(S) \subset \cdots \\ \subset \sum\nolimits_k {Space(S) \subset } \sum\nolimits_{k + 1} {Space(S) \subset } \cdots \\ \end{gathered} \] form an infinite hierarchy. Furthermore, this separation is extended to space classes defined by ATMs with a nonconstant alternation bound A provided that the product $A \cdot S$ grows sublogarithmically. These lower bounds can also be used to show that basic closure properties do not hold for such classes. We obtain that for any $S \in \Omega (1\log ) \cap o(\log )$ and all $k > 1$, $\Sigma _k {Space(S)} $ and $\prod _k {\textit{Space}}(S)$ are not closed under complementation and concatenation. Moreover, $\Sigma _k {{\textit{Space}}(S)} $ is not closed under intersection and $\prod _k {\textit{Space}}(S)$ is not closed under union. It is also shown that ATMs recognizing bounded languages can always be guaranteed to halt. For the class of Z-bounded languages with $Z \leqslant \exp S$, we obtain the equality co-$\Sigma _k {{\textit{Space}}(S)} = \Pi _k {\textit{Space}}(S)$. Finally, for sublogarithmic bounded ATMs, we give a separation between the weak and strong space measure and prove a logarithmic lower space bound for the recognition of nonregular context-free languages.

Alternating Turing machines (ATMs) with bounded number of reversals are considered. It is proved that the machines making fewer than $\log ^{*} n$ reversals can recognize only regular languages. On the other hand, the class of languages that can be recognized by ATMs using $\log ^{*} n$ reversals is very wide. The authors prove that above this limit even a slight increase of the number of reversals leads to a considerably larger class of languages. It is also proved that every $T(n)$-time bounded ATM may be replaced by an equivalent machine working in the same time and making no more than $\log ^{*} (T(n))$ reversals.

Every single-tape Turing machine (TM) of time complexity $T(n) \geqq n^{2}$ can be simulated by a single-tape TM in space $T^{{1 / 2}}(n)$. It is shown that the time of the simulation can be bounded by $T^{3/2}(n)$ in the case of deterministic TMs and by $T(n)$ in the case of nondeterministic ones. Similar results are shown for off-line machines and for machines with multidimensional tape.

The reversal complexity of alternating Turing machines (ATM) is investigated. The strict lower bounds on reversals for recognizing nonregular languages by Sigma /sub k/ machines are settled. Some results relating reversal and space complexities are obtained.>