Meditation Matters: Replies to the Anti-McMindfulness Bandwagon!

A critical reply to the anti-mindfulness critics in the collection, who oppose the popular secularized adoption of mindfulness on various grounds (it is not Buddhism, it is Buddhism, it is a tool of neo-capitalist exploitation, etc.), I argue that mindfulness is a quality of consciousness, opposite mindlessness, that may be cultivated through practice, and is almost always beneficial to those who cultivate it

The Big-O Problem for Max-Plus Automata is Decidable (PSPACE-Complete)

We show that the big-O problem for max-plus automata is decidable and PSPACE-complete. The big-O (or affine domination) problem asks whether, given two max-plus automata computing functions f and g, there exists a constant c such that f < cg+ c. This is a relaxation of the containment problem asking whether f < g, which is undecidable. Our decidability result uses Simon's forest factorisation theorem, and relies on detecting specific elements, that we call witnesses, in a finite semigroup closed under two special operations: stabilisation and flattening

Structural liveness of petri nets is ExpSpace-hard and decidable

Place/transition Petri nets are a standard model for a class of distributed systems whose reachability spaces might be infinite. One of well-studied topics is verification of safety and liveness properties in this model; despite an extensive research effort, some basic problems remain open, which is exemplified by the complexity status of the reachability problem that is still not fully clarified. The liveness problems are known to be closely related to the reachability problem, and various structural properties of nets that are related to liveness have been studied. Somewhat surprisingly, the decidability status of the problem of determining whether a net is structurally live, i.e. whether there is an initial marking for which it is live, remained open for some time; e.g. Best and Esparza (Inf Process Lett 116(6):423–427, 2016. https://doi.org/10.1016/j.ipl.2016.01.011) emphasize this open question. Here we show that the structural liveness problem for Petri nets is ExpSpace-hard and decidable. In particular, given a net N and a semilinear set S, it is decidable whether there is an initial marking of N for which the reachability set is included in S; this is based on results by Leroux (28th annual ACM/IEEE symposium on logic in computer science, LICS 2013, New Orleans, LA, USA, June 25–28, 2013, IEEE Computer Society, pp 23–32, 2013. https://doi.org/10.1109/LICS.2013.7)

The complexity of verifying loop-free programs as differentially private

We study the problem of verifying differential privacy for loop-free programs with probabilistic choice. Programs in this class can be seen as randomized Boolean circuits, which we will use as a formal model to answer two different questions: first, deciding whether a program satisfies a prescribed level of privacy; second, approximating the privacy parameters a program realizes. We show that the problem of deciding whether a program satisfies ε-differential privacy is coNP#P-complete. In fact, this is the case when either the input domain or the output range of the program is large. Further, we show that deciding whether a program is (ε,δ)-differentially private is coNP#P-hard, and in coNP#P for small output domains, but always in coNP#P#P. Finally, we show that the problem of approximating the level of differential privacy is both NP-hard and coNP-hard. These results complement previous results by Murtagh and Vadhan showing that deciding the optimal composition of differentially private components is #P-complete, and that approximating the optimal composition of differentially private components is in P

History-deterministic Vector Addition Systems

We consider history-determinism, a restricted form of non-determinism, for Vector Addition Systems with States (VASS) when used as acceptors to recognise languages of finite words. History-determinism requires that the non-deterministic choices can be resolved on-the-fly; based on the past and without jeopardising acceptance of any possible continuation of the input word. Our results show that the history-deterministic (HD) VASS sit strictly between deterministic and non-deterministic VASS regardless of the number of counters. We compare the relative expressiveness of HD systems, and closure-properties of the induced language classes, with coverability and reachability semantics, and with and without $\varepsilon$-labelled transitions. Whereas in dimension 1, inclusion and regularity remain decidable, from dimension two onwards, HD-VASS with suitable resolver strategies, are essentially able to simulate 2-counter Minsky machines, leading to several undecidability results: It is undecidable whether a VASS is history-deterministic, or if a language equivalent history-deterministic VASS exists. Checking language inclusion between history-deterministic 2-VASS is also undecidable.Comment: This is the full version of a paper published in CONCUR 202

Asymmetric distances for approximate differential privacy

Differential privacy is a widely studied notion of privacy for various models of computation, based on measuring differences between probability distributions. We consider (epsilon,delta)-differential privacy in the setting of labelled Markov chains. For a given epsilon, the parameter delta can be captured by a variant of the total variation distance, which we call lv_{alpha} (where alpha = e^{epsilon}). First we study lv_{alpha} directly, showing that it cannot be computed exactly. However, the associated approximation problem turns out to be in PSPACE and #P-hard. Next we introduce a new bisimilarity distance for bounding lv_{alpha} from above, which provides a tighter bound than previously known distances while remaining computable with the same complexity (polynomial time with an NP oracle). We also propose an alternative bound that can be computed in polynomial time. Finally, we illustrate the distances on case studies

Model Checking Linear Dynamical Systems under Floating-point Rounding

We consider linear dynamical systems under floating-point rounding. In these systems, a matrix is repeatedly applied to a vector, but the numbers are rounded into floating-point representation after each step (i.e., stored as a fixed-precision mantissa and an exponent). The approach more faithfully models realistic implementations of linear loops, compared to the exact arbitrary-precision setting often employed in the study of linear dynamical systems. Our results are twofold: We show that for non-negative matrices there is a special structure to the sequence of vectors generated by the system: the mantissas are periodic and the exponents grow linearly. We leverage this to show decidability of $\omega$-regular temporal model checking against semialgebraic predicates. This contrasts with the unrounded setting, where even the non-negative case encompasses the long-standing open Skolem and positivity problems. On the other hand, when negative numbers are allowed in the matrix, we show that the reachability problem is undecidable by encoding a two-counter machine. Again, this is in contrast to the unrounded setting where point-to-point reachability is known to be decidable in polynomial time

On the complexity of verifying differential privacy

This thesis contributes to the understanding of the computational complexity of verifying differential privacy. The problem is considered in two constrained, but expressive, models; namely labelled Markov chains and randomised circuits. In the setting of labelled Markov chains (LMC) it is shown that most relevant decision problems are undecidable when considered directly and exactly. Given an LMC, and an ε, consider the problem of finding the least value of δ such that the chain is (ε, δ)-differentially private. Finding this value of δ can be expressed as a variant of the total variation distance. Whilst finding the exact value is not possible, it can be approximated, with a complexity between #P and PSPACE. Instead, bisimilarity distances are studied as over-estimate of δ, which can be computed in polynomial time assuming access to an NP oracle and a slightly weaker distance can be computed in polynomial time. One may also wish to estimate the minimal value of ε such that the LMC is ε-differentially private. The question of whether such an ε even exists is studied through the big-O problem. That is, does there exist a constant C such that the probability of each word in one system is at most C times the probability in the other machine. However in general this problem is undecidable but can be decided on unary chains (and is coNP-complete). On chains with bounded language (that is, when there exists w_1,…..,w_m in Σ such that all words are of the form w_1^*…w_m^*) the problem is decidable subject to Schanuel’s conjecture by invoking the first order theory of the reals with exponential function. The minimal such constant C corresponds exactly to exp(ε) and approximating this value is not possible, even when the value is known to exist. A bisimilarity distance to over-estimate exp(ε) can be computed in PSPACE. In the setting of randomised circuits, the complexity of verifying pure differential privacy is fully captured as coNP^#P-complete; formalising the intuition that differential privacy is universal quantification followed by a condition on probabilities. However verifying approximate differential privacy is between coNP^#P and coNP^#P^#P, and coNP^#P-complete when the number of output bits is small (poly-logarithmic) relative to the total size of the circuit. Further, each parameter cannot be approximated given the other in polynomial time (assuming P not equal to NP)

The big-O problem for labelled markov chains and weighted automata

Given two weighted automata, we consider the problem of whether one is big-O of the other, i.e., if the weight of every finite word in the first is not greater than some constant multiple of the weight in the second. We show that the problem is undecidable, even for the instantiation of weighted automata as labelled Markov chains. Moreover, even when it is known that one weighted automaton is big-O of another, the problem of finding or approximating the associated constant is also undecidable. Our positive results show that the big-O problem is polynomial-time solvable for unambiguous automata, coNP-complete for unlabelled weighted automata (i.e., when the alphabet is a single character) and decidable, subject to Schanuel’s conjecture, when the language is bounded (i.e., a subset of w_1^* … w_m^* for some finite words w_1,… ,w_m). On labelled Markov chains, the problem can be restated as a ratio total variation distance, which, instead of finding the maximum difference between the probabilities of any two events, finds the maximum ratio between the probabilities of any two events. The problem is related to ε-differential privacy, for which the optimal constant of the big-O notation is exactly exp(ε)