Dynamic Bayesian networks (DBNs) are compact graphical representations used to model probabilistic systems where interdependent random variables and their distributions evolve over time. In this paper, we study the verification of the evolution of conditional-independence (CI) propositions against temporal logic specifications. To this end, we consider two specification formalisms over CI propositions: linear temporal logic (LTL), and non-deterministic Büchi automata (NBAs). This problem has two variants. Stochastic CI properties take the given concrete probability distributions into account, while structural CI properties are viewed purely in terms of the graphical structure of the DBN. We show that deciding whether a stochastic CI proposition eventually holds is at least as hard as the Skolem problem for linear recurrence sequences, which is a long-standing open problem in number theory. On the other hand, we show that verifying the evolution of structural CI propositions against LTL and NBA specifications is in PSPACE, and is hard for both NP and coNP. We also identify natural restrictions on the graphical structure of the DBN that make the verification of structural CI properties tractable.

In this paper we complete Büchi's proof that there is no decision algorithm for the solubility in integers of arbitrary systems of diagonal quadratic form equations, by proving the assertion that whenever $x_1^2, \cdots, x_5^2$ are five squares such that the second differences satisfy \[x_{k+2}^2 - 2 x_{k+1}^2 + x_k^2 = 2\] for $k = 1,2,3$, then they must be consecutive. This answers a question of J.~Richard~Büchi.

The Skolem Problem asks to determine whether a given linear recurrence sequence (LRS) \(\langle u_n \rangle_{n=0}^\infty\) over the integers has a zero term, that is, whether there exists \(n\) such that \(u_n = 0\). Decidability of the problem is open in general, with the most notable positive result being a decision procedure for LRS of order at most \(4\).

The Skolem Problem asks to determine whether a given linear recurrence sequence (LRS) has a zero term. Showing decidability of this problem is equivalent to giving an effective proof of the Skolem-Mahler-Lech Theorem, which asserts that a non-degenerate LRS has finitely many zeros. The latter result was proven over 90 years ago via an ineffective method showing that such an LRS has only finitely many p-adic zeros. In this paper we consider the problem of determining whether a given LRS has a p-adic zero, as well as the corresponding function problem of computing exact representations of all p-adic zeros. We present algorithms for both problems and report on their implementation. The output of the algorithms is unconditionally correct, and termination is guaranteed subject to the p-adic Schanuel Conjecture (a standard number-theoretic hypothesis concerning the p-adic exponential function). While these algorithms do not solve the Skolem Problem, they can be exploited to find natural-number and rational zeros under additional hypotheses. To illustrate this, we apply our results to show decidability of the Simultaneous Skolem Problem (determine whether two coprime linear recurrences have a common natural-number zero), again subject to the p-adic Schanuel Conjecture.

Standpoint linear temporal logic (SLTL) is a recently introduced extension of classical linear temporal logic (LTL) with standpoint modalities. Intuitively, these modalities allow to express that, from agent a's standpoint, it is conceivable that a given formula holds. Besides the standard interpretation of the standpoint modalities we introduce four new semantics, which differ in the information an agent can extract from the history. We provide a general model checking algorithm applicable to SLTL under any of the five semantics. Furthermore we analyze the computational complexity of the corresponding model checking problems, obtaining PSPACE-completeness in three cases, which stands in contrast to the known EXPSPACE-completeness of the SLTL satisfiability problem.

We consider reachability problems for linear dynamical systems. In dimension d these problems are specified by respective semialgebraic sets S, T ⊆ ℝdof source and target states and a matrix $M \in {\mathbb{Q}^{d \times d}}$. The task is to determine whether there is a point in S whose orbit under M intersects the target T in at least m distinct points. The case m = 1 (mere reachability) can be reduced to mild generalisations of the Skolem and Positivity Problems for linear recurrence sequences, whose decidability has been open for many decades. The situation is markedly different for multiple reachability, where m can be greater than one. In this paper, we prove that multiple reachability is undecidable already in dimension d = 10 with fixed multiplicity m = 9. Since our undecidability construction also shows that decision procedures for dimension d ∈ {3, … , 9} would entail significant new results on effective solutions of Diophantine equations, we subsequently focus on the case d = 2, that is, multiple reachability in the plane. Here we obtain two positive results. We show that multiple reachability is decidable if the matrix M is a rotation and it is also decidable without restriction on M for halfplane targets. The former result relies on a theorem in arithmetic geometry, due to Bombieri and Zannier, concerning intersections of algebraic subgroups with subvarieties.

The Skolem Problem asks to determine whether a given integer linear recurrence sequence (LRS) has a zero term. This problem, whose decidability has been open for many decades, arises across a wide range of topics in computer science, including loop termination, formal languages, automata theory, and probabilistic model checking, amongst many others. In the present paper, we introduce a notion of "large" zeros of (non-degenerate) linear recurrence sequences, i.e., zeros occurring at an index larger than a sixth-fold exponential of the size of the data defining the given LRS . We establish two main results. First, we show that large zeros are very sparse: the set of positive integers that can possibly arise as large zeros of some LRS has null density. This in turn immediately yields a Universal Skolem Set of density one, answering a question left open in the literature. Second, we define an infinite set of prime numbers, termed "good", having density one amongst all prime numbers, with the following property: for any large zero of a given LRS, there is an interval around the large zero together with an upper bound on the number of good primes possibly present in that interval. The bound in question is much lower than one would expect if good primes were distributed similarly as ordinary prime numbers, as per the Cramér model in number theory. We therefore conjecture that large zeros do not exist, which would entail decidability of the Skolem Problem.

Expansions of the monadic second-order (MSO) theory of the structure ⟨ℕ;<⟩ have been a fertile and active area of research ever since the publication of the seminal papers of Büchi and Elgot & Rabin on the subject in the 1960s. In the present paper, we establish decidability of the MSO theory of ⟨ℕ;<,P⟩, where P ranges over a large class of unary "dynamical" predicates, i.e., sets of non-negative values assumed by certain integer linear recurrence sequences. One of our key technical tools is the novel concept of (effective) prodisjunctivity, which we expect may also find independent applications further afield.

We prove that for any integers α, β > 1, the existential fragment of the first-order theory of the structure ⟨ℤ; 0,1,<, +,αℕ,ßℕ⟩ is decidable (where αℕ is the set of positive integer powers of α, and likewise for βℕ). On the other hand, we show by way of hardness that decidability of the existential fragment of the theory of ⟨ℕ; 0,1,<, +,x ↦ αx,x ↦ βχ⟩ for any multiplicatively independent α, β > 1 would lead to mathematical breakthroughs regarding base-α and base-β expansions of certain transcendental numbers. Finally, modifying the original proof of Hieronymi and Schulz we show that for any multiplicatively independent α, β > 1, it is undecidable whether a given formula with at most 3 alternating blocks of quantifiers holds in ⟨ℕ;0,1,<, +,αℕ,ßℕ⟩.

The Monniaux Problem in abstract interpretation asks, roughly speaking, whether the following question is decidable: Given a program P , a safety (e.g., non-reachability) specification \(\varphi\) , and an abstract domain of invariants \(\mathcal {D}\) , does there exist an inductive invariant \(\mathcal {I}\) in \(\mathcal {D}\) guaranteeing that program P meets its specification φ? The Monniaux Problem is of course parameterised by the classes of programs and invariant domains that one considers. In this article, we show that the Monniaux Problem is undecidable for unguarded affine programs and semilinear invariants (unions of polyhedra). Moreover, we show that decidability is recovered in the important special case of simple linear loops.

For which unary predicates P 1 , … , P m is the MSO theory of the structure 〈 N ; < , P 1 , … , P m 〉 decidable? We survey the state of the art, leading us to investigate combinatorial properties of almost-periodic, morphic, and toric words. In doing so, we show that if each P i can be generated by a toric dynamical system of a certain kind, then the attendant MSO theory is decidable. We give various applications of toric words, including the recent result of [1] that the MSO theory of 〈 N ; < , { 2 n : n ∈ N } , { 3 n : n ∈ N } 〉 is decidable.

In discrete-time linear dynamical systems (LDSs), a linear map is repeatedly applied to an initial vector yielding a sequence of vectors called the orbit of the system. A weight function assigning weights to the points in the orbit can be used to model quantitative aspects, such as resource consumption, of a system modelled by an LDS. This paper addresses the problems to compute the mean payoff, the total accumulated weight, and the discounted accumulated weight of the orbit under continuous weight functions and polynomial weight functions as a special case. Besides general LDSs, the special cases of stochastic LDSs and of LDSs with bounded orbits are considered. Furthermore, the problem of deciding whether an energy constraint is satisfied by the weighted orbit, i.e., whether the accumulated weight never drops below a given bound, is analysed.

A linear constraint loop is specified by a system of linear inequalities that define the relation between the values of the program variables before and after a single execution of the loop body. In this paper we consider the problem of determining whether such a loop terminates, i.e., whether all maximal executions are finite, regardless of how the loop is initialised and how the non-determinism in the loop body is resolved. We focus on the variant of the termination problem in which the loop variables range over ℝ. Our main result is that the termination problem is decidable over the reals in dimension 2. A more abstract formulation of our main result is that it is decidable whether a binary relation on ℝ² that is given as a conjunction of linear constraints is well-founded.

We consider numbers of the form S_β(u): = ∑_{n=0}^∞ (u_n)/(βⁿ), where u = ⟨u_n⟩_{n=0}^∞ is an infinite word over a finite alphabet and β ∈ ℂ satisfies |β| > 1. Our main contribution is to present a combinatorial criterion on u, called echoing, that implies that S_β(u) is transcendental whenever β is algebraic. We show that every Sturmian word is echoing, as is the Tribonacci word, a leading example of an Arnoux-Rauzy word. We furthermore characterise ̅{ℚ}-linear independence of sets of the form {1, S_β(u₁),…,S_β(u_k)}, where u₁,…,u_k are Sturmian words having the same slope. Finally, we give an application of the above linear independence criterion to the theory of dynamical systems, showing that for a contracted rotation on the unit circle with algebraic slope, its limit set is either finite or consists exclusively of transcendental elements other than its endpoints 0 and 1. This confirms a conjecture of Bugeaud, Kim, Laurent, and Nogueira.

We investigate the decidability of the monadic second-order (MSO) theory of the structure (N; <, P1,…,Pk), for various unary predicates P1,…,Pk ⊆ N. We focus in particular on 'arithmetic' predicates arising in the study of linear recurrence sequences, such as fixed-base powers Powk = {kn : n ∈ N}, k-th powers Nk = {nk : n ∈ N}, and the set of terms of the Fibonacci sequence Fib = {0, 1, 2, 3, 5, 8, 13,…} (and similarly for other linear recurrence sequences having a single, non-repeated, dominant characteristic root). We obtain several new unconditional and conditional decidability results, a select sample of which are the following:

The matrix semigroup membership problem asks, given square matrices $M,M_1,\ldots,M_k$ of the same dimension, whether $M$ lies in the semigroup generated by $M_1,\ldots,M_k$. It is classical that this problem is undecidable in general but decidable in case $M_1,\ldots,M_k$ commute. In this paper we consider the problem of whether, given $M_1,\ldots,M_k$, the semigroup generated by $M_1,\ldots,M_k$ contains a non-negative matrix. We show that in case $M_1,\ldots,M_k$ commute, this problem is decidable subject to Schanuel's Conjecture. We show also that the problem is undecidable if the commutativity assumption is dropped. A key lemma in our decidability result is a procedure to determine, given a matrix $M$, whether the sequence of matrices $(M^n)_{n\geq 0}$ is ultimately nonnegative. This answers a problem posed by S. Akshay (arXiv:2205.09190). The latter result is in stark contrast to the notorious fact that it is not known how to determine effectively whether for any specific matrix index $(i,j)$ the sequence $(M^n)_{i,j}$ is ultimately nonnegative (which is a formulation of the Ultimate Positivity Problem for linear recurrence sequences).

Abstract We introduce the notion of porous invariants for multipath affine loops over the integers. These are invariants definable in (fragments of) Presburger arithmetic and, as such, lack certain tame geometrical properties, such a convexity and connectedness. Nevertheless, we show that in many cases such invariants can be automatically synthesised, and moreover can be used to settle reachability questions for various non-trivial classes of affine loops and target sets. For the class of $$\mathbb {Z}$$ Z -linear invariants (those defined as conjunctions of linear equations with integer coefficients), we show that a strongest such invariant can be computed in polynomial time. For the more general class of $$\mathbb {N}$$ N -semi-linear invariants (those defined as Boolean combinations of linear inequalities with integer coefficients), such a strongest invariant need not exist. Here we show that for point targets the existence of a separating invariant is undecidable in general. However we show that such separating invariants can be computed either by restricting the number of program variables or by restricting from multipath to single-path loops. Additionally, we consider porous targets, represented as $$\mathbb {Z}$$ Z -semi-linear sets (those defined as Boolean combinations of equations with integer coefficients). We show that an invariant can be computed providing the target spans the whole space. We present our tool porous, which computes porous invariants.

One of the most basic, longstanding open problems in the theory of dynamical systems is whether reachability is decidable for one-dimensional piecewise affine maps with two intervals. In this paper we prove that for injective maps, it is decidable.We also study various related problems, in each case either establishing decidability, or showing that they are closely connected to Diophantine properties of certain transcendental numbers, analogous to the positivity problem for linear recurrence sequences. Lastly, we consider topological properties of orbits of one-dimensional piecewise affine maps, not necessarily with two intervals, and negatively answer a question of Bournez, Kurganskyy, and Potapov, about the set of orbits in expanding maps.

The Positivity Problem for linear recurrence sequences over a ring R of real algebraic numbers is to determine, given an LRS ${\left( {{u_n}} \right)_{n \in \mathbb{N}}}$ over R, whether un≥ 0 for all n. It is known to be Turing-equivalent to the following reachability problem: given a linear dynamical system (M, s) Rd×d×Rdand a halfspace H ⊆ ℝd, determine whether the orbit ${\left( {{M^n}s} \right)_{n \in \mathbb{N}}}$ ever enters H. The more general model-checking problem for LDS is to determine, given (M, s) and an ω-regular property φ over semialgebraic predicates T1,…, Tℓ⊆ ℝd, whether the orbit of (M, s) satisfies φ.In this paper, we establish the following1)The Positivity Problem for LRS over real algebraic numbers reduces to the Positivity Problem for LRS over the integers; and2)The model-checking problem for LDS with diagonalisable M is decidable subject to a Positivity oracle for simple LRS over the integers.In other words, the full semialgebraic model-checking problem for diagonalisable linear dynamical systems is no harder than the Positivity Problem for simple integer linear recurrence sequences. This is in sharp contrast with the situation for arbitrary (not necessarily diagonalisable) LDS and arbitrary (not necessarily simple) integer LRS, for which no such correspondence is expected to hold.

Abstract 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 $$\omega $$ ω -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 with the unrounded setting where point-to-point reachability is known to be decidable in polynomial time.

We consider the problem of deciding the existence of real roots of real-valued exponential polynomials with algebraic coefficients. Such functions arise as solutions of linear differential equations with real algebraic coefficients. We focus on two problems: theZero Problem, which asks whether an exponential polynomial has a real root, and theInfinite Zeros Problem, which asks whether such a function has infinitely many real roots. Our main result is that for differential equations of order at most 8 the Zero Problem is decidable, subject to Schanuel’s Conjecture, while the Infinite Zeros Problem is decidable unconditionally. We show moreover that a decision procedure for the Infinite Zeros Problem at order 9 would yield an algorithm for computing the Lagrange constant of any given real algebraic number to arbitrary precision, indicating that it will be very difficult to extend our decidability results to higher orders.

A polynomial program is one in which all assignments are given by polynomial expressions and in which all branching is nondeterministic (as opposed to conditional). Given such a program, an algebraic invariant is one that is defined by polynomial equations over the program variables at each program location. Müller-Olm and Seidl have posed the question of whether one can compute the strongest algebraic invariant of a given polynomial program. In this article, we show that, while strongest algebraic invariants are not computable in general, they can be computed in the special case of affine programs, that is, programs with exclusively linear assignments. For the latter result, our main tool is an algebraic result of independent interest: Given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate.

We consider the model-checking problem for parametric probabilistic dynamical systems, formalised as Markov chains with parametric transition functions, analysed under the distribution-transformer semantics (in which a Markov chain induces a sequence of distributions over states). We examine the problem of synthesising the set of parameter valuations of a parametric Markov chain such that the orbits of induced state distributions satisfy a prefix-independent ω-regular property. Our main result establishes that in all non-degenerate instances, the feasible set of parameters is (up to a null set) semialgebraic, and can moreover be computed (in polynomial time assuming that the ambient dimension, corresponding to the number of states of the Markov chain, is fixed).

It is a longstanding open problem whether there is an algorithm to decide the Skolem Problem for linear recurrence sequences (LRS) over the integers, namely whether a given such sequence has a zero term (i.e., whether un = 0 for some n). A major breakthrough in the early 1980s established decidability for LRS of order 4 or less, i.e., for LRS in which every new term depends linearly on the previous four (or fewer) terms. The Skolem Problem for LRS of order 5 or more, in particular, remains a major open challenge to this day.

The celebrated Skolem-Mahler-Lech Theorem states that the set of zeros of a linear recurrence sequence is the union of a finite set and finitely many arithmetic progressions. The corresponding computational question, the Skolem Problem, asks to determine whether a given linear recurrence sequence has a zero term. Although the Skolem-Mahler-Lech Theorem is almost 90 years old, decidability of the Skolem Problem remains open. The main contribution of this paper is an algorithm to solve the Skolem Problem for simple linear recurrence sequences (those with simple characteristic roots). Whenever the algorithm terminates, it produces a stand-alone certificate that its output is correct - a set of zeros together with a collection of witnesses that no further zeros exist. We give a proof that the algorithm always terminates assuming two classical number-theoretic conjectures: the Skolem Conjecture (also known as the Exponential Local-Global Principle) and the p-adic Schanuel Conjecture. Preliminary experiments with an implementation of this algorithm within the tool Skolem point to the practical applicability of this method.

We study fundamental reachability problems on pseudo-orbits of linear dynamical systems. Pseudo-orbits can be viewed as a model of computation with limited precision and pseudo-reachability can be thought of as a robust version of classical reachability. Using an approach based on $o$-minimality of $\reals_{\exp}$ we prove decidability of the discrete-time pseudo-reachability problem with arbitrary semialgebraic targets for diagonalisable linear dynamical systems. We also show that our method can be used to reduce the continuous-time pseudo-reachability problem to the (classical) time-bounded reachability problem, which is known to be conditionally decidable.

We study the Escape Problem for discrete-time linear dynamical systems over compact semialgebraic sets. We establish a uniform upper bound on the number of iterations it takes for every orbit of a rational matrix to escape a compact semialgebraic set defined over rational data. Our bound is doubly exponential in the ambient dimension, singly exponential in the degrees of the polynomials used to define the semialgebraic set, and singly exponential in the bitsize of the coefficients of these polynomials and the bitsize of the matrix entries. We show that our bound is tight by providing a matching lower bound.

The concept of decomposition in computer science and engineering is considered a fundamental component of computational thinking and is prevalent in design of algorithms, software construction, hardware design, and more. We propose a simple and natural formalization of sequential decomposition, in which a task is decomposed into two sequential sub-tasks, with the first sub-task to be executed before the second sub-task is executed. These tasks are specified by means of input/output relations. We define and study decomposition problems, which is to decide whether a given specification can be sequentially decomposed. Our main result is that decomposition itself is a difficult computational problem. More specifically, we study decomposition problems in three settings: where the input task is specified explicitly, by means of Boolean circuits, and by means of automatic relations. We show that in the first setting decomposition is NP-complete, in the second setting it is NEXPTIME-complete, and in the third setting there is evidence to suggest that it is undecidable. Our results indicate that the intuitive idea of decomposition as a system-design approach requires further investigation. In particular, we show that adding a human to the loop by asking for a decomposition hint lowers the complexity of decomposition problems considerably.

We consider the MSO model-checking problem for simple linear loops, or equivalently discrete-time linear dynamical systems, with semialgebraic predicates (i.e., Boolean combinations of polynomial inequalities on the variables). We place no restrictions on the number of program variables, or equivalently the ambient dimension. We establish decidability of the model-checking problem provided that each semialgebraic predicate either has intrinsic dimension at most 1, or is contained within some three-dimensional subspace. We also note that lifting either of these restrictions and retaining decidability would necessarily require major breakthroughs in number theory.

Termination analysis of linear loops plays a key rôle in several areas of computer science, including program verification and abstract interpretation. Already for the simplest variants of linear loops the question of termination relates to deep open problems in number theory, such as the decidability of the Skolem and Positivity Problems for linear recurrence sequences, or equivalently reachability questions for discrete-time linear dynamical systems. In this article, we introduce the class of o-minimal invariants , which is broader than any previously considered, and study the decidability of the existence and algorithmic synthesis of such invariants as certificates of non-termination for linear loops equipped with a large class of halting conditions. We establish two main decidability results, one of them conditional on Schanuel’s conjecture is transcendental number theory.

Abstract We introduce the notion of porous invariants for multipath (or branching/nondeterministic) affine loops over the integers; these invariants are not necessarily convex, and can in fact contain infinitely many ‘holes’. Nevertheless, we show that in many cases such invariants can be automatically synthesised, and moreover can be used to settle (non-)reachability questions for various interesting classes of affine loops and target sets.

We study a parametric version of the Kannan-Lipton Orbit Problem for linear dynamical systems. We show decidability in the case of one parameter and Skolem-hardness with two or more parameters. More precisely, consider a $d$-dimensional square matrix $M$ whose entries are algebraic functions in one or more real variables. Given initial and target vectors $u,v\in \mathbb{Q}^d$, the parametric point-to-point orbit problem asks whether there exist values of the parameters giving rise to a concrete matrix $N \in \mathbb{R}^{d\times d}$, and a positive integer $n\in \mathbb{N}$, such that $N^nu = v$. We show decidability for the case in which $M$ depends only upon a single parameter, and we exhibit a reduction from the well-known Skolem Problem for linear recurrence sequences, suggesting intractability in the case of two or more parameters.

We study decision problems for sequences which obey a second-order holonomic recurrence of the form f(n + 2) = P(n) f(n + 1) + Q(n) f(n) with rational polynomial coefficients, where P is non-constant, Q is non-zero, and the degree of Q is smaller than or equal to that of P. We show that existence of infinitely many zeroes is decidable. We give partial algorithms for deciding the existence of a zero, positivity of all sequence terms, and positivity of all but finitely many sequence terms. If Q does not have a positive integer zero then our algorithms halt on almost all initial values (f(1), f(2)) for the recurrence. We identify a class of recurrences for which our algorithms halt for all initial values. We further identify a class of recurrences for which our algorithms can be extended to total ones.

It is a longstanding open problem whether there is an algorithm to decide the Skolem Problem for linear recurrence sequences, namely whether a given such sequence has a zero term. In this paper we introduce the notion of a Universal Skolem Set: an infinite subsetSof the positive integers such that there is an effective procedure that inputs a linear recurrence sequence u = (u(n))n ≥ 0and decides whether u(n) = 0 for some n ∈S. The main technical contribution of the paper is to exhibit such a set.

We study fundamental decision problems on linear dynamical systems in discrete time. We focus on pseudo-orbits, the collection of trajectories of the dynamical system for which there is an arbitrarily small perturbation at each step. Pseudo-orbits are generalizations of orbits in the topological theory of dynamical systems. We study the pseudo-orbit problem, whether a state belongs to the pseudo-orbit of another state, and the pseudo-Skolem problem, whether a hyperplane is reachable by an ε-pseudo-orbit for every ε. These problems are analogous to the well-studied orbit problem and Skolem problem on unperturbed dynamical systems. Our main results show that the pseudo-orbit problem is decidable in polynomial time and the Skolem problem on pseudo-orbits is decidable. The former extends the seminal result of Kannan and Lipton from orbits to pseudo-orbits. The latter is in contrast to the Skolem problem for linear dynamical systems, which remains open for proper orbits.

We study the computational complexity of the Escape Problem for discrete-time linear dynamical systems over compact semialgebraic sets, or equivalently the Termination Problem for affine loops with compact semialgebraic guard sets. Consider the fragment of the theory of the reals consisting of negation-free $\exists \forall$-sentences without strict inequalities. We derive several equivalent characterisations of the associated complexity class which demonstrate its robustness and illustrate its expressive power. We show that the Compact Escape Problem is complete for this class.

An infinite sequence $\langle{u_n}\rangle_{n\in\mathbb{N}}$ of real numbers is holonomic (also known as P-recursive or P-finite) if it satisfies a linear recurrence relation with polynomial coefficients. Such a sequence is said to be positive if each $u_n \geq 0$, and minimal if, given any other linearly independent sequence $\langle{v_n}\rangle_{n \in\mathbb{N}}$ satisfying the same recurrence relation, the ratio $u_n/v_n$ converges to $0$. In this paper, we focus on holonomic sequences satisfying a second-order recurrence $g_3(n)u_n = g_2(n)u_{n-1} + g_1(n)u_{n-2}$, where each coefficient $g_3, g_2,g_1 \in \mathbb{Q}[n]$ is a polynomial of degree at most $1$. We establish two main results. First, we show that deciding positivity for such sequences reduces to deciding minimality. And second, we prove that deciding minimality is equivalent to determining whether certain numerical expressions (known as periods, exponential periods, and period-like integrals) are equal to zero. Periods and related expressions are classical objects of study in algebraic geometry and number theory, and several established conjectures (notably those of Kontsevich and Zagier) imply that they have a decidable equality problem, which in turn would entail decidability of Positivity and Minimality for a large class of second-order holonomic sequences.

We consider the problem of deciding ω-regular properties on infinite traces produced by linear loops. Here we think of a given loop as producing a single infinite trace that encodes information about the signs of program variables at each time step. Formally, our main result is a procedure that inputs a prefix-independent ω-regular property and a sequence of numbers satisfying a linear recurrence, and determines whether the sign description of the sequence (obtained by replacing each positive entry with “+”, each negative entry with “−”, and each zero entry with “0”) satisfies the given property. Our procedure requires that the recurrence be simple, i.e., that the update matrix of the underlying loop be diagonalisable. This assumption is instrumental in proving our key technical lemma: namely that the sign description of a simple linear recurrence sequence is almost periodic in the sense of Muchnik, Sem'enov, and Ushakov. To complement this lemma, we give an example of a linear recurrence sequence whose sign description fails to be almost periodic. Generalising from sign descriptions, we also consider the verification of properties involving semi-algebraic predicates on program variables.

We exhibit an algorithm to compute the strongest algebraic (or polynomial) invariants that hold at each location of a given guard-free linear hybrid automaton (i.e., a hybrid automaton having only unguarded transitions, all of whose assignments are given by affine expressions, and all of whose continuous dynamics are given by linear differential equations). Our main tool is a control-theoretic result of independent interest: given such a linear hybrid automaton, we show how to discretise the continuous dynamics in such a way that the resulting automaton has precisely the same algebraic invariants.

We consider the problem of synthesising polynomial ranking functions for single-path loops over the reals with continuous semi-algebraic update function and compact semi-algebraic guard set. We show that a loop of this form has a polynomial ranking function if and only if it terminates. We further show that termination is decidable for such loops in the special case where the update function is affine.

We consider reachability in dynamical systems with discrete linear updates, but with fixed digital precision, i.e., such that values of the system are rounded at each step. Given a matrix M ∈ ℚ^{d × d}, an initial vector x ∈ ℚ^{d}, a granularity g ∈ ℚ_+ and a rounding operation [⋅] projecting a vector of ℚ^{d} onto another vector whose every entry is a multiple of g, we are interested in the behaviour of the orbit 𝒪 = ⟨[x], [M[x]],[M[M[x]]],… ⟩, i.e., the trajectory of a linear dynamical system in which the state is rounded after each step. For arbitrary rounding functions with bounded effect, we show that the complexity of deciding point-to-point reachability - whether a given target y ∈ ℚ^{d} belongs to 𝒪 - is PSPACE-complete for hyperbolic systems (when no eigenvalue of M has modulus one). We also establish decidability without any restrictions on eigenvalues for several natural classes of rounding functions.

Continuous linear dynamical systems are used extensively in mathematics, computer science, physics, and engineering to model the evolution of a system over time. A central technique for certifying safety properties of such systems is by synthesising inductive invariants. This is the task of finding a set of states that is closed under the dynamics of the system and is disjoint from a given set of error states. In this paper we study the problem of synthesising inductive invariants that are definable in o-minimal expansions of the ordered field of real numbers. In particular, assuming Schanuel's conjecture in transcendental number theory, we establish effective synthesis of o-minimal invariants in the case of semi-algebraic error sets. Without using Schanuel's conjecture, we give a procedure for synthesizing o-minimal invariants that contain all but a bounded initial segment of the orbit and are disjoint from a given semi-algebraic error set. We further prove that effective synthesis of semi-algebraic invariants that contain the whole orbit, is at least as hard as a certain open problem in transcendental number theory.

The Skolem Problem asks, given a linear recurrence sequence (un), whether there exists n ∈ N such that un = 0. In this paper we consider the following specialisation of the problem: given in addition c ∈ N, determine whether there exists n ∈ N of the form n = lpk, with k, l ≤ c and p any prime number, such that un = 0.

Consider a discrete dynamical system given by a square matrix $M \in \mathbb{Q}^{d \times d}$ and a starting point $s \in \mathbb{Q}^d$. The orbit of such a system is the infinite trajectory $\langle s, Ms, M^2s, \ldots\rangle$. Given a collection $T_1, T_2, \ldots, T_m \subseteq \mathbb{R}^d$ of semialgebraic sets, we can associate with each $T_i$ an atomic proposition $P_i$ which evaluates to true at time $n$ if, and only if, $M^ns \in T_i$. This gives rise to the LTL Model-Checking Problem for discrete linear dynamical systems: given such a system $(M,s)$ and an LTL formula over such atomic propositions, determine whether the orbit satisfies the formula. The main contribution of the present paper is to show that the LTL Model-Checking Problem for discrete linear dynamical systems is decidable in dimension 3 or less.

The Continuous Polytope Escape Problem (CPEP) asks whether every trajectory of a linear differential equation initialised within a convex polytope eventually escapes the polytope. We provide a polynomial-time algorithm to decide CPEP for compact polytopes. We also establish a quantitative uniform upper bound on the time required for every trajectory to escape the given polytope. In addition, we establish iteration bounds for termination of discrete linear loops via reduction to the continuous case.

Automated invariant generation is a fundamental challenge in program analysis and verification, going back many decades, and remains a topic of active research. In this talk I'll present a select overview and survey of work on this problem, and discuss unexpected connections to other fields including algebraic geometry, group theory, and quantum computing. (No previous knowledge of these topics will be assumed.) This is joint work with Ehud Hrushovski, Amaury Pouly, and James Worrell.

We consider the decidability of state-to-state reachability in linear time-invariant control systems over discrete time. We analyse this problem with respect to the allowable control sets, which in general are assumed to be defined by boolean combinations of linear inequalities. Decidability of the version of the reachability problem in which control sets are affine subspaces of Rn is a fundamental result in control theory. Our first result is that reachability is undecidable if the set of controls is a finite union of affine subspaces. We also consider versions of the reachability problem in which (i) the set of controls consists of a single affine subspace together with the origin and (ii) the set of controls is a convex polytope. In these two cases we respectively show that the reachability problem is as hard as Skolem's Problem and the Positivity Problem for linear recurrence sequences (whose decidability has been open for several decades). Our main contribution is to show decidability of a version of the reachability problem in which control sets are convex polytopes, under certain spectral assumptions on the transition matrix.

We consider the Membership and the Half-Space Reachability problems for matrices in dimensions two and three. Our first main result is that the Membership Problem is decidable for finitely generated sub-semigroups of the Heisenberg group over rational numbers. Furthermore, we prove two decidability results for the Half-Space Reachability Problem. Namely, we show that this problem is decidable for sub-semigroups of GL(2,Z) and of the Heisenberg group over rational numbers.

We consider the problem of deciding termination of single-path while loops with integer variables, affine updates, and affine guard conditions. The question is whether such a loop terminates on all integer initial values. This problem is known to be decidable for the subclass of loops whose update matrices are diagonalisable, but the general case has remained open since being conjectured decidable by Tiwari in 2004. In this paper we show decidability of determining termination for arbitrary update matrices, confirming Tiwari's conjecture. For the class of loops considered in this paper, the question of deciding termination on a specific initial value is a longstanding open problem in number theory. The key to our decision procedure is in showing how to circumvent the difficulties inherent in deciding termination on a fixed initial value.

The Semialgebraic Orbit Problem is a fundamental reachability question that arises in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. An instance of the problem comprises a dimension d in N, a square matrix A in Q^{d x d}, and semialgebraic source and target sets S,T subseteq R^d. The question is whether there exists x in S and n in N such that A^nx in T. The main result of this paper is that the Semialgebraic Orbit Problem is decidable for dimension d <= 3. Our decision procedure relies on separation bounds for algebraic numbers as well as a classical result of transcendental number theory - Baker’s theorem on linear forms in logarithms of algebraic numbers. We moreover argue that our main result represents a natural limit to what can be decided (with respect to reachability) about the orbit of a single matrix. On the one hand, semialgebraic sets are arguably the largest general class of subsets of R^d for which membership is decidable. On the other hand, previous work has shown that in dimension d=4, giving a decision procedure for the special case of the Orbit Problem with singleton source set S and polytope target set T would entail major breakthroughs in Diophantine approximation.

We consider the decidability of the membership problem for matrix-exponential semigroups: Given k ∈ N and square matrices A 1 , … , A k , C , all of the same dimension and with real algebraic entries, decide whether C is contained in the semigroup generated by the matrix exponentials exp ( A i t ), where i ∈ { 1,… , k } and t ≥ 0. This problem can be seen as a continuous analog of Babai et al.’s and Cai et al.’s problem of solving multiplicative matrix equations and has applications to reachability analysis of linear hybrid automata and switching systems. Our main results are that the semigroup membership problem is undecidable in general, but decidable if we assume that A 1 , … , A k commute. The decidability proof is by reduction to a version of integer programming that has transcendental constants. We give a decision procedure for the latter using Baker’s theorem on linear forms in logarithms of algebraic numbers, among other tools. The undecidability result is shown by reduction from Hilbert’s Tenth Problem.

It is known that Metric Temporal Logic (MTL) is strictly less expressive than the Monadic First-Order Logic of Order and Metric (FO[<, +1]) when interpreted over timed words; this remains true even when the time domain is bounded a priori. In this work, we present an extension of MTL with the same expressive power as FO[<, +1] over bounded timed words (and also, trivially, over time-bounded signals). We then show that expressive completeness also holds in the general (time-unbounded) case if we allow the use of rational constants $q \in \mathbb{Q}$ in formulas. This extended version of MTL therefore yields a definitive real-time analogue of Kamp's theorem. As an application, we propose a trace-length independent monitoring procedure for our extension of MTL, the first such procedure in a dense real-time setting.

We study the growth behaviour of rational linear recurrence sequences. We show that for low-order sequences, divergence is decidable in polynomial time. We also exhibit a polynomial-time algorithm which takes as input a divergent rational linear recurrence sequence and computes effective fine-grained lower bounds on the growth rate of the sequence.

The termination analysis of linear loops plays a key rôle in several areas of computer science, including program verification and abstract interpretation. Such deceptively simple questions also relate to a number of deep open problems, such as the decidability of the Skolem and Positivity Problems for linear recurrence sequences, or equivalently reachability questions for discrete-time linear dynamical systems. In this paper, we introduce the class of o-minimal invariants, which is broader than any previously considered, and study the decidability of the existence and algorithmic synthesis of such invariants as certificates of non-termination for linear loops equipped with a large class of halting conditions. We establish two main decidability results, one of them conditional on Schanuel's conjecture in transcendental number theory.

The concept of decomposition in computer science and engineering is considered a fundamental component of computational thinking and is prevalent in design of algorithms, software construction, hardware design, and more. We propose a simple and natural formalization of sequential decomposition, in which a task is decomposed into two sequential sub-tasks, with the first sub-task to be executed out before the second sub-task is executed. These tasks are specified by means of input/output relations. We define and study decomposition problems, which is to decide whether a given specification can be sequentially decomposed. Our main result is that decomposition itself is a difficult computational problem. More specifically, we study decomposition problems in three settings: where the input task is specified explicitly, by means of Boolean circuits, and by means of automatic relations. We show that in the first setting decomposition is NP-complete, in the second setting it is NEXPTIME-complete, and in the third setting there is evidence to suggest that it is undecidable. Our results indicate that the intuitive idea of decomposition as a system-design approach requires further investigation. In particular, we show that adding human to the loop by asking for a decomposition hint lowers the complexity of decomposition problems considerably.

We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose assignments are given by affine expressions). Our main tool is an algebraic result of independent interest: given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate.

Freeze LTL is a temporal logic with registers that is suitable for specifying properties of data words. In this paper we study the model checking problem for Freeze LTL on one-counter automata. This problem is known to be undecidable in general and PSPACE-complete for the special case of deterministic one-counter automata. Several years ago, Demri and Sangnier investigated the model checking problem for the flat fragment of Freeze LTL on several classes of counter automata and posed the decidability of model checking flat Freeze LTL on one-counter automata as an open problem. In this paper we resolve this problem positively, utilising a known reduction to a reachability problem on one-counter automata with parameterised equality and disequality tests. Our main technical contribution is to show decidability of the latter problem by translation to Presburger arithmetic.

The Polytope Escape Problem for continuous linear dynamical systems consists of deciding, given an affine function f:Rd -> Rd and a convex polytope P⊆ Rd, both with rational descriptions, whether there exists an initial point x0 in P such that the trajectory of the unique solution to the differential equation: ·x(t)=f(x(t)) x 0= x0 is entirely contained in P. We show that this problem is reducible in polynomial time to the decision version of linear programming with real algebraic coefficients. The latter is a special case of the decision problem for the existential theory of real closed fields, which is known to lie between NP and PSPACE. Our algorithm makes use of spectral techniques and relies, among others, on tools from Diophantine approximation.

The Orbit Problem consists of determining, given a matrix A in R^dxd and vectors x,y in R^d, whether there exists n in N such that A^n=y. This problem was shown to be decidable in a seminal work of Kannan and Lipton in the 1980s. Subsequently, Kannan and Lipton noted that the Orbit Problem becomes considerably harder when the target y is replaced with a subspace of R^d. Recently, it was shown that the problem is decidable for vector-space targets of dimension at most three, followed by another development showing that the problem is in PSPACE for polytope targets of dimension at most three. In this work, we take a dual look at the problem, and consider the case where the initial vector x is replaced with a polytope P_1, and the target is a polytope P_2. Then, the question is whether there exists n in N such that A^n P_1 intersection P_2 does not equal the empty set. We show that the problem can be decided in PSPACE for dimension at most three. As in previous works, decidability in the case of higher dimensions is left open, as the problem is known to be hard for long-standing number-theoretic open problems. Our proof begins by formulating the problem as the satisfiability of a parametrized family of sentences in the existential first-order theory of real-closed fields. Then, after removing quantifiers, we are left with instances of simultaneous positivity of sums of exponentials. Using techniques from transcendental number theory, and separation bounds on algebraic numbers, we are able to solve such instances in PSPACE.

This volume contains the proceedings of the 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), held at Reykjavík University in Iceland from 20 to 23 June 2017. LICS is an annual international forum on the broad range of topics that lie at the intersection of computer science and mathematical logic. In addition to the main symposium, seven workshops were co-located with LICS 2017: • INFINITY: Verification of Infinite-State Systems • LearnAut: Learning and Automata • LCC: Logic and Computational Complexit • LMW: Logic Mentoring Workshop • LOLA: Syntax and Semantics of Low-Level Languages • Metafinite model theory and definability and complexity of numeric graph parameters • WiL: Women in Logic

The Orbit Problem consists of determining, given a linear transformation A on d-dimensional rationals Q^d, together with vectors x and y, whether the orbit of x under repeated applications of A can ever reach y. This problem was famously shown to be decidable by Kannan and Lipton in the 1980s. In this paper, we are concerned with the problem of synthesising suitable invariants P which are subsets of R^d, i.e., sets that are stable under A and contain x and not y, thereby providing compact and versatile certificates of non-reachability. We show that whether a given instance of the Orbit Problem admits a semialgebraic invariant is decidable, and moreover in positive instances we provide an algorithm to synthesise suitable invariants of polynomial size. It is worth noting that the existence of semilinear invariants, on the other hand, is (to the best of our knowledge) not known to be decidable.

Freeze LTL is a temporal logic with registers that is suitable for specifying properties of data words. In this paper we study the model checking problem for Freeze LTL on one-counter automata. This problem is known to be undecidable in full generality and PSPACE-complete for the special case of deterministic one-counter automata. Several years ago, Demri and Sangnier investigated the model checking problem for the flat fragment of Freeze LTL on several classes of counter automata and posed the decidability of model checking flat Freeze LTL on one-counter automata as an open problem. In this paper we resolve this problem positively, utilising a known reduction to a reachability problem on one-counter automata with parameterised equality and disequality tests. Our main technical contribution is to show decidability of the latter problem by translation to Presburger arithmetic.

Herman's self-stabilisation algorithm, introduced 25 years ago, is a well-studied synchronous randomised protocol for enabling a ring of $N$ processes collectively holding any odd number of tokens to reach a stable state in which a single token remains. Determining the worst-case expected time to stabilisation is the central outstanding open problem about this protocol. It is known that there is a constant $h$ such that any initial configuration has expected stabilisation time at most $h N^2$. Ten years ago, McIver and Morgan established a lower bound of $4/27 \approx 0.148$ for $h$, achieved with three equally-spaced tokens, and conjectured this to be the optimal value of $h$. A series of papers over the last decade gradually reduced the upper bound on $h$, with the present record (achieved in 2014) standing at approximately $0.156$. In this paper, we prove McIver and Morgan's conjecture and establish that $h = 4/27$ is indeed optimal.

The Continuous Skolem Problem asks whether a real-valued function satisfying a linear differential equation has a zero in a given interval of real numbers. This is a fundamental reachability problem for continuous linear dynamical systems, such as linear hybrid automata and continuoustime Markov chains. Decidability of the problem is currently open — indeed decidability is open even for the sub-problem in which a zero is sought in a bounded interval. In this paper we show decidability of the bounded problem subject to Schanuel's Conjecture, a unifying conjecture in transcendental number theory. We furthermore analyse the unbounded problem in terms of the frequencies of the differential equation, that is, the imaginary parts of the characteristic roots. We show that the unbounded problem can be reduced to the bounded problem if there is at most one rationally linearly independent frequency, or if there are two rationally linearly independent frequencies and all characteristic roots are simple. We complete the picture by showing that decidability of the unbounded problem in the case of two (or more) rationally linearly independent frequencies would entail a major new effectiveness result in Diophantine approximation, namely computability of the Diophantine-approximation types of all real algebraic numbers.

The continuous evolution of a wide variety of systems, including continuous-time Markov chains and linear hybrid automata, can be described in terms of linear differential equations. In this paper we study the decision problem of whether the solution x(t) of a system of linear differential equations dx/dt = Ax reaches a target halfspace infinitely often. This recurrent reachability problem can equivalently be formulated as the following Infinite Zeros Problem: does a real-valued function f: R≥0 → R satisfying a given linear differential equation have infinitely many zeros? Our main decidability result is that if the differential equation has order at most 7, then the Infinite Zeros Problem is decidable. On the other hand, we show that a decision procedure for the Infinite Zeros Problem at order 9 (and above) would entail a major breakthrough in Diophantine Approximation, specifically an algorithm for computing the Lagrange constants of arbitrary real algebraic numbers to arbitrary precision.

We consider a continuous analogue of (Babai et al. 1996)'s and (Cai et al. 2000)'s problem of solving multiplicative matrix equations. Given k + 1 square matrices A1, ..., Ak, C, all of the same dimension, whose entries are real algebraic, we examine the problem of deciding whether there exist non-negative reals t1, ..., tk such that

We establish a relationship between reachability problems in timed automata and space-bounded counter automata. We show that reachability in timed automata with three or more clocks is logarithmic-space inter-reducible with reachability in space-bounded counter automata with two counters. We moreov er show the logarithmic-space equivalence of reachability in two-clock timed automata and space-bounded one-counter automata. This last reduction has recently been employed by Fearnley and Jurdziński to settle the computational complexity of reachability in two-clock timed automata.

We consider higher-dimensional versions of Kannan and Lipton’s Orbit Problem—determining whether a target vector space ν may be reached from a starting point x under repeated applications of a linear transformation A . Answering two questions posed by Kannan and Lipton in the 1980s, we show that when ν has dimension one, this problem is solvable in polynomial time, and when ν has dimension two or three, the problem is in NP RP .

Metric temporal logic (MTL) is one of the most prominent specification formalisms for real-time systems. Over infinite timed words, full MTL is undecidable, but satisfiability for a syntactially defined safety fragment, called safety MTL, was proved decidable several years ago. Satisfiability for safety MTL is also known to be equivalent to a fair termination problem for a class of channel machines with insertion errors. However, hitherto, its precise computational complexity has remained elusive, with only a nonelementary lower bound. Via another equivalent problem, namely termination for a class of rational relations, we show that satisfiability for safety MTL is A ckermann -complete (i.e., among the easiest nonprimitive recursive problems). This is surprising since decidability was originally established using Higman’s Lemma, suggesting a much higher nonmultiply recursive complexity.

We consider the complexity of deciding the truth of first-order existential sentences of linear arithmetic with divisibility over both the integers and the p-adic numbers. We show that if an existential sentence of Presburger arithmetic with divisibility is satisfiable then the smallest satisfying assignment has size at most exponential in the size of the formula, showing that the decision problem for such sentences is in NEXPTIME. Establishing this upper bound requires subtle adaptations to an existing decidability proof of Lipshitz. We consider also the first-order linear theory of the p-adic numbers. Here divisibility can be expressed via the valuation function. The decision problem for existential sentences over the p-adic numbers is an important component of the decision procedure for existential Presburger arithmetic with divisibility. The problem is known to be NP-hard and in EXPTIME, as a second main contribution, we show that this problem lies in the Counting Hierarchy, and therefore in PSPACE.

We consider polyhedral versions of Kannan and Lip-ton's Orbit Problem [14, 13]—determining whether a target polyhedron V may be reached from a starting point x under repeated applications of a linear transformation A in an ambient vector space ℚm. In the context of program verification, very similar reachability questions were also considered and left open by Lee and Yannakakis in [15], and by Braverman in [4]. We present what amounts to a complete characterisation of the decidability landscape for the Polyhedron-Hitting Problem, expressed as a function of the dimension m of the ambient space, together with the dimension of the polyhedral target V: more precisely, for each pair of dimensions, we either establish decidability, or show hardness for longstanding number-theoretic open problems.

A fundamental problem in program verification concerns the termination of simple linear loops of the form: where x is a vector of variables, u, a, and c are integer vectors, and A and B are integer matrices. Assuming the matrix A is diagonalisable, we give a decision procedure for the problem of whether, for all initial integer vectors u, such a loop terminates. The correctness of our algorithm relies on sophisticated tools from algebraic and analytic number theory, Diophantine geometry, and real algebraic geometry. To the best of our knowledge, this is the first substantial advance on a 10-year-old open problem of Tiwari [38] and Braverman [8].

We investigate the Matrix Powering Positivity Problem, PosMatPow: given an m X m square integer matrix M, a linear function f: Z^{m X m} -> Z with integer coefficients, and a positive integer n (encoded in binary), determine whether f(M^n) \geq 0. We show that for fixed dimensions m of 2 and 3, this problem is decidable in polynomial time.

We consider two decision problems for linear recurrence sequences (LRS) over the integers, namely the Positivity Problem (are all terms of a given LRS positive?) and the Ultimate Positivity Problem (are all but finitely many terms of a given LRS positive?). We show decidability of both problems for LRS of order 5 or less, with complexity in the Counting Hierarchy for Positivity, and in polynomial time for Ultimate Positivity. Moreover, we show by way of hardness that extending the decidability of either problem to LRS of order 6 would entail major breakthroughs in analytic number theory, more precisely in the field of Diophantine approximation of transcendental numbers.

Metric Temporal Logic (MTL) is a generalisation of Linear Temporal Logic in which the Until and Since modalities are annotated with intervals that express metric constraints. Hirshfeld and Rabinovich have shown that over the reals, firstorder logic with binary order relation <; and unary function +1 is strictly more expressive than MTL with integer constants. Indeed they prove that no temporal logic whose modalities are definable by formulas of bounded quantifier depth can be expressively complete for FO(<;, +1). In this paper we show that if we allow unary functions +q, q ∈ Q, in first-order logic and correspondingly allow rational constants in MTL, then the two logics have the same expressive power. This gives the first generalisation of Kamp's theorem on the expressive completeness of LTL for FO(<;) to the quantitative setting. The proof of this result involves a generalisation of Gabbay's notion of separation to the metric setting.

We consider higher-dimensional versions of Kannan and Lipton's Orbit Problem---determining whether a target vector space V may be reached from a starting point x under repeated applications of a linear transformation A. Answering two questions posed by Kannan and Lipton in the 1980s, we show that when V has dimension one, this problem is solvable in polynomial time, and when V has dimension two or three, the problem is in NPRP.

Bounded model checking (BMC) is a highly successful bug-finding method that examines paths of bounded length for violations of a given regular or w-regular specification. A completeness threshold for a given model M and specification φ is a bound k such that, if no counterexample to φ of length k or less can be found in M, then M in fact satisfies φ. The quest for `small' completeness thresholds in BMC goes back to the very inception of the technique, over a decade ago, and remains a topic of active research. For a fixed specification, completeness thresholds are typically expressed in terms of key attributes of the models under consideration, such as their diameter (length of the longest shortest path) and especially their recurrence diameter (length of the longest loop-free path). A recent research paper identified a large class of LTL specifications having completeness thresholds linear in the models' recurrence diameter [7]. However, the authors left open the question of whether linearity is in general even decidable. In the present paper, we settle the problem in the affirmative, by showing that the linearity problem for both regular and ω-regular specifications (provided as automata and Buchi automata respectively) is PSPACE-complete. Moreover, we establish the following dichotomies: for regular specifications, completeness thresholds are either linear or exponential, whereas for ω-regular specifications, completeness thresholds are either linear or at least quadratic.

Abstract A channel machine consists of a finite controller together with several fifo channels; the controller can read messages from the head of a channel and write messages to the tail of a channel. In this paper we focus on channel machines with insertion errors , i.e., machines in whose channels messages can spontaneously appear. We consider the invariance problem: does a given insertion channel machine have an infinite computation all of whose configurations satisfy a given predicate? We show that this problem is primitive-recursive if the predicate is closed under message losses. We also give a non-elementary lower bound for the invariance problem under this restriction. Finally, using the previous result, we show that the satisfiability problem for the safety fragment of Metric Temporal Logic is non-elementary.

Abstract Herman’s algorithm is a synchronous randomized protocol for achieving self-stabilization in a token ring consisting of N processes. The interaction of tokens makes the dynamics of the protocol very difficult to analyze. In this paper we study the distribution of the time to stabilization, assuming that there are three tokens in the initial configuration. We show for arbitrary N and for an arbitrary timeout t that the probability of stabilization within time t is minimized by choosing as the initial three-token configuration the configuration in which the tokens are placed equidistantly on the ring. Our result strengthens a corollary of a theorem of McIver and Morgan (Inf. Process Lett. 94(2): 79–84, 2005 ), which states that the expected stabilization time is minimized by the equidistant configuration.

A fundamental problem in numerical computation and computational geometry is to determine the sign of arithmetic expressions in radicals. Here we consider the simpler problem of deciding whether $\sum_{i=1}^m C_i A_i^{X_i}$ is zero for given rational numbers $A_i$, $C_i$, $X_i$. It has been known for almost twenty years that this can be decided in polynomial time. In this paper we improve this result by showing membership in uniform TC0. This requires several significant departures from Blömer's polynomial-time algorithm as the latter crucially relies on primitives, such as gcd computation and binary search, that are not known to be in TC0.

Alternating timed automata are a powerful extension of classical Alur-Dill timed automata that are closed under all Boolean operations. They have played a key role, among others, in providing verification algorithms for prominent specification formalisms such as Metric Temporal Logic. Unfortunately, when interpreted over an infinite dense time domain (such as the reals), alternating time automata have an undecidable language emptiness problem. The main result of this paper is that, over bounded time domains, language emptiness for alternating timed automata is decidable (but nonelementary). The proof involves showing decidability of a class of parametric McNaughton games that are played over timed words and that have winning conditions expressed in the monadic logic of order augmented with the distance-one relation. As a corollary, we establish the decidability of the time-bounded model-checking problem for Alur-Dill timed automata against specifications expressed as alternating timed automata.

A channel machine consists of a finite controller together with several fifo channels; the controller can read messages from the head of a channel and write messages to the tail of a channel. In this paper, we focus on channel machines with insertion errors, i.e., machines in whose channels messages can spontaneously appear. Such devices have been previously introduced in the study of Metric Temporal Logic. We consider the termination problem: are all the computations of a given insertion channel machine finite? We show that this problem has non-elementary, yet primitive recursive complexity.

In an influential paper titled "The benefits of relaxing punctuality" [2], Alur, Feder, and Henzinger introduced Metric Interval Temporal Logic (MITL) as a fragment of the real-time logic metric temporal logic (MTL) in which exact or punctual timing constraints are banned. Their main result showed that model checking and satisfiability for MITL are both EXPSPACE-Complete. Until recently, it was widely believed that admitting even the simplest punctual specifications in any linear-time temporal logic would automatically lead to undecidability. Although this was recently disproved, until now no punctual fragment of MTL was known to have even primitive recursive complexity (with certain decidable fragments having provably non-primitive recursive complexity). In this paper we identify a "co-flat' subset of MTL that is capable of expressing a large class of punctual specifications and for which model checking (although not satisfiability) has no complexity cost over MITL. Our logic is moreover qualitatively different from MITL in that it can express properties that are not timed-regular. Correspondingly, our decision procedures do not involve translating formulas into finite-state automata, but rather into certain kinds of reversal-bounded Turing machines. Using this translation we show that the model checking problem for our logic is EXPSPACE-Complete.

Metric Temporal Logic (MTL) is a prominent specification formalism for real-time systems. In this paper, we show that the satisfiability problem for MTL over finite timed words is decidable, with non-primitive recursive complexity. We also consider the model-checking problem for MTL: whether all words accepted by a given Alur-Dill timed automaton satisfy a given MTL formula. We show that this problem is decidable over finite words. Over infinite words, we show that model checking the safety fragment of MTL--which includes invariance and time-bounded response properties--is also decidable. These results are quite surprising in that they contradict various claims to the contrary that have appeared in the literature.

Metric temporal logic (MTL) is a prominent specification formalism for real-time systems. In this paper, we show that the satisfiability problem for MTL over finite timed words is decidable, with non-primitive recursive complexity. We also consider the model-checking problem for MTL: whether all words accepted by a given Alur-Dill timed automaton satisfy a given MTL formula. We show that this problem is decidable over finite words. Over infinite words, we show that model checking the safety fragment of MTL-which includes invariance and time-bounded response properties-is also decidable. These results are quite surprising in that they contradict various claims to the contrary that have appeared in the literature. The question of the decidability of MTL over infinite words remains open.

Abstract We present a framework for model checking concurrent software systems which incorporates both states and events. Contrary to other state/event approaches, our work also integrates two powerful verification techniques, counterexample-guided abstraction refinement and compositional reasoning. Our specification language is a state/event extension of linear temporal logic, and allows us to express many properties of software in a concise and intuitive manner. We show how standard automata-theoretic LTL model checking algorithms can be ported to our framework at no extra cost, enabling us to directly benefit from the large body of research on efficient LTL verification. We also present an algorithm to detect deadlocks in concurrent message-passing programs. Deadlock- freedom is not only an important and desirable property in its own right, but is also a prerequisite for the soundness of our model checking algorithm. Even though deadlock is inherently non-compositional and is not preserved by classical abstractions, our iterative algorithm employs both (non-standard) abstractions and compositional reasoning to alleviate the state-space explosion problem. The resulting framework differs in key respects from other instances of the counterexample-guided abstraction refinement paradigm found in the literature. We have implemented this work in the magic verification tool for concurrent C programs and performed tests on a broad set of benchmarks. Our experiments show that this new approach not only eases the writing of specifications, but also yields important gains both in space and in time during verification. In certain cases, we even encountered specifications that could not be verified using traditional pure event-based or state-based approaches, but became tractable within our state/event framework. We also recorded substantial reductions in time and memory consumption when performing deadlock-freedom checks with our new abstractions. Finally, we report two bugs (including a deadlock) in the source code of Micro-C/OS versions 2.0 and 2.7, which we discovered during our experiments.

REVIEWSthe old techniques are the ones that are discussed most of the time, even when new testing techniques and tools are covered.It seems to me that there are two types of software testing book.The first type focuses on the practical aspects and tends to include guidelines and templates, and it may even define new names for already-invented concepts.The other type tries to be objective and teach software quality and testing without the 'hard sell' arguments.This particular book is somewhere in-between, i.e. it is focused on the practical parts of software testing, but it is also objective when it comes to techniques and concepts.Hence, it can be recommended, not only to practitioners of software testing, but also to students who want to read a stepby-step guide to software testing.If you have read a lot of software testing books, there is obviously not much new for you in this one.However, if you want to read about software testing in a task-oriented way, and get plenty of examples of templates, this is the book for you.In addition, the well-written part on modern testing tools (Section V), in combination with the bonus CD-ROM, was highly interesting and, in fact, fun to read.

We consider the language inclusion problem for timed automata: given two timed automata A and B, are all the timed traces accepted by B also accepted by A? While this problem is known to be undecidable, we show here that it becomes decidable if A is restricted to having at most one clock. This is somewhat surprising, since it is well-known that there exist timed automata with a single clock that cannot be complemented. The crux of our proof consists in reducing the language inclusion problem to a reachability question on an infinite graph; we then construct a suitable well-quasi-order on the nodes of this graph, which ensures the termination of our search algorithm. We also show that the language inclusion problem is decidable if the only constant appearing among the clock constraints of A is zero. Moreover, these two cases are essentially the only decidable instances of language inclusion, in terms of restricting the various resources of timed automata.

We present an algorithm to detect deadlocks in concurrent message-passing programs. Even though deadlock is inherently noncompositional and its absence is not preserved by standard abstractions, our framework employs both abstraction and compositional reasoning to alleviate the state space explosion problem. We iteratively construct increasingly more precise abstractions on the basis of spurious counterexamples to either detect a deadlock or prove that no deadlock exists. Our approach is inspired by the counterexample-guided abstraction refinement paradigm. However, our notion of abstraction as well as our schemes for verification and abstraction refinement differs in key respects from existing abstraction refinement frameworks. Our algorithm is also compositional in that abstraction, counterexample validation, and refinement are all carried out component-wise and do not require the construction of the complete state space of the concrete system under consideration. Finally, our approach is completely automated and provides diagnostic feedback in case a deadlock is detected. We have implemented our technique in the MAGIC verification tool and present encouraging results (up to 20 times speed-up in time and 4 times less memory consumption) with concurrent message-passing C programs. We also report a bug in the real-time operating system MicroC/OS version 2.70.

We consider several questions related to the use of digitization techniques for timed automata. These very successful techniques reduce dense-time language inclusion problems to discrete time, but are applicable only when the implementation is closed under digitization and the specification is closed under inverse digitization. We show that, for timed automata, the former (whether the implementation is closed under digitization) is decidable, but not the latter. We also investigate digitization questions in connection with the robust semantics for timed automata. The robust modeling approach introduces a timing fuzziness through the semantic removal of equality testing. Since its introduction half a decade ago, research into the robust semantics has suggested that it yields roughly the same theory as the standard semantics. This paper shows that, surprisingly, this is not the case: the robust semantics is significantly less tractable, and differs from the standard semantics in many key respects. In particular, the robust semantics yields an undecidable (nonregular) discrete-time theory, in stark contrast with the standard semantics. This makes it virtually impossible to apply digitization techniques together with the robust semantics. On the positive side, we show that the robust languages of timed automata remain recursive.