Reachability in Dynamical Systems with Rounding
Christel Baier, Florian Funke, Simon Jantsch, Toghrul Karimov, Engel Lefaucheux, Joël Ouaknine, Amaury Pouly, David Purser, Markus A. Whiteland
RReachability in Dynamical Systems with Rounding
Christel Baier
Technische Universität Dresden, Germany
Florian Funke
Technische Universität Dresden, Germany
Simon Jantsch
Technische Universität Dresden, Germany
Toghrul Karimov
Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
Engel Lefaucheux
Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
Joël Ouaknine
Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, GermanyDepartment of Computer Science, Oxford University, UK
Amaury Pouly
Université de Paris, CNRS, IRIF, F-75006, Paris, France
David Purser
Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
Markus A. Whiteland
Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
Abstract
We consider reachability in dynamical systems with discrete linear updates, but with fixed digitalprecision, 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 theorbit O = h [ x ] , [ M [ x ]] , [ M [ M [ x ]]] , . . . i , i.e., the trajectory of a linear dynamical system in which thestate is rounded after each step. For arbitrary rounding functions with bounded effect, we showthat the complexity of deciding point-to-point reachability—whether a given target y ∈ Q d belongsto 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 ofrounding functions. Theory of computation
Keywords and phrases dynamical systems, rounding, reachability
Funding
This work was funded by DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science ), the Cluster of Excellence EXC 2050/1 (CeTI, project ID390696704, as part of Germany’s Excellence Strategy), DFG-projects BA-1679/11-1 and BA-1679/12-1, and the Research Training Group QuantLA (GRK 1763).
Joël Ouaknine : Supported by ERC grant AVS-ISS (648701).
Amaury Pouly : Supported by CODYS project ANR-18-CE40-0007. a r X i v : . [ c s . CC ] S e p Reachability in Dynamical Systems with Rounding A discrete-time linear dynamical system in ambient space Q d is specified via a linear trans-formation together with a starting point. The state of the system is then updated at eachstep by applying the linear transformation, giving rise to an orbit (or infinite trajectory) in Q d .One of the most well-known questions for such systems is the Skolem Problem , whichasks whether the orbit ever hits a given ( d − This problem haslong eluded decidability, although instances of dimension d ≤ point-to-point reachability , known to bedecidable in polynomial time [22]. In both cases, however, one assumes arbitrary precision,which arguably is unrealistic for simulations carried out on digital computers. In this paper,we therefore turn our attention to instances of these problems in which the numerical stateof the system is rounded to finite precision at each time step. This leads us to the followingdefinition: (cid:73) Problem (Rounded Point-to-Point Reachability (Rounded P2P)) . Given a matrix M ∈ Q d × d ,an initial vector x ∈ Q d , a target vector y ∈ Q d , a granularity g ∈ Q + , and a roundingoperation [ · ] projecting a vector of Q d onto another vector whose every entry is a multipleof g , let the orbit O of this system be the infinite sequence h [ x ] , [ M [ x ]] , [ M [ M [ x ]]] , . . . i , i.e., x (0) = [ x ] and x ( i +1) = [ M x ( i ) ] . The Rounded Point-to-Point Reachability (RoundedP2P) Problem asks whether [ y ] ∈ O . Main contributions.
We make the following contributions, summarised in Figure 1: We introduce a family of natural problems, Rounded P2P (parameterised by the roundingfunction), which to the best of our knowledge has not previously been studied. We show that for hyperbolic systems (i.e., those whose associated linear transformationhas no eigenvalue of modulus 1) the Rounded P2P Problem is solvable—and is in fact
PSPACE -complete—for any ‘reasonable’ (i.e., bounded-effect) rounding function. Itis interesting to note, in contrast, that exact P2P reachability is known to be solvablein polynomial time. Our approach to solving the Rounded P2P Problem relies on theobservation that, outside a ball of exponential size, the change in magnitude of the systemstate at each step dwarfs any effect due to rounding. It thus suffices to exhaustivelyexamine the effect of the dynamics inside an exponentially bounded state space. In the general case (without any restriction on the magnitude of eigenvalues), the effectof rounding may forever remain non-negligible, requiring a careful analysis. We have notbeen able to solve the problem in full generality, but we do provide a complete solutionfor certain natural classes of rounding functions. More precisely, assume that the lineartransformation has been converted to Jordan normal form (now requiring us to work withcomplex algebraic numbers). We can then solve the Rounded P2P Problem under twonatural classes of rounding functions: (a)
Polar rounding functions: given a complex number of the form Ae iθ , such functionsround A and θ independently. In such instances we can handle in EXPSPACE all The Skolem Problem is usually formulated in terms of linear recurrence sequences, but is equivalent tothe description given here. Historically this problem has been known as the orbit problem , however there are now multiple ‘orbitproblems’ (polytope reachability, hyperplane reachability, (semi-)algebraic set reachability,... etc.) andso we specify point-to-point reachability. . Baier et al. 3
Rounding type HyperbolicSystems No restrictions on eigenvaluesJordan normal form(Note: no hardness) GeneralPolar Ae iθ PSPACE -complete,Section 3
EXPSPACE , Section 4.1 Open but
PSPACE -hardArgand truncation or expansion
EXPSPACE , Section 4.2Argand minimal error Open (difficultieshighlighted in Section 5)Arbitrary bounded-effect Open (Open Problem 21)
Figure 1
Decidability and complexity table for the Rounded P2P Problem. reasonable rounding functions on A , and what we view as the only natural roundingfunction on θ . (b) Argand rounding: given a complex number of the form a + bi , the Argand truncation will round a and b independently downwards (in magnitude), ensuring that themodulus never increases. Similarly, the Argand expansion (which rounds a and b independently upwards) guarantees that the modulus can only increase. Under suchrounding functions, we show decidability in EXPSPACE . We highlight some limitations of our methods, identifying a simple but technicallychallenging open problem, which points to some of the key difficulties in solving theRounded P2P Problem in full generality. More precisely, we consider minimal errorrounding for a simple rotation in two-dimensional space, for which Rounded P2P ispresently open. (cid:73)
Remark.
It is worth noting that the rounded versions of the Skolem Problem (does therounded orbit ever hit a ( d − d -dimensional half-space?) remain at least as hard as theirexact integer counterparts, since over the integers rounding has no effect; the decidabilityof these problems therefore remains open. However, the rounded versions of reaching abounded polytope or a bounded semialgebraic set (problems not known to be decidable inthe exact setting [14, 4]) reduce to a finite number of Rounded P2P reachability queries(since a bounded set can contain only finitely many rounded points). These observationstogether motivate our focus, in the present paper, on the Rounded P2P Problem.It is interesting to consider rounded reachability problems in the stochastic setting, i.e.,Markov chains. One observes that the state space [ S ] = { [ x ] ∈ [0 , d | x sub-stochastic } is finite, which entails decidability of virtually any reachability problem, including Skolemand Positivity. This is somewhat arresting, since without rounding reachability problemsare known to be exactly as hard for stochastic systems as for general systems [3]. In anyevent, one should note that ensuring that for all x ∈ [ S ], [ M x ] ∈ [ S ] requires some care, asarbitrary rounding does not necessarily preserve (sub-)stochasticity. Related work
With the emerging use of numerical computations during the 80s, doubts were raisedconcerning the transferability of results about dynamical systems obtained by simulation infinite-state machines. In this direction, the sensitivity that a rounding function may have onthe long-term behaviour of a dynamical system is studied in [5]. How rounded orbits can besimulated by actual orbits of the dynamical system is investigated in [20, 29].
Reachability in Dynamical Systems with Rounding
The series of papers [6, 7, 8, 9] examines which statistical properties of a discrete dynamicalsystem are preserved under the introduction of a rounding function, a good summary of whichcan be found in Blank’s book [10, Chapter 5]. As the rounding is refined, some propertiesof the discretized orbits follow probabilistic laws asymptotically, as shown in [16, 17]. Thepaper [18] studies how volatile statistical notions are in the presence of finite precision (suchas the mean distance of two orbits of discrete dynamical systems).Another line of research focuses on discretized rotations in Z and higher-dimensionallattices [24, 1]. A connection from roundoff problems in the 2-dimensional case to expandingmaps on the p -adic integers is described in [11, 36]. Building on this, [35] conjecturesperiodicity of all orbits of these discretized rotations in Z . It is shown in [2] that there areinfinitely many periodic orbits, and [31] attempts to concisely describe points leading toperiodic orbits.In the context of model checking, continuous dynamical systems have been translated intodiscrete models, mainly timed automata that approximate the behaviour of the original system[26, 13, 32]. On a more general level, one can observe a growing interest in the systematicstudy of roundoff errors inherent in finite precision computations [19, 33, 21, 25, 27, 15]. Let N , Z , Q , R , A be the naturals, integers, rationals, reals, and algebraic numbers respectively. Rounding real numbers
Let g ∈ R + be a granularity. We define our rounding functions taking values to integers, i.e., g = 1. For g = 1 we consider [ x ] = g · [ x/g ]. Given a set S , we let [ S ] = { [ x ] | x ∈ S } . The floor function b x c and ceiling functions d x e are well-known rounding functions inmathematics and computer science. We recall two further rounding functions: Minimal error rounding rounds to the nearest value: [ x ] = arg min y ∈ Z | x − y | . If | x − y | =0 . Truncation (‘towards zero rounding’, to cut off the remaining bits): if x > b x c else d x e , or expansion : if x > d x e else b x c .Whenever possible, we prefer to analyse the problems without choosing a specific roundingfunction, relying only upon the property of bounded effect : (cid:73) Definition 1.
A real rounding function [ · ] : R → R has bounded effect if there exists ∆ such that | x − [ x ] | ≤ ∆ for all x . Rounding complex numbers
Complex numbers have both a real and imaginary part. Thus one can consider roundingeach of the components separately, which we call
Argand rounding . Consider x = a + bi with a, b ∈ R , then let [ x ] = [ a ] + [ b ] i , where [ · ] can be any real rounding function (leading to Argand truncation , Argand expansion and
Argand minimal error rounding functions).However, complex numbers can also be readily represented using polar coordinates asfollows: a number is represented as x = Ae iθ , where A is the modulus and θ is the anglebetween the 2-d coordinates (1 ,
0) and ( a, b ) (when represented as a + bi ). Then, a polarrounding function rounds A and θ independently, i.e. [ x ] = [ A ] e i [ θ ] . The rounding of [ A ] canbe any real rounding function. For the rounding of the angle we always assume minimalerror rounding. That is, given granularity θ g = πR for some R ∈ N , then [ θ ] is a multiple of θ g with minimal error and arbitrary but deterministic tie breaking. . Baier et al. 5 We generalise non-specific bounded-effect rounding to the complex numbers. (cid:73)
Definition 2.
A complex rounding function [ · ] : C → C has bounded effect on the modulus if there exists ∆ such that || x | − | [ x ] || ≤ ∆ for all x . Argand and polar roundings are both defined by applying bounded-effect real roundingfunctions to each component, and have bounded effect under Definition 2. However, notethe distinction with Definition 1; polar rounding can exhibit arbitrary large effects (in thefollowing sense: given any ∆ >
0, one can always find x ∈ C such that | x − [ x ] | > ∆), butnevertheless has only bounded effect on the modulus . (cid:73) Definition 3 ( [ K ] -Ball) . Given a complex rounding function [ · ] and an integer K let a [ K ] -ball be the set of admissible points of modulus at most K , i.e., { [ x ] | x ∈ C , | [ x ] | ≤ K } . Rounding vectors
In general, a rounding function on K induces a rounding function on vectors K d , where[( x , . . . , x d )] = ([ x ] , . . . , [ x d ]), although not all rounding functions on vectors need take thisform. We generalise non-specific bounded-effect rounding to vectors. (cid:73) Definition 4.
A rounding function [ · ] : K d → K d has bounded effect on the modulus if thereexists ∆ such that || x | k − | [ x ] | k | ≤ ∆ for all x and every k ∈ { , , . . . , d } . Finally, we assume that all of our rounding functions can be computed in polynomial timeand are fixed (rather than inputs) in our problems, and thus ∆ is also a fixed parameter.
In this section we establish our first main result for hyperbolic systems, which we first define: (cid:73)
Definition 5 (Hyperbolic System [23, Section 1.2]) . A linear map represented by the matrix M ∈ R d × d is hyperbolic if all of its eigenvalues have modulus different from one. (cid:73) Theorem 6.
The Rounded P2P Problem is
PSPACE -complete for hyperbolic linear mapsrepresented by rational matrices and real rounding functions with bounded effect.
We first demonstrate that the problem is in
PSPACE for matrices in Jordan normalform, to which we will reduce the general case in a second step. As the passage to Jordannormal form inevitably introduces complex numbers,
PSPACE membership will be shownfor Jordan normal form matrices over the algebraic numbers and, accordingly, complexrounding functions with bounded effect on the modulus. To complete the picture we showhardness for hyperbolic systems (in fact, the hardness result applies even for non-hyperbolicsystems, that is for matrices whose eigenvalues may include 1).
PSPACE
We now prove the membership part of Theorem 6 under the additional assumption that thematrices are in Jordan normal form. (cid:73)
Lemma 7.
The Rounded P2P Problem decidable in
PSPACE for any complex roundingfunction with bounded effect on the modulus ∆ and hyperbolic matrices M ∈ A d × d in Jordannormal form. Reachability in Dynamical Systems with Rounding
Proof.
We consider a single Jordan block of dimension d with eigenvalue λ . If the matrix M has multiple Jordan blocks, the algorithm can be run in lock step for each block. Hence,without loss of generality we let M = λ λ ... λ . The idea will be to show that for | λ | >
1, for values large enough growth will outstripthe rounding, and the orbit will grow beyond the target, never to return. If | λ | < | λ | = 1 here.Formally, in each dimension k ∈ { , . . . , d } we compute a radius C k , defining a [ C k ]-ballof radius C k about 0, containing x k and y k such that for all z in the orbit O if z k [ C k ]-ballthen [ M z ] k [ C k ]-ball. That is, if the orbit has left the ball, it will never come back. Thealgorithm proceeds by simulating the orbit from x until one of the following occurs. y is found, in which case return yes , ora point repeats, in which case return no , ora point x ( i ) is found such that (cid:12)(cid:12) ( x ( i ) ) k (cid:12)(cid:12) ≥ C k for some k , in which case return no .Since B = [ { x ∈ R d | for all k | x k | ≤ C k } ] is finite, one of the three must occur. Rememberingall previous points would require too much space. Therefore we record a counter of thenumber of steps taken and once this exceeds the maximum number of points then we knowsome point must have been repeated (possibly many times by this point). Let C = max i C k ,then the bounding hyper-cube of B has (2 C/g ) d points, hence B has fewer points. We showthis number has at most exponential size in the description length of the input, and hencecan be represented in PSPACE . (cid:73) Case 1 (suppose | λ | > ). For the d th component we have ( x ( i +1) ) d = [ λ ( x ( i ) ) d ]. Thereis a bounded effect of the rounding ∆, ensuring (cid:12)(cid:12) ( x ( i +1) ) d (cid:12)(cid:12) ≥ | λ | (cid:12)(cid:12) ( x ( i ) ) d (cid:12)(cid:12) − ∆. So when | λ | (cid:12)(cid:12) ( x ( i ) ) d (cid:12)(cid:12) − ∆ > (cid:12)(cid:12) ( x ( i ) ) d (cid:12)(cid:12) , this component must grow. Let ‘ = max { , ∆ , | y | , . . . , | y d |} .We define the radius C d := ∆ | λ |− + ‘ , which satisfies the desired property described above.Now suppose that the radius C k is defined so that C k ≤ ‘ P d − k +1 j =0 ( | λ |− ) j (holds for k = d )and assume that (cid:12)(cid:12) ( x ( i ) ) j (cid:12)(cid:12) ≤ C j for each j ∈ { k, . . . , d } . For the k − x ( i +1) ) k − = [ λ ( x ( i ) ) k − +1( x ( i ) ) k ]. Since (cid:12)(cid:12) ( x ( i ) ) k (cid:12)(cid:12) ≤ C k , we have (cid:12)(cid:12) ( x ( i +1) ) k − (cid:12)(cid:12) ≥| λ | (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) − ∆ − C k , and there is growth when | λ | (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) − ∆ − C k > (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) , i.e.,when (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) > ∆+ C k | λ |− . So, we may define C k − := ∆+ C k | λ |− + ‘ , which satisfies the propertydescribed above, and moreover, C k − ≤ C k | λ |− + ‘ ≤ ‘ P d − ( k − j =0 ( | λ |− ) j due to our choiceof ‘ . Repeat for all remaining components k − , . . . , C k ≤ ‘ P dj =0 ( | λ |− ) j ≤ ‘ ( d + 1)(1 + ( | λ |− ) d ) for each k , and the claim follows. (cid:73) Case 2 (suppose | λ | < ). We require the ball to have the property that if the orbit leaves,it will never come back. However for | λ | <
1, while initially there may be some growth (dueto other components), once large enough | λ | will dominate and the modulus will decrease.Therefore, we want to ensure we choose the ball large enough that the orbit will never leavethe ball in the first place. The following definitions of the radii C j can easily be altered tofurnish this requirement. By running processes in lock step, here and elsewhere, we mean running all of the processes simultaneously(interleaving instructions for each process) until either x ( i ) = y or one of the processes concludes non-reachability. . Baier et al. 7 Consider the last component d : we have (cid:12)(cid:12) ( x ( i +1) ) d (cid:12)(cid:12) ≤ | λ | (cid:12)(cid:12) ( x ( i ) ) d (cid:12)(cid:12) + ∆. Set again ‘ =max { , ∆ , | y | , . . . , | y d |} and define C d := ∆1 −| λ | + ‘ ; if (cid:12)(cid:12) ( x ( i ) ) d (cid:12)(cid:12) ≤ C d , then (cid:12)(cid:12) ( x ( i +1) ) d (cid:12)(cid:12) ≤ C d .Having fixed C k for k ∈ { k, . . . , d } , consider component k −
1: We have ( x ( i +1) ) k − =[ λ ( x ( i ) ) k − + ( x ( i ) ) k ], and so (cid:12)(cid:12) ( x ( i +1) ) k − (cid:12)(cid:12) ≤ | λ | (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) + (cid:12)(cid:12) ( x ( i ) ) k (cid:12)(cid:12) + ∆. Let us define C k − := C k +∆1 −| λ | + ‘ . Now if (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) ≤ C k − then (cid:12)(cid:12) ( x ( i +1) ) k − (cid:12)(cid:12) ≤ C k − . Repeat foreach remaining component. It can be shown, similar to the previous case, that C k ≤ ‘ ( d + 1)(1 + ( −| λ | ) d ) for each k , and this concludes the proof. (cid:74) Reducing the general form to Jordan normal form
In the previous section we assumed that the matrix is always in Jordan normal form, whichis a significant restriction. In this section we will not assume Jordan normal form, whichmeans we cannot make any assumption about the rounding, other than being of boundedeffect, to prove Theorem 6. After a change of basis properties such as ‘rounding towardszero’ may not be preserved.
Proof (upper bound of Theorem 6).
Let ∆ be the fixed, bounded effect on the modulus of[ · ]. Let us consider hyperbolic M = P JP − ∈ Q d × d . We ask whether x ( i +1) = y for some i .Observe that x ( i +1) = [ M x ( i ) ] = M x ( i ) + e ( M x ( i ) ) where e ( x ) := [ x ] − x ∈ [ − ∆ , ∆] d for any x since [ · ] has bounded effect. Now if we define z ( i ) := P − x ( i ) we have that z ( i +1) = P − x ( i +1) = P − ( M x ( i ) + e ( M x ( i ) )) = Jz ( i ) + P − e ( P Jz ( i ) ) = (cid:76) Jz ( i ) (cid:77) where (cid:76) z (cid:77) := z + P − e ( P z ) for any z . The question x ( i ) ? = y for some i now becomesequivalent to z ( i ) ? = P − y . But note that the system for z ( i ) is in Jordan normal formand the rounding function (cid:76) · (cid:77) has bounded effect on the modulus, with bound ∆ ≤ max ≤ k ≤ d max e ∈ [ − ∆ , ∆] d ( P − e ) k . Since ∆ is fixed and P − is computable in polynomialtime [12], then ∆ is of polynomial size. Hence, we have produced in polynomial time aninstance of the Rounded P2P problem with a matrix in Jordan normal form. As the proof ofLemma 7 shows that this problem is solvable in PSPACE even if ∆ is given as input, wecan conclude that the
PSPACE upper bound holds also for the general case. (cid:74)
We will prove
PSPACE -hardness (i.e., the lower bound of Theorem 6) by reduction fromquantified boolean formula (QBF), which is
PSPACE -complete [34]. We do this by firstencoding a simple programming language into the rounded P2P Problem. Then, we showthat reachability in this language can solve QBF. Whilst a direct reduction is possible, weprovide exposition via the language for two reasons; first, we will show that the language isrobust to choice of rounding function (Remark 9), and secondly the reduction results in aninstance where all eigenvalues have modulus 1, but by a small perturbation, we observe thatthe problem remains hard when all of the eigenvalues do not have modulus 1 (Remark 10).The language will consist of m instructions, operating over d variables. Each instruction isa boolean map f i : [0 , d → [0 , d , where each dimension i is updated using a logical formulaof the d inputs. Each of the m instructions is conducted in turn and updating the d variablesis simultaneous in each step. Thus, references to variable in a function are the evaluationin the previous step. Once the m instructions are complete, the system returns to the firstinstruction and repeats ( x ( i ) = ( f m ◦ f m − ◦ · · · ◦ f ◦ f )( x ( i − ), see also Algorithm 1).An instruction is encoded into the rounded dynamical system using a map f i : N d → N d for 0 ≤ i ≤ m −
1, where instructions are of the form ( f i ( x )) j = b ( p j · x ) c where p j in Reachability in Dynamical Systems with Rounding
Algorithm 1
System behaviour of the language
Input: x ∈ [0 , d initial vector, y ∈ [0 , d target vector while x = y do x ← f ( x ) x ← f ( x ) e.g. = x ← x ∨ ( x ∧ x ) x ← if ( x ∨ x ) then x else x x ← true ... x d ← x ... x ← f m ( x ) end Q d . We demonstrate how to encode the required logical operations in a rounded dynamicalsystem: and ( x i ← x j ∧ x k = j x j + x k k ), or ( x i ← x j ∨ x k = j x j + x k k ), negation( x i ← ¬ x j = b − x j c ), resetting a variable to false ( x i ← b c ), copying a variable withoutchange ( x i ← b x i c ) or moving/duplicating a variable ( x i ← b x j c ). To enable this, we willassume there is always access to the constant 1 (or true ) by an implicit dimension, fixed to1. In multiple steps any logical formula can be evaluated. This can be done with auxiliaryvariables to store partial computations, where the instructions will in fact be multi-stepinstructions making use of a finite collection of auxiliary variables which will not be referencedexplicitly. Meanwhile any unused variables can be copied without change. In particularthe syntax x ← if ( x ) then x else x can be encoded, by equivalence with the logicalformula x ← (( x = ⇒ x ) ∧ ( ¬ x = ⇒ x )).We ask, given some initial configuration x (0) , and a target y : does there exist i such that x ( i ) = y . If there was just one step function, the system dynamics would be a direct instanceof the rounded orbit semantics. When there are m functions, we remark the sequence offunctions can be encoded by taking m copies of each variable, and each function f i , cantransfer the function from one copy to the next, zeroing the previous set of variables. Thatis, let M = f m f f ... f m − . Then the initial configuration becomes ( x (0) , , . . . , y, , . . . , (cid:73) Lemma 8.
Reachability in this language can solve QBF.
Proof.
Formally we write a program in our language to decide the truth of a formula ofthe form ∀ x ∃ x ∀ x . . . ∃ x n ψ ( x , . . . , x n ), where ψ is a quantifier free boolean formula. Forconvenience we assume it starts with ∀ , ends with ∃ and alternates. Formulae not in thisform can be padded if necessary with variables which do not occur in the formula ψ . . Baier et al. 9 The program will have the following variables: x , . . . , x n , ˆ ψ, s , . . . , s n , s , . . . , s n and c , . . . , c n . The bits x , . . . , x n represent the current allocation to the corresponding bitvariables of ψ , and ˆ ψ will store the current evaluation of ψ ( x , . . . , x n ). To cycle through allallocations to x , . . . , x n , the variables will be treated as a binary number and incremented byone many times, for this purpose the bits c , . . . , c n represent the carry bits when incrementing x , . . . , x n .The intuition of s zi is the following: for fixed x , . . . , x i − it stores the evaluation of Qx i +1 Q x i +2 . . . ∃ x n ψ ( x , . . . , x i − , z, x i +1 , . . . , x n ) where Q, Q ∈ {∃ , ∀} as required by theformula. Therefore the overall formula is true if and only if s ∧ s is eventually true.We define 3 + n instructions, and each run through f → f n will cover exactly oneallocation to x , . . . , x n , with the next run through covering the next allocation that one getsby incrementing the rightmost bit. Once x i +1 has been in both the 1 state and the 0 statefor all values below, we have enough information to set s x i i +1 . This is set when the carry-bit c i +1 is one, which indicates that x i +1 has visited both 0 and 1 and is being returned back to0 (thus setting x i +1 = · · · = x n back to 0).We let the initial configuration be (0 , . . . , Step . Step . Step . Evaluate ψ Update either s n or s n Start incrementing x n f ( · ) = n ˆ ψ ← ψ ( x , . . . , x n ) f ( · ) = s n ← if ( x n = 0) then ˆ ψ else s n s n ← if ( x n = 1) then ˆ ψ else s n f ( · ) = ( x n ← ¬ x n c n ← x n Step n − i , for i = n − to . If there is a carry, update s zi and continue incrementing i even ( x i universally quantified): i odd ( x i existentially quantified): f n − i ( · ) = f n − i ( · ) = x i ← if ( c i +1 ) then ¬ x i else x i c i ← c i +1 ∧ x i c i +1 ← s i ← if ( c i +1 ∧ ¬ x i ) then s i +1 ∧ s i +1 else s i s i ← if ( c i +1 ∧ x i ) then s i +1 ∧ s i +1 else s i x i ← if ( c i +1 ) then ¬ x i else x i c i ← c i +1 ∧ x i c i +1 ← s i ← if ( c i +1 ∧ ¬ x i ) then s i +1 ∨ s i +1 else s i s i ← if ( c i +1 ∧ x i ) then s i +1 ∨ s i +1 else s i Step n . Set every variable to 1 if QBF satisfied. After this step, the program returns to f . f n ( · ) = n v ← if ( s ∧ s ) then else v (for all variables v )The (3+n)th step ensures that configuration (1 , . . . ,
1) will be reached if and only if the givenQBF formula is satisfied. (cid:74)(cid:73)
Remark 9 (Choice of rounding function).
The presentation here relies on specific choices ofrounding function, but we observe that the language can easily exchange several differentnatural rounding functions, so the reduction is robust. The rounding is only useful in the and and or instructions. The floor function can be replaced by essentially any other rounding.For example x j ∨ x k = l x j + x k m and x j ∧ x k = l − x j + x k m . Similarly, when [ · ] is minimalerror rounding then x j ∨ x k = [ x j + x k ] and x j ∧ x k = [ x j + x k ] (the break point is not used).Thus, the problem will also be hard for any of these roundings. (cid:73) Remark 10 (Perturbation: ensuring the eigenvalues are not modulus 1).
Observe that underthe perturbation that multiplies each operation by 1.1 (before taking floor) we obtainthe same resulting operation. For example x i ← x j ∨ x k = j x j + x k k is equivalent to x i ← x j ∨ x k = j ( x j + x k ) ∗ . k . Hence, if the resulting matrix M has eigenvalues 1, taking1 . M (or similar value to 1.1) will result in a matrix that does not with the same orbit;which shows that hardness is retained for matrices in which no eigenvalue has modulus 1. (cid:73) Remark 11 (Dimension).
The hardness result needs reachability instances of unboundeddimension. For a QBF formula with n variables and ‘ logical operations, the resultinginstance of rounded P2P has dimension (3 n + 1 + ‘ )(4 n + 15 + ‘ ). In this section we consider certain cases when the eigenvalues can be of modulus one. Inparticular we work in the Jordan normal form and show that the problem can be solved forcertain types of rounding. We fall short of arbitrary deterministic rounding, which would berequired to show the problem in full generality through the Jordan normal form approach.First, we show decidability for polar-rounding, along with an example with numbersrequiring exponential space by the time the system becomes periodic—seeming to implyany ‘wait and see’ approach would require
EXPSPACE . We also show decidability forcertain types of Argand rounding, in particular truncation and expansion, but minimal-errorrounding remains open (which we discuss further in Section 5).
We restrict ourselves to a Jordan block M of dimension d , with eigenvalue λ of modulus 1.Since the polar rounding function has bounded effect on the modulus , the remaining blocks,which need not be of modulus 1 can be solved (Lemma 7) by running this algorithm in lockstep with the algorithm for those blocks. All together, this gives us: (cid:73) Theorem 12.
The Rounded P2P Problem is decidable in
EXPSPACE for the polarrounding function with θ g = πR , R ≥ and matrices M ∈ A d × d in Jordan normal form. To prove Theorem 12 we show that each dimension d, d − , . . . , y k (the target value in dimension k ). Let h a, b i be the smallest angle between vectors a and b – this is a value in [0 , π ] and, inparticular, it is always positive. It is used as a measure of alignment: the more a and b arealigned the smaller h a, b i is. We will assume that the system will round up if [ x ] − x = 0 . k ∈ { , . . . , d } is just rotating after position N , if for all i ≥ N :( x ( i +1) ) k = [ λ ( x ( i ) ) k ]. Note that dimension d is just rotating after 0, by definition. Ourgoal is to show that every dimension k will eventually be just rotating (for which we wouldrequire it to have modulus | y k | ) or reach a point that lets us conclude it has permanentlydiverged past y k . So we assume, henceforth, that dimension k is just rotating.We let φ ( i ) = (cid:10) λ ( x ( i ) ) k − , ( x ( i ) ) k (cid:11) . As ( x ( i +1) ) k − = [ λ ( x ( i ) ) k − + ( x ( i ) ) k ], small valuesof φ ( i ) (between 0 and π/
2) lead to an increase in modulus of ( x ( i +1) ) k − , whereas largevalues (between π/ π ) lead to a decrease when (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) is sufficiently large relativeto (cid:12)(cid:12) ( x ( i ) ) k (cid:12)(cid:12) . Our analysis relies on the fact that φ ( i ) can never increase: . Baier et al. 11 (cid:73) Lemma 13.
Suppose that dimension k is just rotating after step N . Then, for all i ≥ N +1 : φ ( i ) ≥ φ ( i + 1) . If dimension k − k and its modulus in some step, we canconclude that k − (cid:73) Lemma 14.
Suppose that dimension k is just rotating after step N , that φ ( N ) = φ ( N +1) and (cid:12)(cid:12) ( x ( N ) ) k − (cid:12)(cid:12) = (cid:12)(cid:12) ( x ( N +1) ) k − (cid:12)(cid:12) . Then, dimension k − is just rotating after N . If the precondition of Lemma 14 holds, we move to the next dimension k −
2. Otherwise, wewant to give a bound such that whenever (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) exceeds it, we can conclude that it neverdecreases back to | y k − | . We first introduce the angle γ ( i ) = (cid:10) λ ( x ( i ) ) k − + ( x ( i ) ) k , λ ( x ( i ) ) k − (cid:11) .The angle γ ( i ) decreases with increasing (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) , as dimension k is just rotating and hencedoes not change in modulus. We observe that γ ( i ) ≤ φ ( i ) for all i . The following shows thatan increase in modulus caused by crossing an ‘axis’ (i.e. if γ ( i ) > π/
2) can only happen once,as in the next step, the angle will have decreased. (cid:73)
Lemma 15.
Let a = λ ( x ( i ) ) k − and b = ( x ( i ) ) k . Suppose that θ g ≤ π/ , γ ( i ) > π/ and | a + b | > | a | . Then h λ [ a + b ] , [ λb ] i ≤ π/ , entailing γ ( i + 1) ≤ φ ( i + 1) ≤ π/ . Furthermore, a decrease cannot be followed by an increase, unless the angle changes: (cid:73)
Lemma 16.
Suppose dimension k is just rotating after N . It is not possible for i − ≥ N ,to have φ ( i −
1) = φ ( i ) = φ ( i + 1) > π/ and (cid:12)(cid:12) ( x ( i +1) ) k − (cid:12)(cid:12) > (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) < (cid:12)(cid:12) ( x ( i − ) k − (cid:12)(cid:12) . Finally, we place a limit on the number of consecutive increases until we can decide thatdimension k − (cid:73) Lemma 17.
Let a = λ ( x ( N ) ) k − and b = ( x ( N ) ) k for some N > . Suppose that k is justrotating after N , | a + b | > | a | + 0 . and | a | ≥ √ | b | . Then, for all i > N : (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) > | a | . With Lemmata 14-17 we are in a position to prove Theorem 12 (the proofs of the precedinglemmata, and the
EXPSPACE analysis can be found in Appendix B).
Proof of Theorem 12.
As described above, we consider each dimension separately, startingwith k = d , and assume by induction that the previous dimension is just rotating. Wedescribe an algorithm that tracks the value of φ and operates according to Figure 2. Eachrealisable value of φ relates to a copy of Figure 2 (we only draw one example of φ satisfying φ > π/ φ ≤ π/ φ > π/ φ D), the other which indicatesthe previous was not decrementing (including first arrival) ( φ I).The algorithm moves on each update step according to the arrow, which denotes whetherthe update is modulus increasing M ↑ , decreasing M ↓ or stationary M S . Similarly φ may decrease φ ↓ or stay stationary φS , but never increase (Lemma 13). Whenever φ decreases we make progress through the DAG to a lower value of φ . All combinations { M ↑ , M ↓ , M S } × { φ ↓ , φS } are accounted for at each state.Progress is made whenever we move through the DAG towards a stopping criterion. Forself-loops a bound is provided (in blue) on the maximum time spent in this state. Sincefor each dimension we will ultimately end up in just rotating, or be able to stop early, theproblem is decidable. (cid:74) φ I φ D stopjustrotating φφ > π/ φ ≤ π/ φ b i gg e r φ s m a ll e r φ ↓ φ ↓ M ↓ φS M ↑ φSM ↓ φS M ↑ φS | x k − | > max { | x k | / √ , | y k − |} , γ ≤ π/ M ↑ φSφ ↓ M ↑ φSγ > π/ M ↑ φS impossibleby Lemma 16 M ↓ φS impossible(as φ ≤ π/ M ↑ φS and | x k − | > | y k − | (as M ↓ unreachable) MSφS (by Lemma 14)at most (cid:12)(cid:12)(cid:12) x ( N ) k − (cid:12)(cid:12)(cid:12) (cannot decrease below 0) at most max { | x k | / √ , | y k − |} (stop transition available)at most | y k − | (stop transition available) Figure 2
State diagram for φ whilst considering dimension k −
1, assuming k is just rotating . (cid:73) Example 18 (System requiring
EXPSPACE to be periodic) . If φ ≤ π/
2, the considereddimension will either diverge at some point, or become periodic. This depends, essentially, onwhether (cid:12)(cid:12) ( x ( i ) ) k (cid:12)(cid:12) cos( φ ) < .
5, in which case the rounding will not lead to an increase when (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) is sufficiently large relative to (cid:12)(cid:12) ( x ( i ) ) k (cid:12)(cid:12) . We give an example where (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) grows to (cid:12)(cid:12) ( x ( i ) ) k (cid:12)(cid:12) , and requires numbers of doubly exponential size (and exponential space) in d before becoming periodic. We assume that θ g = π/ M be a single Jordan block of dimension d with eigenvalue λ = e iπ/ . The angle φ ( i ) remains constant, but the modulus grows while (cid:12)(cid:12) ( x ( i ) ) k (cid:12)(cid:12) < (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) ≤ (cid:12)(cid:12) ( x ( i ) ) k (cid:12)(cid:12) . We start at the point x (0) = ((3 + d, , . . . , (6 , , (5 , , (4 , Ae iθ is written ( A, θ ). This system is periodic, with maximalcomponent x ( N ) = ((4 (2 d − ) , , . . . , (4 · · , , (4 · , , (4 , , (4 , is largerthan a 32-bit number. This idea is illustrated in Figure 3, where y represents ( x ( i ) ) k and isjust rotating, and x represents ( x ( i ) ) k − , which grows to | y | .Despite Example 18, which shows that waiting until becoming periodic may need expo-nential space, we conjecture the Rounded P2P can be solved in PSPACE . This is becauseif ( x ( i ) ) exceeds a value representable in polynomial space we expect it will never return tothe target y (a value representable in polynomial space). However, we are unable to showat the moment that it never gets very large and subsequently returns to a small value. We now consider Argand truncation based rounding showing decidability in
EXPSPACE .The rounding function is of the form [ a + bi ] = [ a ] + [ b ] i where, for x ∈ R , [ x ] = b x c if x ≥ x ] = d x e if x <
0, which has a non-increasing effect on the modulus. (cid:73)
Theorem 19.
The Rounded P2P Problem is decidable in
EXPSPACE for deterministicArgand rounding function with a non-increasing effect on the modulus and matrices M ∈ A d × d in Jordan normal form. . Baier et al. 13 y y xλx λx + y [ λx + y ] y y xλx λx + y [ λx + y ] Rotation by arg( λ )Offset by y RoundingRounding Tie-breakRounding Point
Figure 3
Example where the system may become large before being periodic (see Example 18).
As a key ingredient of Theorem 19 we will make use of the following theorem: (cid:73)
Theorem 20 ([28, Corollary 3.12, p.41]) . Both x is a rational multiple of π and sin( x ) isrational only at sin( x ) = 0 , , or . Both x is a rational multiple of π and cos( x ) are rationalonly at cos( x ) = 0 , , or . Both x is a rational multiple of π and tan( x ) are rational only at tan( x ) = 0 , or ± . Proof sketch of Theorem 19.
Without loss of generality we consider only a single Jordanblock with | λ | = 1, as the remaining blocks can be handled in lock step (using the algorithmof Lemma 7 if the eigenvalue is not of modulus one). Consider the d th component. At eachstep, whenever rounding takes place, then there is some decrease in the modulus. Thus,either the coordinate hits zero (and stays forever), or it stabilises and becomes periodic (withno rounding ever occurring again). The d th coordinate can be simulated until this happens.At this point, if its modulus is not | y d | , y will not be reached in the future and we return no .If dimension x d reaches zero, then this dimension from some point on becomes irrelevantand the instance can be reduced to an instance of dimension d −
1. Note that this case mustoccur if arg( λ ) is not a root of unity as an irrational point is found infinitely often.In the case where x d does not reach zero, then it is periodic at some modulus. Thisimplies it never rounds again, and so surely hits integer points at every step. We show thatthis can only occur if arg( λ ) is a multiple of π/
2. Assume that arg( λ ) is not a multipleof π/
2: the rotation of a point with integer coordinate to integer coordinate leads to theconclusion of either rational tangent or rational sine and cosine. By Theorem 20 a rationaltangent alongside a rational angle (arg( λ ) is a root of unity) implies that the angle must bea multiple of π/
4. It is not π/
4, as there is no Pythagorean triangle with angle π/
4. ByTheorem 20 rational sine and cosine and rational angle concludes the angle must be a multipleof π/
2. Finally, we show that when arg( λ ) is a multiple of π/ d −
1, and hence we can put a bound on how far we need to simulate. (cid:74)(cid:73)
Remark (Argand expansion in Jordan Normal Form).
Instead of considering the roundingfunction to always decrease the modulus, we consider the rounding function to always increasethe modulus. Then, by the same rationality argument either arg( λ ) is a multiple of π/ λ ) is not a multiple of π/ (a) r = 10, θ = π/
42 (b) r = 15, θ = π/
91 (c) r = 10, θ = (0 . π (d) r = 20, θ = π/ Figure 4
Rotational examples. We start with all points in the circle of radius r , and consider theeffect of rotating every point by θ , followed by minimal error rounding. This can be seen as viewingthe combined orbits, starting at several points. Redder points are added in later generations. and rounding is applied infinitely often. We observe that rounding infinitely often resultsin divergence. Suppose instead the modulus converges, in supremum, to C . However the[ C ]-ball is finite, thus rounding infinitely often must eventually exhaust the set, contradictingsupremacy. Since divergence occurs in the d th component the system can be iterated untileither x ( i ) = y or ( x ( i ) ) d exceeds y d . (Unless ( x (0) ) d = y d = 0, in which case the d thcomponent can be deleted.) In this section we consider the following open problem, which already exhibits a technicaldifficulty for a relatively simple instance. (cid:73)
Open Problem 21.
Under which deterministic bounded-effect rounding functions does theRounded P2P Problem become decidable (even when restricted to Jordan normal form)?
In particular we emphasize that even decidability of the Rounded P2P Problem inthe case of a 2D rotation matrix remains open. This should be compared to the papers[24, 11, 36, 35, 2, 31], which consider linear maps on R that are close to rotations, and thefloor rounding b·c is used to induce discretized maps on Z . The conjecture made in [35]that all orbits of these maps are eventually periodic (and thus finite) is, to the best of ourknowledge, still open in general. This lack of understanding of the dynamics of rotations evenon a 2-dimensional lattice is striking and hints at an intrinsic level of difficulty in dealingwith eigenvalues of modulus 1.We ran experiments on the behaviour of rounded orbits induced by rotations in the plane.Four prototypical results are depicted in Figure 4. We note that in every one of our examplesthe orbits eventually become periodic. Moreover, all experiments fall into the four categoriesof Figure 4, i.e., where the resulting set consists of (a) a square with cut-off corners, (b) thissame square, but with a central square cut out, and (c) all points within the circle with someseemingly randomly added points outside (in the case of an irrational multiple of π ), (d) theinitial circle with added ‘tentacles’ occuring in intervals corresponding to the rotational angle(in the case of a rational multiple of π ). We have been unable construct a rotation with aninfinite rounded orbit.One could hope that other kinds of rounding functions simplify the analysis of the orbits.We have shown truncation based rounding, for example, either helps converge towards zero, ordiverge towards infinity, and this can be exploited (particularly at the bottom dimension of aJordan block). However, roundings which may either round up or down greatly complicate the . Baier et al. 15 analysis. Nevertheless, we conjecture that all rounded orbits obtained by rotation eventuallybecome periodic. Random rounding functions
Orbit problems for rounding functions which behave probab-ilistically are another line of open problems and are a natural candidate for future work.
References Shigeki Akiyama, Tibor Borbély, Horst Brunotte, Attila Pethő, and Jörg Thuswaldner.Generalized radix representations and dynamical systems. I.
Acta Mathematica Hungarica ,108:207 – 238, 08 2005. doi:10.1007/s10474-005-0221-z . Shigeki Akiyama and Attila Pethő. Discretized rotation has infinitely many periodic orbits.
Nonlinearity , 26(3):871–880, 2013. doi:10.1088/0951-7715/26/3/871 . S Akshay, Timos Antonopoulos, Joël Ouaknine, and James Worrell. Reachability problems forMarkov chains.
Information Processing Letters , 115(2):155–158, 2015. Shaull Almagor, Joël Ouaknine, and James Worrell. The semialgebraic orbit problem. InRolf Niedermeier and Christophe Paul, editors, , volume 126of
LIPIcs , pages 6:1–6:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. C. Beck and G. Roepstorff. Effects of phase space discretization on the long-time behaviorof dynamical systems.
Physica D: Nonlinear Phenomena , 25(1):173 – 180, 1987. doi:10.1016/0167-2789(87)90100-X . Michael Blank. Ergodic properties of discretizations of dynamic systems.
Dokl. Akad. NaukSSSR , 278(4):779 – 782, 1984. Michael Blank. Ergodic properties of a method of numerical simulation of chaotic dynamicalsystems.
Mathematical Notes of the Academy of Sciences of the USSR , 45:267–273, 1989. doi:10.1007/BF01158885 . Michael Blank. Small perturbations of chaotic dynamical systems.
Russian MathematicalSurveys , 44(6):1–33, dec 1989. doi:10.1070/rm1989v044n06abeh002302 . Michael Blank. Pathologies generated by round-off in dynamical systems.
Physica D: NonlinearPhenomena , 78(1):93 – 114, 1994. doi:10.1016/0167-2789(94)00103-0 . Michael Blank.
Discreteness and Continuity in Problems of Chaotic Dynamics . Translationsof mathematical monographs. American Mathematical Society, 1997. D. Bosio and F. Vivaldi. Round-off errors and p -adic numbers. Nonlinearity , 13(1):309–322,1999. doi:10.1088/0951-7715/13/1/315 . Jin-yi Cai. Computing Jordan normal forms exactly for commuting matrices in polynomialtime.
Int. J. Found. Comput. Sci. , 5(3/4):293–302, 1994. doi:10.1142/S0129054194000165 . Rebekah Carter and Eva M. Navarro-López. Dynamically-driven timed automaton abstractionsfor proving liveness of continuous systems. In Marcin Jurdziński and Dejan Ničković, editors,
Formal Modeling and Analysis of Timed Systems , pages 59–74, Berlin, Heidelberg, 2012.Springer Berlin Heidelberg. doi:10.1007/978-3-642-33365-1_6 . Ventsislav Chonev, Joël Ouaknine, and James Worrell. The polyhedron-hitting problem.In Piotr Indyk, editor,
Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium onDiscrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015 , pages 940–956.SIAM, 2015. Eva Darulova, Anastasiia Izycheva, Fariha Nasir, Fabian Ritter, Heiko Becker, and RobertBastian. Daisy - framework for analysis and optimization of numerical programs (tool paper).In Dirk Beyer and Marieke Huisman, editors,
Tools and Algorithms for the Constructionand Analysis of Systems , pages 270–287, Cham, 2018. Springer International Publishing. doi:10.1007/978-3-319-89960-2_15 . Phil Diamond and Igor Vladimirov. Asymptotic independence and uniform distribution ofquantization errors for spatially discretized dynamical systems.
International Journal ofBifurcation and Chaos , 8:1479–1490, 1998. doi:10.1142/S0218127498001133 . Phil Diamond and Igor Vladimirov. Set-valued Markov chains and negative semitrajectoriesof discretized dynamical systems.
Journal of Nonlinear Science , 12:113–141, 2002. doi:10.1007/s00332-001-0450-4 . S.P. Dias, L. Longa, and E. Curado. Influence of the finite precision on the simulations ofdiscrete dynamical systems.
Communications in Nonlinear Science and Numerical Simulation ,16(3):1574 – 1579, 2011. doi:10.1016/j.cnsns.2010.07.003 . Eric Goubault and Sylvie Putot. Static analysis of finite precision computations. InRanjit Jhala and David Schmidt, editors,
Verification, Model Checking, and Abstract In-terpretation , pages 232–247, Berlin, Heidelberg, 2011. Springer Berlin Heidelberg. doi:10.1007/978-3-642-18275-4_17 . Stephen Hammel, James Yorke, and Celso Grebogi. Numerical orbits of chaotic pro-cesses represent true orbits.
Bull. Amer. Math. Soc. , 19:465–, 04 1988. doi:10.1090/S0273-0979-1988-15701-1 . Anastasiia Izycheva and Eva Darulova. On sound relative error bounds for floating-pointarithmetic. In
Proceedings of the 17th Conference on Formal Methods in Computer-AidedDesign , FMCAD ’17, page 15–22, Austin, Texas, 2017. FMCAD Inc. doi:10.23919/FMCAD.2017.8102236 . Ravindran Kannan and Richard J. Lipton. Polynomial-time algorithm for the orbit problem.
J. ACM , 33(4):808–821, 1986. doi:10.1145/6490.6496 . Anatole Katok and Boris Hasselblatt.
Introduction to the Modern Theory of DynamicalSystems . Encyclopedia of Mathematics and its Applications. Cambridge University Press,1995. doi:10.1017/CBO9780511809187 . John Lowenstein, Spyros Hatjispyros, and Franco Vivaldi. Quasi-periodicity, global stabilityand scaling in a model of Hamiltonian round-off.
Chaos: An Interdisciplinary Journal ofNonlinear Science , 7(1):49–66, 1997. doi:10.1063/1.166240 . Victor Magron, George Constantinides, and Alastair Donaldson. Certified roundoff errorbounds using semidefinite programming.
ACM Trans. Math. Softw. , 43(4), 2017. doi:10.1145/3015465 . Oded Maler and Grégory Batt. Approximating continuous systems by timed automata. InJasmin Fisher, editor,
Formal Methods in Systems Biology , pages 77–89, Berlin, Heidelberg,2008. Springer Berlin Heidelberg. doi:10.1007/978-3-540-68413-8_6 . Mariano Moscato, Laura Titolo, Aaron Dutle, and César A. Muñoz. Automatic estimation ofverified floating-point round-off errors via static analysis. In Stefano Tonetta, Erwin Schoitsch,and Friedemann Bitsch, editors,
Computer Safety, Reliability, and Security , pages 213–229,Cham, 2017. Springer International Publishing. doi:10.1007/978-3-319-66266-4_14 . Ivan Niven.
Irrational Numbers . Number 11 in The Carus Mathematical Monographs. TheMathematical Association of America, 1956. doi:10.5948/9781614440116 . Helena E. Nusse and James A. Yorke. Is every approximate trajectory of some processnear an exact trajectory of a nearby process?
Comm. Math. Phys. , 114(3):363–379, 1988. doi:10.1007/BF01242136 . Joël Ouaknine and James Worrell. On linear recurrence sequences and loop termination.
ACMSIGLOG News , 2(2):4–13, 2015. Attila Pethö, Jörg M. Thuswaldner, and Mario Weitzer. The finiteness property for shift radixsystems with general parameters.
Integers , 19:A50, 2019. URL: http://math.colgate.edu/%7Eintegers/t50/t50.Abstract.html . Stefano Schivo and Romanus Langerak.
Discretization of Continuous Dynamical SystemsUsing UPPAAL , pages 297–315. Lecture Notes in Computer Science. Springer, 9 2017. doi:10.1007/978-3-319-68270-9_15 . . Baier et al. 17 Alexey Solovyev, Charles Jacobsen, Zvonimir Rakamarić, and Ganesh Gopalakrishnan. Rigor-ous estimation of floating-point round-off errors with symbolic Taylor expansions. In NikolajBjørner and Frank de Boer, editors,
FM 2015: Formal Methods , pages 532–550, Cham, 2015.Springer International Publishing. doi:10.1007/978-3-319-19249-9_33 . Larry J. Stockmeyer and Albert R. Meyer. Word problems requiring exponential time:Preliminary report. In Alfred V. Aho, Allan Borodin, Robert L. Constable, Robert W. Floyd,Michael A. Harrison, Richard M. Karp, and H. Raymond Strong, editors,
Proceedings of the5th Annual ACM Symposium on Theory of Computing, April 30 - May 2, 1973, Austin, Texas,USA , pages 1–9. ACM, 1973. doi:10.1145/800125.804029 . Franco Vivaldi. The arithmetic of discretized rotations.
AIP Conference Proceedings , 826, 032006. doi:10.1063/1.2193120 . Franco Vivaldi and Igor Vladimirov. Pseudo-randomness of round-off errors in discretizedlinear maps on the plane.
International Journal of Bifurcation and Chaos , 13(11):3373–3393,2003. doi:https://doi.org/10.1142/S0218127403008557 . A Additional material for Section 3.2,
PSPACE -hardnessA.1 Perturbation
We expand on Remark 10, observing that multiplying by 1 . x i , x j , x k are each in { , } we have,1 ← b ∗ . c x i ← x j ∨ x k = j x j + x k k = j ( x j + x k ) ∗ . k x i ← x j ∧ x k = j x j + x k k j ( x j + x k ) ∗ . k x i ← ¬ x j = b − x j c = b (1 − x j ) ∗ . c . x i ← b c = b (0) ∗ . c x i ← b x i c = b ( x i ) ∗ . c x i ← b x j c = b ( x j ) ∗ . c . A.2 Dimension of Rounded P2P instance in proof of
PSPACE -hardness
It is clear that the required reduction is polynomial, we precisely characterise the dimensionof the resulting system here. (cid:73)
Proposition 22.
The resulting instance of rounded P2P has dimension (3 n + 1 + ‘ )(4 n +15 + ‘ ) , if ψ has ‘ logical operations. Proof of Proposition 22.
The functions f , . . . , f n hide the inner workings of the reductionto the Rounded P2P Problem, by contracting steps and auxiliary variables and illustratingthe effect using logic, rather than the floor of a linear combination.Following the routine steps to translate the logical commands into the Rounded P2PProblem, we see that: f depends on the formula ψ to evaluate. If ψ is a formula with ‘ logical operators wehave: ‘ steps, resolving each logical operator according to topological ordering ‘ auxiliary variables to store partial computations. f takes two steps and four extra variables. f n − i takes 3 steps for each i and 8 extra variables. The 8 variables can be shared forall functions. f n takes 3 steps and 2 × t extra variables, where t is the total number of main variables.However it can be simplified to 2 steps, and 1 extra variable (by noticing it is equivalentto v ← ( s ∧ s ) ∨ v ).Thus the total number of steps is ‘ + 2 + 3( n −
1) + 2 = 3 n + 1 + ‘ steps. The total number ofvariables is ‘ + t + 4 + 8 + 1 plus 1 to store true , so total of t + 14 + ‘ . Note that t = 4 n + 1,total 4 n + 15 + ‘ . Thus when exploded as per Section 3.2, there are (3 n + 1 + ‘ )(4 n + 15 + ‘ )dimensions in the Rounded P2P Problem. (cid:74) B Additional material for Section 4.1, Polar rounding in Jordannormal form (cid:73)
Lemma 23.
Assume b = [ b ] , then h [ a ] , b i ≥ h [ a + b ] , b i Proof of Lemma 23.
Assume without loss of generality (by rotation) that arg( b ) = 0. Thus b = ( x,
0) for x ≥
0. A point at a = ( u, v ) is translated to ( u + x, v ), thus the angle betweenthe x -axis is smaller. Hence h a, b i ≥ h a + b, b i .Now assume first that arg( a ) ∈ [0 , π ]. Then arg( a ) = h a, b i and arg( a ) ≥ h a + b, b i =arg( a + b ) ≥
0. From the fact that b = [ b ] it follows that arg( b ) = 0 is a viable anglein our rounding. As we assume minimal error rounding on the angle, it follows thatarg([ a ]) ≥ arg([ a + b ]) and hence h [ a ] , b i ≥ h [ a + b ] , b i .If arg( a ) ∈ [ − π, − arg( a ) = h a, b i and arg( a ) ≤ − h a + b, b i = arg( a + b ) ≤ h [ a ] , b i ≥ h [ a + b ] , b i . (cid:74)(cid:73) Lemma 13.
Suppose that dimension k is just rotating after step N . Then, for all i ≥ N +1 : φ ( i ) ≥ φ ( i + 1) . Proof of Lemma 13.
We show that for all i ≥ N + 1: D λ ( x ( i ) ) k − , ( x ( i ) ) k E ≥ D λ ( x ( i +1) ) k − , ( x ( i +1) ) k E We first make the following calculating: D [ λ ( x ( i ) ) k − ] , ( x ( i ) ) k E ≥ D [ λ ( x ( i ) ) k − + ( x ( i ) ) k ] , ( x ( i ) ) k E (Lemma 23)= D ( x ( i +1) ) k − , ( x ( i ) ) k E (by definition)= D [ λ ( x ( i +1) ) k − ] , [ λ ( x ( i ) ) k ] E (1)= D [ λ ( x ( i +1) ) k − ] , ( x ( i +1) ) k E ( k is just rotating)Equation (1): Since ( x ( i +1) ) k − and ( x ( i ) ) k are both at admissible angles, their rotations areat the same point between two admissible angles. Thus the rotation-effect of the roundingwill be the same for both values.Observe that [ λ ( x ( i +1) ) k − ] and ( x ( i +1) ) k both lie on admissible angles. Consider θ = h [ λ [ a ]] , λ [ a ] i , observe this angle (and the direction of the angle) is the same, nomatter the value of a . This angle corresponds with (cid:10) [ λ ( x ( i +1) ) k − ] , λ ( x ( i +1) ) k − (cid:11) and (cid:10) [ λ ( x ( i ) ) k − ] , λ ( x ( i ) ) k − (cid:11) . Further θ ≤ θ g / (cid:73) Case 1 (Suppose (cid:10) [ λ ( x ( i ) ) k − ] , ( x ( i ) ) k (cid:11) < π ). The calculation above also shows thatarg([ λ ( x ( i ) ) k − ]) relative to arg(( x ( i ) ) k ) is positive if and only if arg([ λ ( x ( i +1) ) k − ]) relative . Baier et al. 19 to arg(( x ( i +1) ) k − ) is positive. That is, not only does the angle-distance decrease, but therelative position of the two points stays the same.Because of this, and the fact that both ( x ( i ) ) k − and ( x ( i +1) ) k − lie on admissible angles,the angle effect of rounding λ ( x ( i ) ) k − relative to ( x ( i ) ) k is the same as the angle effect ofrounding λ ( x ( i +1) ) k − relative to ( x ( i +1) ) k . Hence we can conclude: D λ ( x ( i ) ) k − , ( x ( i ) ) k E ≥ D λ ( x ( i +1) ) k − , ( x ( i +1) ) k E (cid:73) Case 2 (Suppose (cid:10) [ λ ( x ( i ) ) k − ] , ( x ( i ) ) k (cid:11) = π and (cid:10) [ λ ( x ( i +1) ) k − ] , ( x ( i +1) ) k (cid:11) < π ). Given Rθ g = π = (cid:10) [ λ ( x ( i ) ) k − ] , ( x ( i ) ) k (cid:11) and (cid:10) [ λ ( x ( i ) ) k − ] , λ ( x ( i ) ) k − (cid:11) ≤ θ g / φ ( i ) = D [ λ ( x ( i ) ) k − ] , ( x ( i ) ) k E − D [ λ ( x ( i ) ) k − ] , λ ( x ( i ) ) k − E ≥ ( R − θ g + θ g / (cid:10) [ λ ( x ( i +1) ) k − ] , ( x ( i +1) ) k (cid:11) ≤ ( R − θ g we have φ ( i + 1) ≤ D [ λ ( x ( i +1) ) k − ] , ( x ( i +1) ) k E + D [ λ ( x ( i +1) ) k − ] , λ ( x ( i +1) ) k − E ≤ ( R − θ g + θ g / φ ( i ) ≥ ( R − θ g + θ g / ≥ φ ( i + 1). (cid:73) Case 3 (Suppose (cid:10) [ λ ( x ( i ) ) k − ] , ( x ( i ) ) k (cid:11) = (cid:10) [ λ ( x ( i +1) ) k − ] , ( x ( i +1) ) k (cid:11) = π ). The rotationby (cid:10) [ λ ( x ( i ) ) k − ] , λ ( x ( i ) ) k − (cid:11) in either direction results in the angle (when renormalised into[0 , π ]) of φ ( i ) = D [ λ ( x ( i ) ) k − ] , ( x ( i ) ) k E − D [ λ ( x ( i ) ) k − ] , λ ( x ( i ) ) k − E Similarly, since the effect is the same at i + 1 we have φ ( i + 1) = D [ λ ( x ( i +1) ) k − ] , ( x ( i +1) ) k E − D [ λ ( x ( i +1) ) k − ] , λ ( x ( i +1) ) k − E Since (cid:10) [ λ ( x ( i ) ) k − ] , ( x ( i ) ) k (cid:11) = (cid:10) [ λ ( x ( i +1) ) k − ] , ( x ( i +1) ) k (cid:11) and (cid:10) [ λ ( x ( i ) ) k − ] , λ ( x ( i ) ) k − (cid:11) = (cid:10) [ λ ( x ( i +1) ) k − ] , λ ( x ( i +1) ) k − (cid:11) we have φ ( i ) = φ ( i + 1). (cid:74)(cid:73) Lemma 14.
Suppose that dimension k is just rotating after step N , that φ ( N ) = φ ( N +1) and (cid:12)(cid:12) ( x ( N ) ) k − (cid:12)(cid:12) = (cid:12)(cid:12) ( x ( N +1) ) k − (cid:12)(cid:12) . Then, dimension k − is just rotating after N . Proof of Lemma 14.
We show that for all i ≥ N : ( x ( i +1) ) k − = [ λ ( x ( i ) ) k − ]. First let i = N . We have D λ ( x ( N +1) ) k − , ( x ( N +1) ) k E = D λ ( x ( N ) ) k − , ( x ( N ) ) k E (by assumption)= D λ [ λ ( x ( N ) ) k − ] , [ λ ( x ( N ) ) k ] E (1)= D λ [ λ ( x ( N ) ) k − ] , ( x ( N +1) ) k E ( k is just rotating)Equation (1) holds because ( x ( N ) ) k − and ( x ( N ) ) k are both at admissible points, and hencerounding after rotating by λ has the same effect on both sides.It follows that arg( λ ( x ( N +1) ) k − ) = arg( λ [ λ ( x ( N ) ) k − ]). As (cid:12)(cid:12) ( x ( N +1) ) k − (cid:12)(cid:12) = (cid:12)(cid:12) ( x ( N ) ) k − (cid:12)(cid:12) by assumption, we can conclude that ( x ( N +1) ) k − = [ λ ( x ( N ) ) k − ].The next step from N +1 to N +2 is just a rotation of this case, and hence we can concludeby induction. (cid:74)(cid:73) Lemma 24.
For R ≥ : l π/ θ g m θ g + θ g ≤ π/ , where θ g = πR . aa + bb θ αθ − θ − θ h a, b i θ β Figure 5
Situation in the upper left quadrant ( φ ( i ) > π/
2) after an increasing step. α ≤ π/ φ ( i + 1) ≤ π/ Proof of Lemma 24. l π/ θ g m θ g + θ g → π/ θ g → R → ∞ ). Enumeration of the first100 cases concludes less than π/ π/ (cid:74)(cid:73) Lemma 15.
Let a = λ ( x ( i ) ) k − and b = ( x ( i ) ) k . Suppose that θ g ≤ π/ , γ ( i ) > π/ and | a + b | > | a | . Then h λ [ a + b ] , [ λb ] i ≤ π/ , entailing γ ( i + 1) ≤ φ ( i + 1) ≤ π/ . Proof of Lemma 15.
Refer to Figure 5, let α = h a + b, b i , β = h a, b i− π/ θ = π/ − β = π − h a, b i .First we claim α ≤ π/
4: If θ > π/ α ≤ π/ − θ , we have α ≤ π/
4. Otherwisesuppose θ ≤ π/ θ = α . Because | a + b | > | a | we have θ < θ so α ≤ π/ θ g = π/
2, which implies θ g / π/ α ≤ θ g /
2. Since h a + b, b i < θ g / h [ a + b ] , b i = 0 since b is already rounded and [ λa + b ] is within θ g /
2. then h λ [ a + b ] , [ λb ] i ≤ θ g / θ g = π/R , R ≥
3. Then h a + b, b i ≤ π/ h [ a + b ] , b i ≤ l π/ θ g m θ g and h λ [ a + b ] , [ λb ] i ≤ l π/ θ g m θ g + θ g /
2, then by Lemma 24 l π/ θ g m θ g + θ g / ≤ π/ (cid:74)(cid:73) Lemma 25.
Let a i = λ ( x ( i ) ) k − and b i = ( x ( i ) ) k .Suppose h a i , b i i > π/ and h a i + b i , a i i ≤ π/ and | a i +1 | = | [ a i + b i ] | < | a i | (i.e. (cid:12)(cid:12) ( x ( i +1) ) k − (cid:12)(cid:12) < (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) ).If h a i +1 , b i +1 i = h a i , b i i and h a i +1 + b i +1 , a i +1 i ≤ π/ then | a i +2 | ≤ | a i +1 | (entailing (cid:12)(cid:12) ( x ( i +2) ) k − (cid:12)(cid:12) ≤ (cid:12)(cid:12) ( x ( i +1) ) k − (cid:12)(cid:12) ). Proof of Lemma 25.
The situation is depicted in Figure 6. By rotational symmetry, assume a i = ( x, b i let b = ( − l, m ). Let r, t besuch that: | a i + b | = r (inner black circle) and | [ a i + b ] | = t < | a i | (= x ).Then by rotational symmetry again (using h a i +1 , b i +1 i = h a i , b i i ), assume a i +1 = ( t, | ( t,
0) + b | ≤ r (red/dashed lines), and since [ r ] = t and minimal errorrounding is used | [( t,
0) + b ] | ≤ t . We have a i +2 = λ [ a i +1 + b ] = λ [( t,
0) + b ], hence | a i +2 | ≤ t . (cid:74) . Baier et al. 21 h a, b i r , x possible t Figure 6
Situation in the upper right quadrant after a decreasing step, indicating that if the nextstep stays inside the upper right quadrant, it cannot increase. Orange/dotted can be excluded asthen h a + b, a i > π/ (cid:73) Lemma 16.
Suppose dimension k is just rotating after N . It is not possible for i − ≥ N ,to have φ ( i −
1) = φ ( i ) = φ ( i + 1) > π/ and (cid:12)(cid:12) ( x ( i +1) ) k − (cid:12)(cid:12) > (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) < (cid:12)(cid:12) ( x ( i − ) k − (cid:12)(cid:12) . Proof of Lemma 16.
Recall γ ( i ) = (cid:10) λ ( x ( i ) ) k − + ( x ( i ) ) k , λ ( x ( i ) ) k − (cid:11) .Suppose following a modulus decreasing transition (hence φ ( i ) ≥ π/
2) there is φ ( i ) = φ ( i + 1) and (cid:12)(cid:12) ( x ( i +1) ) k − (cid:12)(cid:12) > (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) the last step was a decrease with γ ( i − ≤ π/ γ ( i ) ≤ π/ (cid:12)(cid:12) ( x ( i +1) ) k − (cid:12)(cid:12) > (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) ).the last step was a decrease with γ ( i − > π/ γ ( i ) ≤ π/ , this cannothappen without a change of angle (contradicting φ ( i −
1) = φ ( i )).this step has γ ( i ) > π/
2, then by Lemma 15 an increase ( (cid:12)(cid:12) ( x ( i +1) ) k − (cid:12)(cid:12) > (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) )would cause the angle to decrease (contradicting φ ( i ) = φ ( i + 1)). (cid:74)(cid:73) Lemma 26.
Let a = λ ( x ( N ) ) k − and b = ( x ( N ) ) k for some N > , and let γ = h a + b, a i .Suppose that k is just rotating after N , γ < π/ and | a + b | > | a | + 0 . .Then, for all i > N : (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) > | a | . Proof.
Recall that φ ( i ) = (cid:10) λ ( x ( i ) ) k − , ( x ( i ) ) k (cid:11) and let φ = φ ( N ). By Lemma 13 we have: φ ( i ) ≤ φ for all i ≥ N . We show the claim by induction on i . If i = N + 1, it holdsby assumption. Else, we can assume that (cid:12)(cid:12) ( x ( i − ) k − (cid:12)(cid:12) > | a | . We let d = ( x ( i − ) k − , e = ( x ( i − ) k and we aim to show that | d + e | ≥ | a + b | . | d + e | ≥ | a + b |⇐⇒ | d + e | ≥ | a + b | ⇐ = | d | + | b | − | d | | b | cos( π − φ ) ≥ | a | + | b | − | a | | b | cos( π − φ ) (*) ⇐⇒ ( | a | + C ) − | a | + C ) | b | cos( π − φ ) ≥ | a | − | a | | b | cos( π − φ ) (**) ⇐⇒ | a | C + C − C | b | cos( π − φ ) ≥ ⇐⇒ | a | C + C ≥ C | b | cos( π − φ ) ⇐ = | a | ≥ | b | cos( π − φ ) a , a + ba + b b b c o s ( π − φ ) | a + b | c o s ( γ ) γ φ π − φ Figure 7
To see that | a | ≥ | b | cos( π − φ ) holds, it is enough to observe that | a | = | b | cos( π − φ ) + | a + b | cos( γ ) (see Figure 7), and 0 < γ < π . The step ( ∗ ) is valid as k is just rotating, whichimplies | b | = | e | , together with φ ( i − ≤ φ , which implies cos( π − φ ) ≥ cos( π − φ ( i − ∗∗ ) we use that | d | = (cid:12)(cid:12) ( x ( i − ) k − (cid:12)(cid:12) = | a | + C , for some positive C .Now, from | d + e | ≥ | a + b | > | a | + 0 . | [ d + e ] | = (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) > | a | . (cid:74)(cid:73) Lemma 27.
Let a, b be algebraic numbers, and assume that | a + b | > | a | ≥ √ | b | .Then, h a + b, a i ≤ π/ . Proof.
Let γ = h a + b, a i and assume, for contradiction, that π/ < γ ≤ π . We have: | b | = | a + b | + | a | − | a | | a + b | cos( γ ) > | a | − | a | | a + b | cos( γ ) > | a | The last step follows by cos( γ ) <
0. But then we have | b | > √ | a | , which contradicts theassumptions. (cid:74)(cid:73) Lemma 17.
Let a = λ ( x ( N ) ) k − and b = ( x ( N ) ) k for some N > . Suppose that k is justrotating after N , | a + b | > | a | + 0 . and | a | ≥ √ | b | . Then, for all i > N : (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) > | a | . Proof of Lemma 17.
Direct corollary of Lemma 26 and Lemma 27. (cid:74)(cid:73)
Proposition 28.
The problem (of Theorem 12) is in
EXPSPACE . Proof of Proposition 28.
This proof gives a more detailed analysis of the algorithm aspresented in Theorem 12, showing that it only uses exponential space. As in Theorem 12 weconsider each dimension separately, starting with dimension d . For each dimension k we willestablish upper bounds T k on the number of steps that we need while considering dimension k , and U k on the maximum numeric value in dimension k that we need to consider.We assume that we are at step N and have concluded that dimension k just rotatesat the right modulus (to conclude this for dimension d we need only one step). Weuse the same stopping criteria as in Theorem 12 to conclude that we no longer reach y . Observe that in Theorem 12 the value of (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) never increases beyond the value K = max( | y k | , | y k − | , (cid:12)(cid:12) ( x ( N ) ) k − (cid:12)(cid:12) ); thus if (cid:12)(cid:12) ( x ( i ) ) k − (cid:12)(cid:12) > K we can conclude that we can . Baier et al. 23 stop. This uses that the previous dimension is rotating on the “right” modulus, that is | y k | = (cid:12)(cid:12) ( x ( i ) ) k (cid:12)(cid:12) holds.The [ K ]-ball (see Definition 3) has K · πθ g admissible points and hence, by the aboveargument, after this many steps we will either have: left the ball and concluded that wecan stop, become just rotating in the current dimension, or decreased φ ( i ). As there are πθ g possible values of φ , in total we will spend at most K · ( πθ g ) steps in dimension k − T k − = K · ( πθ g ) + T k . This lets us assumethat N , the step at which we start considering dimension k −
1, satisfies: N ≤ d · T k . We notethat (cid:12)(cid:12) ( x ( N ) ) k − (cid:12)(cid:12) is at most (cid:12)(cid:12) ( x (0) ) k − (cid:12)(cid:12) + N · U k , as U k is an upper bound on dimension k .Hence we put U k − = (cid:12)(cid:12) ( x (0) ) k − (cid:12)(cid:12) + d · T k · U k Using these upper bounds we now calculate how large the values U j may get withdecreasing j (we start at j = d ). In order not to distinguish the initial and target values ofeach dimension, we overestimate by using y s = P dj =0 | y ( j ) | and i s = P dj =0 (cid:12)(cid:12) ( x (0) ) j (cid:12)(cid:12) . T d = 1 , U d = i s T k − = K · (cid:18) πθ g (cid:19) + T k ≤ ( y s + U k − ) · (cid:18) πθ g (cid:19) + T k U k − = i s + d · T k · U k ≤ i s + d · ( y s + U k ) · (cid:18) πθ g (cid:19) + T k +1 ! · U k ≤ i s · · d · y s · (cid:18) πθ g (cid:19) | {z } F · U k The last step uses that T k +1 ≤ U k . Given the input, F is fixed and of pseudopolynomial size.We can conclude that: U d − j ≤ ( F · i s ) j As F · i s is exponential in the input, it follows that U is at most double-exponential in theinput. Hence, it requires at most exponentially many bits to express. (cid:74) C Additional material for Section 4.2, Argand truncation orexpansion in Jordan normal form
In this subsection we assume all angles are given in degrees as we will make useof rationality arguments on the angles. This is simply a stylistic choice, since it would beequivalent to consider rational multiples of π . (cid:73) Theorem 19.
The Rounded P2P Problem is decidable in
EXPSPACE for deterministicArgand rounding function with a non-increasing effect on the modulus and matrices M ∈ A d × d in Jordan normal form. Proof of Theorem 19.
Without loss of generality we consider only the case where | λ | = 1,since if | λ | 6 = 1 the algorithm of Theorem 6 can be used on each such block in lock step. Consider the d th component. At each step, whenever rounding takes place, then there issome decrease in the modulus. Thus, either the coordinate hits zero (and stays forever), or itstabilises and becomes periodic (with no rounding ever occurring again).The d th coordinate can be simulated until this happens. Then clearly it must match thetarget y d occasionally, otherwise the answer is no.In the following we argue either it stabilises at zero in which case it is trivial. Or itbecomes periodic with non-zero modulus and that this occurs if and only if arg( λ ) is amultiple of 90 degrees (and otherwise must reduce to zero). (cid:73) Case 1 ( ( x ( i ) ) d reaches zero). In this case it is stable, and the next coordinate is not effectedby this coordinate (from some point on). Then the problem can be reduced to a smallerinstance, by deleting the d th coordinate. (cid:73) Case 2 (Periodic at modulus > and arg( λ ) is not a root of unity). This case does notoccur. If arg( λ ) is not a root of unity then λ i x is dense on the circle of radius | x | , and musteventually hit a point with non-integer coordinates. Such points must be rounded, decreasingthe modulus, contradicting stability, and thus periodicity. (cid:73) Case 3 (Periodic at modulus > and arg( λ ) is a root of unity). If arg( λ ) is a root of unityit is rational ( λ n = 1 implies n arg( λ ) = k
360 for some k ∈ N ). We show the only angle thatdoes not tend to zero is a multiple of 90.The following arguments assume we start at a + bi and move to c + di by a rotation ofarg( λ ), the proofs will be based on the rationality/irrationality of the angle and tan , sin , cosof the angle. To do this we assume both are in the upper right quadrant, as by rotating bothby 90,180 or 270 to get it there will have the same argument regarding the irrationality.Suppose we move from a + 0 i to c + di . Recall the modulus is fixed, so c + d = a . Theangle formed by this is arg( λ ) and the tangent is dc . Since c, d are rounded to a rational (butno rounding takes place) then the tangent is rational. By Theorem 20, the only point witharg( λ ) rational and tan(arg( λ )) rational are arg( λ ) = 45 and 90. Note that it cannot be 45,because then c + di = c + ci , and we have p ( c + c ) = a . There is no integer solution tothis equation (no Pythagorean triangle has angle 45 degrees). Hence to move from axis tonon-axis the only acceptable angle is a multiple of 90 degrees (which indeed is not non-axis).However, as a result of the finite period before stabilising and becoming periodic, theorbit could already be at a non-axis point, and move entirely within non-axis points. Supposewe move from a + bi to c + di , with angle arg( λ ). Note that a + bi = C exp( iθ ) and c + di = C exp( i ( θ +arg( λ ))) and hence c + di = ( a + bi )(exp( i arg( λ ))) = ( a + bi )(cos(arg( λ ))+ i sin(arg( λ ))). Then we have c = Re( c + di ) = Re(( a + bi )(cos(arg( λ )) + i sin(arg( λ )))) = a cos(arg( λ )) − b sin(arg( λ ))and d = Im( c + di ) = Im(( a + bi )(cos(arg( λ )) + i sin(arg( λ )))) = b cos(arg( λ )) + a sin(arg( λ )) . Note then that c + bda = ( a + b a ) cos(arg( λ )) and d − bca = ( a + b a ) sin(arg( λ )) , but since a, b, c, d are rational we have cos(arg( λ )) and sin(arg( λ )) rational. Recall, byTheorem 20, the only point cos(arg( λ )) and arg( λ ) are rational is arg( λ ) multiple of 30 (butnot 60) or 90 and the only point sin(arg( λ )) and arg( λ ) are rational is arg( λ ) multiple of 60or 90. Thus arg( λ ) is a multiple of 90. . Baier et al. 25 In this case there is no rounding whatsoever. Indeed in this case the final coordinate, x d , is periodic after the first rounding step, with period at most 4. If it starts at a + bi ,it goes through (at most) − b + ai, − a + bi, b − ai before returning back to a + bi . Thenwe show the penultimate coordinate, x d − , grows from some point on giving a stoppingcriterion (either x ( i ) = y at some point, or ( x ( i ) ) d − > y d − and never comes back). Theinitial point is invariant-under-rounding (i.e. when x = [ x ]), it is rotated by 90 degrees toanother invariant-under-rounding point and then adds a point from the previous component(which is already invariant-under-rounding), resulting in an invariant-under-rounding point.Therefore we can use standard techniques to show that ( x ( n ) ) d − = λ ( x ( n − ) d − + ( x ( n − ) d must grow; To see this note that ( x ( n ) ) d − = λ n ( x (0) ) d − + nλ n − ( x (0) ) d , which diverges as n → ∞ . Thus the analysis of components 1 . . . d − d never increases. Hence,an upper bound for the value in this dimension is U d = (cid:12)(cid:12) ( x (0) ) d (cid:12)(cid:12) . This implies that we neverexit the [ U d ]-ball, and hence, after at most T d = | [ U d ] | ≤ (2 U d /g ) d steps we can concludewhether ( x (0) ) d becomes periodic at 0, or some other modulus. In the latter case we must bein Case 3 , and we can conclude that dimension d − y d − ]-ball, which is of single exponential size.In the first case, we proceed to dimension d −
1, to which the same analysis applies, asnow dimension d is at modulus 0 and does not influence the dynamics any more. Dimension d − U d − = (cid:12)(cid:12) ( x (0) ) d − (cid:12)(cid:12) + T d · U d .To simplify the calculations, we want to assume that T k − > T k and hence set: T k − = | [ U k − ] | + T k ≤ (2 U k − /g ) d + T k . This gives us P dj = k T j ≤ d · T k . We will use d · T k as anoverestimate for the number of steps that were taken before reaching dimension k −
1. Inorder not having to distinguish the different initial values, we overestimate by assuming value i s = P dj =0 (cid:12)(cid:12) ( x (0) ) j (cid:12)(cid:12) in every dimension. Overall, this leads us to the following equations for U k and T k : U d = i s , T d = [ U d ] ≤ (2 U d /g ) d (1) T k − = [ U k − ] + T k ≤ (cid:18) U k − g (cid:19) d + T k (2) U k − = i s + d · T k · U k (3) ≤ i s + d · (cid:18) U k g (cid:19) d + T k +1 ! · U k (4) ≤ i s · d · (2 /g ) d · | {z } F ( U k ) d +1 (5)Step 5 uses that T k +1 < U k (see Step 3) and d >
0. Given the input, F is fixed and singleexponential. It follows that: U d − j ≤ ( F · i s ) ( d +1) j As F · i s is single exponential, it follows that U is at most double exponential in the inputand hence expressible in single exponential space.is at most double exponential in the inputand hence expressible in single exponential space.