This paper introduces a generic framework that provides sufficient conditions for guaranteeing polynomial-time decidability of fixed-negation fragments of first-order theories that adhere to certain fixed-parameter tractability requirements. It enables deciding sentences of such theories with arbitrary existential quantification, conjunction and a fixed number of negation symbols in polynomial time. It was recently shown by Nguyen and Pak [SIAM J. Comput. 51(2): 1--31 (2022)] that an even more restricted such fragment of Presburger arithmetic (the first-order theory of the integers with addition and order) is NP-hard. In contrast, by application of our framework, we show that the fixed negation fragment of weak Presburger arithmetic, which drops the order relation from Presburger arithmetic in favour of equality, is decidable in polynomial time. We give two further examples of instantiations of our framework, showing polynomial-time decidability of the fixed negation fragments of weak linear real arithmetic and of the restriction of Presburger arithmetic in which each inequality contains at most one variable.

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.

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.

In this paper we provide a characterization of the complexity class PSPACE by using a purely continuous model defined with polynomial ordinary differential equations.

We investigate the Membership Problem for hypergeometric sequences: given a hypergeometric sequence ❬un❭∞n=0 of rational numbers and a target t∈Q, decide whether t occurs in the sequence. We show decidability of this problem under the assumption that in the defining recurrence p(n)un = q(n)un-1, the roots of the polynomials p(x) and q(x) are all rational numbers. Our proof relies on bounds on the density of primes in arithmetic progressions. We also observe a relationship between the decidability of the Membership problem (and variants) and the Rohrlich-Lang conjecture in transcendence theory.

We investigate the complexity of computing the Zariski closure of a finitely generated group of matrices. The Zariski closure was previously shown to be computable by Derksen, Jeandel, and Koiran, but the termination argument for their algorithm appears not to yield any complexity bound. In this paper we follow a different approach and obtain a bound on the degree of the polynomials that define the closure. Our bound shows that the closure can be computed in elementary time. We also obtain upper bounds on the length of chains of linear algebraic groups, where all the groups are generated over a fixed number field.

We consider the decidability of state-to-state reachability in linear time-invariant control systems over continuous time. We analyze this problem with respect to the allowable control sets, which are assumed to be the image under a linear map of the unit hypercube (i.e. zonotopes). This naturally models bounded (sometimes called saturated) controls. 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 decidability in two dimensions (n = 2) if matrix A satisfies some spectral conditions and conditional decidablility in general. If the transformation matrix A is diagonal with rational entries (or rational multiples of the same algebraic number) then the reachability problem is decidable. If the transformation matrix A only has real eigenvalues, the reachability problem is conditionally decidable. The time-bounded reachability problem is conditionally decidable and unconditionally decidable in two dimensions. Some of our results rely on the decidability of certain logical theories --- namely the theory of the reals with exponential (Rexp) and with bounded sine (Rexp,sin)--- which have been proven decidable conditional on Schanuel's Conjecture --- a unifying conjecture in transcendence theory. We also obtain a hardness result for a mild generalization of the problem where the target is a simple set (hypercube of dimension n - 1 or hyperplane) instead of a point. In this case, we show that the problem is at least as hard as the Continuous Positivity problem if the control set is a singleton, or the Nontangential Continuous Positivity problem if the control set is [-1, 1].

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 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.

An astonishing fact was established by Lee A. Rubel (1981): there exists a fixed non-trivial fourth-order polynomial differential algebraic equation (DAE) such that for any positive continuous function $\varphi$ on the reals, and for any positive continuous function $\epsilon(t)$, it has a $\mathcal{C}^\infty$ solution with $| y(t) - \varphi(t) | < \epsilon(t)$ for all $t$. Lee A. Rubel provided an explicit example of such a polynomial DAE. Other examples of universal DAE have later been proposed by other authors. However, Rubel's DAE \emph{never} has a unique solution, even with a finite number of conditions of the form $y^{(k_i)}(a_i)=b_i$. The question whether one can require the solution that approximates $\varphi$ to be the unique solution for a given initial data is a well known open problem [Rubel 1981, page 2], [Boshernitzan 1986, Conjecture 6.2]. In this article, we solve it and show that Rubel's statement holds for polynomial ordinary differential equations (ODEs), and since polynomial ODEs have a unique solution given an initial data, this positively answers Rubel's open problem. More precisely, we show that there exists a \textbf{fixed} polynomial ODE such that for any $\varphi$ and $\epsilon(t)$ there exists some initial condition that yields a solution that is $\epsilon$-close to $\varphi$ at all times. In particular, the solution to the ODE is necessarily analytic, and we show that the initial condition is computable from the target function and error function.

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 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.

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.

An astonishing fact was established by Lee A. Rubel (1981): there exists a fixed non-trivial fourth-order polynomial differential algebraic equation (DAE) such that for any positive continuous function phi on the reals, and for any positive continuous function epsilon(t), it has a C^infinity solution with | y(t) - phi(t) | < epsilon(t) for all t. Lee A. Rubel provided an explicit example of such a polynomial DAE. Other examples of universal DAE have later been proposed by other authors. However, while these results may seem very surprising, their proofs are quite simple and are frustrating for a computability theorist, or for people interested in modeling systems in experimental sciences. First, the involved notions of universality is far from usual notions of universality in computability theory. In particular, the proofs heavily rely on the fact that constructed DAE does not have unique solutions for a given initial data. This is very different from usual notions of universality where one would expect that there is clear unambiguous notion of evolution for a given initial data, for example as in computability theory. Second, the proofs usually rely on solutions that are piecewise defined. Hence they cannot be analytic, while analycity is often a key expected property in experimental sciences. Third, the proofs of these results can be interpreted more as the fact that (fourth-order) polynomial algebraic differential equations is a too loose a model compared to classical ordinary differential equations. In particular, one may challenge whether the result is really a universality result. The question whether one can require the solution that approximates phi to be the unique solution for a given initial data is a well known open problem [Rubel 1981, page 2], [Boshernitzan 1986, Conjecture 6.2]. In this article, we solve it and show that Rubel's statement holds for polynomial ordinary differential equations (ODEs), and since polynomial ODEs have a unique solution given an initial data, this positively answers Rubel's open problem. More precisely, we show that there exists a fixed polynomial ODE such that for any phi and epsilon(t) there exists some initial condition that yields a solution that is epsilon-close to phi at all times. The proof uses ordinary differential equation programming. We believe it sheds some light on computability theory for continuous-time models of computations. It also demonstrates that ordinary differential equations are indeed universal in the sense of Rubel and hence suffer from the same problem as DAEs for modelization: a single equation is capable of modelling any phenomenon with arbitrary precision, meaning that trying to fit a model based on polynomial DAEs or ODEs is too general (if ithas a sufficient dimension).

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.

The outcomes of this article are twofold. Implicit complexity. We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class P of languages computable in polynomial time in terms of differential equations with polynomial right-hand side. This result gives a purely continuous elegant and simple characterization of P. We believe it is the first time complexity classes are characterized using only ordinary differential equations. Our characterization extends to functions computable in polynomial time over the reals in the sense of Computable Analysis. Our results may provide a new perspective on classical complexity, by giving a way to define complexity classes, like P, in a very simple way, without any reference to a notion of (discrete) machine. This may also provide ways to state classical questions about computational complexity via ordinary differential equations. Continuous-Time Models of Computation. Our results can also be interpreted in terms of analog computers or analog models of computation: As a side effect, we get that the 1941 General Purpose Analog Computer (GPAC) of Claude Shannon is provably equivalent to Turing machines both in terms of computability and complexity, a fact that has never been established before. This result provides arguments in favour of a generalised form of the Church-Turing Hypothesis, which states that any physically realistic (macroscopic) computer is equivalent to Turing machines both in terms of computability and complexity.

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.

The outcomes of this article are twofold. Implicit complexity. We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class P of languages computable in polynomial time in terms of differential equations with polynomial right-hand side. This result gives a purely continuous elegant and simple characterization of P. We believe it is the first time complexity classes are characterized using only ordinary differential equations. Our characterization extends to functions computable in polynomial time over the reals in the sense of Computable Analysis. Our results may provide a new perspective on classical complexity, by giving a way to define complexity classes, like P, in a very simple way, without any reference to a notion of (discrete) machine. This may also provide ways to state classical questions about computational complexity via ordinary differential equations. Continuous-Time Models of Computation. Our results can also be interpreted in terms of analog computers or analog models of computation: As a side effect, we get that the 1941 General Purpose Analog Computer (GPAC) of Claude Shannon is provably equivalent to Turing machines both in terms of computability and complexity, a fact that has never been established before. This result provides arguments in favour of a generalised form of the Church-Turing Hypothesis, which states that any physically realistic (macroscopic) computer is equivalent to Turing machines both in terms of computability and complexity.

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

In this paper we prove that computing the solution of an initial-value problem y = p(y) with initial condition y(t0) = y0 ∈ Rd at time t0 + T with precision 2−μ where p is a vector of polynomials can be done in time polynomial in the value of T, μ and Y = [equation]. Contrary to existing results, our algorithm works over any bounded or unbounded domain. Furthermore, we do not assume any Lipschitz condition on the initial-value problem.