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...ge as the theory of integers has many surprises of its own. A notorious example is the solution of multivariate polynomial equations—or, more generally, quantifier elimination—decidable for the reals =-=[36]-=-, but not for integers [41]. Another classical example of an integer-specific problem is the Collatz problem (or “3x + 1 problem”) [24]. Lagarias’ excellent volume shows clearly that this problem is r...

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...ication of this restriction (and propose a relaxation). We use the notation [a, b] for an interval of integers, namely {a, a + 1, . . . , b}. Counter Machines The counter machine model, due to Minsky =-=[29]-=-, is well known. The details of the definition vary in the literature, but the differences are rarely essential. The following description conforms (up to non-critical details) with [5]. A counter mac...

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... sequence converges (to a particular zone) is not recursive. Koiran, Cosnard, and Garzon showed that such simulations can be done already in two dimensions, but not in one [22]. Blondel and Tsiklitis =-=[6]-=- survey applications of discrete computability and complexity to dynamical systems, including the above-cited results. The author’s interest in the mortality problem and its variants arose from their ...

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...s has many surprises of its own. A notorious example is the solution of multivariate polynomial equations—or, more generally, quantifier elimination—decidable for the reals [36], but not for integers =-=[41]-=-. Another classical example of an integer-specific problem is the Collatz problem (or “3x + 1 problem”) [24]. Lagarias’ excellent volume shows clearly that this problem is related both to Dynamical Sy...

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...putation is iterated until an end-condition is met, have gained much interest in program analysis and several heuristic approaches have been proposed (e.g., various constructions of ranking functions =-=[3, 7, 11, 31]-=-). Note that these works concentrate on integer data, and on functions which are linear, piecewise linear, or defined by linear constraints (which is a wider class). In [37], Tiwari draws on inspirati...

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...ing dynamical-system style problems to a discrete setting. There are kinds of dynamical systems even more remote from our subject, which the interested reader may find in (e.g.) Brin and Stuck’s book =-=[9]-=-. Part of this work has been previously presented at STACS 2013 [2]. Preliminary definitions A closed (respectively open) half-space of R n is the set defined by {x ∈ R n : cx + d ≥ 0} (respectively...

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...about an even smaller class of functions, specifically functions which, in each region, take the form f(x) = (x1 + b1, x2 + b2). Such functions resemble the class studied by Asarin, Maler and Pnueli =-=[1]-=- in a continuous setting. They showed that undecidability of the reachability problem begins at three dimensions. What is the situation for functions over Zn and the mortality problem? 195.4 An appli...

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... remainders, which make it easier to encode computations and simulate counter machines. Our first result is 1 An anecdotal evidence of this difference is that for Petri nets, halting is EXPSPACE-hard =-=[15, 25]-=-, while mortality is PTIME [28, 16]. 5THEOREM 2.4. Global convergence and mortality over Z2 of piecewise affine functions with integer coefficients is a Π0 2-complete problem. Proof. Like the GCP, ou...

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...discrete (but analog models are studied and cross-fertilization with Dynamical System Theory is evident; see for example [33]). Taking classical (discrete) computability to continuous dynamics, Moore =-=[30]-=- discusses the significance of undecidability (in the Turing sense) to dynamical systems. He shows that a TM can be simulated by a piecewise-affine map on the plane—the method is quite similar to the ...

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... for the sake of completeness (but also to make it easier to verify that they do hold in our setting). We rely on some properties of matrices and matrix powers. They may be found in textbooks such as =-=[35]-=-. THEOREM 6.1. There is an algorithm for deciding global convergence over Z n of an affine function f(x) = Ax + b. Proof. Note, first, that f(0) = b. Hence, for global convergence we must have b = 0...

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...putation is iterated until an end-condition is met, have gained much interest in program analysis and several heuristic approaches have been proposed (e.g., various constructions of ranking functions =-=[3, 7, 11, 31]-=-). Note that these works concentrate on integer data, and on functions which are linear, piecewise linear, or defined by linear constraints (which is a wider class). In [37], Tiwari draws on inspirati...

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...anking functions [3, 7, 11, 31]). Note that these works concentrate on integer data, and on functions which are linear, piecewise linear, or defined by linear constraints (which is a wider class). In =-=[37]-=-, Tiwari draws on inspiration from Dynamical System Theory to solve a termination problem for loops with an affine-linear update function—however, over the reals. Consequently, Braverman [8] tackled t...

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...putation is iterated until an end-condition is met, have gained much interest in program analysis and several heuristic approaches have been proposed (e.g., various constructions of ranking functions =-=[3, 7, 11, 31]-=-). Note that these works concentrate on integer data, and on functions which are linear, piecewise linear, or defined by linear constraints (which is a wider class). In [37], Tiwari draws on inspirati...

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... one-dimensional piecewise-affine case is PSPACE-complete. 2A problem similar to global convergence to zero, also found in the dynamical system literature, is global convergence to (any) fixed point =-=[22]-=-. In fact, the pattern of “iterate until stable” (rather than until a known value, such as 0, is reached) is ubiquitous in Computer Science and the problem of termination is of obvious interest. This ...

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...at work in dynamical systems subject to some noise. There is research about the computational power of robust systems [11, 32] and about “robust” versions of decision problems about dynamical systems =-=[12, 28, 29]-=-. 42 Some extensions, or variants, of the orbit problem prove harder (in terms of computational complexity, or in terms of establishing results about them). A celebrated open problem is the decidabili...

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... problem (or “3x + 1 problem”) [24]. Lagarias’ excellent volume shows clearly that this problem is related both to Dynamical System Theory and to Computability Theory. Regarding Computability, Conway =-=[12, 13]-=- and several subsequent works [21, 10, 14, 26, 23] proved undecidability results for generalized Collatz problems by showing how to simulate a counter machine. We shall make essential use of this idea...

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...articular case, results would not be different if it were restricted to Q n (there are problems where such a restriction is significant, even when studying functions with rational coefficients; e.g., =-=[8]-=-). THEOREM 1.2. [5] The following problems are undecidable for all n ≥ 2: Given a piecewise affine function f : R n → R n (with rational coefficients), is it globally convergent? Is it mortal? Global ...

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...rograms, while halting from all initial states, still diverge if started in a configuration that is not reachable in a proper computation (i.e., from an initial state) 1 . For Turing machines, Hooper =-=[19]-=- and Herman [18] proved the undecidability of mortality (under two different definitions of the state space). Kurtz and Simon proved the following THEOREM 1.5 ([23]). The mortality problem for counter...

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...e). f is mortal if for every initial point x0 ∈ X, the trajectory xt+1 = f(xt) reaches the origin (i.e., ∃t ≥ 0 : xt = 0). These problems were studied from a Computability viewpoint by Blondel et al. =-=[5]-=-. They considered piecewise affine functions f, where the coefficients are all rational (this is important since we are considering computability in the traditional, discrete sense). The domain of the...

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... to the origin, forgetting the fractional part. Table 5.4 gives the details. 6 Decidability for affine functions Affine-linear transformations have been much studied in Dynamical System Theory, e.g., =-=[17]-=-. The setting there is that of a continuous state space, but the techniques are hardly affected by the restriction to Zn . I have chosen to include full proofs for the theorems below, however, as they...

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...ine functions ,the problem: does the sequence beginning with x0 reach a given value y? Is known as the orbit problem and was solved in polynomial time for linear transformations over the rationals in =-=[20]-=-. I hope that the results presented are of interest, but also the techniques and connections made to Collatz-like problems and to automata that capture PSPACE. There is certainly much terrain yet to b...

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...mortality for restricted classes of functions). 3For completeness, we should also mention the works on dynamics of algebraic or rational maps over the integers (or more general number fields), e.g., =-=[34]-=-. These represent a different direction of transferring dynamical-system style problems to a discrete setting. There are kinds of dynamical systems even more remote from our subject, which the interes...

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...eshold is regarding the number of regions. One may expect to further develop a tradeoff between the number of regions and dimension, of the kind investigated for a number of undecidable problems, see =-=[27]-=-. At this stage, we only discuss the question of minimizing the number of regions (irrespective of dimension). 9Consider the function f defined in Section 2. How many regions does it have? There are ...

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...egister). 44 6 Appendix: proof of Lemma 2.11 To obtain the bounds stated in Lemma 2.11, the following steps will be taken. First, we recall a particular universal counter machine constructed by Korec =-=[35]-=-. Then, we redo the construction of Theorem 2.7 in a more ad-hoc fashion, to get optimised results for this particular universal machine. Finally, we reduce the number of registers in the machine. Kor...

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...on problem called “mortality” which is quite different from our problem; it deals with a set of matrices, and asks whether the zero matrix is a product of some sequence of matrices from this set. See =-=[46, 29, 3]-=-. One may phrase the essence of the difference between the problems in that our mortality problem ask whether all trajectories lead to zero, while their problem asks whether there is a trajectory lead...

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...a continuous statespace, whereas in the Theory of Computation, most models are discrete (but analog models are studied and cross-fertilization with Dynamical System Theory is evident; see for example =-=[33]-=-). Taking classical (discrete) computability to continuous dynamics, Moore [30] discusses the significance of undecidability (in the Turing sense) to dynamical systems. He shows that a TM can be simul...

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...o encode computations and simulate counter machines. Our first result is 1 An anecdotal evidence of this difference is that for Petri nets, halting is EXPSPACE-hard [15, 25], while mortality is PTIME =-=[28, 16]-=-. 5THEOREM 2.4. Global convergence and mortality over Z2 of piecewise affine functions with integer coefficients is a Π0 2-complete problem. Proof. Like the GCP, our problem is clearly a “∀∃” problem...

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... remainders, which make it easier to encode computations and simulate counter machines. Our first result is 1 An anecdotal evidence of this difference is that for Petri nets, halting is EXPSPACE-hard =-=[15, 25]-=-, while mortality is PTIME [28, 16]. 5THEOREM 2.4. Global convergence and mortality over Z2 of piecewise affine functions with integer coefficients is a Π0 2-complete problem. Proof. Like the GCP, ou...

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...o encode computations and simulate counter machines. Our first result is 1 An anecdotal evidence of this difference is that for Petri nets, halting is EXPSPACE-hard [15, 25], while mortality is PTIME =-=[28, 16]-=-. 5THEOREM 2.4. Global convergence and mortality over Z2 of piecewise affine functions with integer coefficients is a Π0 2-complete problem. Proof. Like the GCP, our problem is clearly a “∀∃” problem...

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...at work in dynamical systems subject to some noise. There is research about the computational power of robust systems [11, 32] and about “robust” versions of decision problems about dynamical systems =-=[12, 28, 29]-=-. 42 Some extensions, or variants, of the orbit problem prove harder (in terms of computational complexity, or in terms of establishing results about them). A celebrated open problem is the decidabili...

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...he problem in case that the “chamber” is a linear subspace of dimension up to 3. Recent progress on this problem is reported in [16]. The case where the target is a closed half-space is considered in =-=[45]-=-. See also the survey [44]. In a variant studied by Cortier [20], a point in Nn is subjected to an arbitrary number of iterations of an affine-linear function f1, and then to iterations of another fun...

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...e “chamber” is a linear subspace of dimension up to 3. Recent progress on this problem is reported in [16]. The case where the target is a closed half-space is considered in [45]. See also the survey =-=[44]-=-. In a variant studied by Cortier [20], a point in Nn is subjected to an arbitrary number of iterations of an affine-linear function f1, and then to iterations of another function f2; the reachability...

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... problem (or “3x + 1 problem”) [24]. Lagarias’ excellent volume shows clearly that this problem is related both to Dynamical System Theory and to Computability Theory. Regarding Computability, Conway =-=[12, 13]-=- and several subsequent works [21, 10, 14, 26, 23] proved undecidability results for generalized Collatz problems by showing how to simulate a counter machine. We shall make essential use of this idea...

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...ard representation of g, whether every trajectory of g reaches 1. THEOREM 2.3. [36] GCP is Π02-complete. 2An anecdotal evidence of this difference is that for Petri nets, halting is EXPSPACE-complete =-=[23, 38, 8, 21]-=-, while mortality is PTIME [41, 24]. 7 Note that the GCP is really a mortality problem (with the end-state defined as 1 instead of 0), but the functions considered in the GCP are not piecewise affine;...

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...ecidability for any number of functions is claimed in [27], while undecidability for Zn, n ≥ 2, follows from results on the matrix mortality problem (see below). A beautiful paper by Bell and Potapov =-=[2]-=- shows how various reachability problems are related to questions about sets of matrices and to Post’s correspondence problem. In particular, show that given 5 integer matrices, M1, . . . ,M5, of dime...

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...ard representation of g, whether every trajectory of g reaches 1. THEOREM 2.3. [36] GCP is Π02-complete. 2An anecdotal evidence of this difference is that for Petri nets, halting is EXPSPACE-complete =-=[23, 38, 8, 21]-=-, while mortality is PTIME [41, 24]. 7 Note that the GCP is really a mortality problem (with the end-state defined as 1 instead of 0), but the functions considered in the GCP are not piecewise affine;...

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...er a function defined by f(x) = Mx where M is a square (k×k) integer matrix; given an initial value u, decide whether the sequence (M iu)i≥0 ever hits a given hyperplane a · x = 0. Vyalyi and Tarasov =-=[55]-=- point out that this problem and Kannan and Lipton’s orbit problem can both be seen as special cases of the problem of reaching a given convex region, which they call the chamber hitting problem, and ...

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...agarias’ excellent volume shows clearly that this problem is related both to Dynamical System Theory and to Computability Theory. Regarding Computability, Conway [12, 13] and several subsequent works =-=[21, 10, 14, 26, 23]-=- proved undecidability results for generalized Collatz problems by showing how to simulate a counter machine. We shall make essential use of this idea, building on the reductions in [23], which were t...

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...mension up to 3. Recent progress on this problem is reported in [16]. The case where the target is a closed half-space is considered in [45]. See also the survey [44]. In a variant studied by Cortier =-=[20]-=-, a point in Nn is subjected to an arbitrary number of iterations of an affine-linear function f1, and then to iterations of another function f2; the reachability problem (for a single target point) i...

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...us case (i.e., using functions whose constant term is zero; in other words, linear functions rather than affine-linear). For a recent progress (solving the problem under a different restriction), see =-=[51]-=-. In the program-analysis context, the case R = 2 represents loops with a single “if statement” in their bodies (where the test is a linear inequality). The termination 16 problem of such loops has be...

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...quivalent to leaving the active region. This problem has been settled in part (specifically, for the homogenous case) by Braverman [8]. For R = 2 the problem has been recently shown to be undecidable =-=[4]-=-. There, the dimension was not bounded. In fact [4] reduces from mortality of counter machines, and the dimension of the dynamical system constructed is related to the size of the counter machine. Usi...

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...agarias’ excellent volume shows clearly that this problem is related both to Dynamical System Theory and to Computability Theory. Regarding Computability, Conway [12, 13] and several subsequent works =-=[21, 10, 14, 26, 23]-=- proved undecidability results for generalized Collatz problems by showing how to simulate a counter machine. We shall make essential use of this idea, building on the reductions in [23], which were t...

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...agarias’ excellent volume shows clearly that this problem is related both to Dynamical System Theory and to Computability Theory. Regarding Computability, Conway [12, 13] and several subsequent works =-=[21, 10, 14, 26, 23]-=- proved undecidability results for generalized Collatz problems by showing how to simulate a counter machine. We shall make essential use of this idea, building on the reductions in [23], which were t...

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...oth be seen as special cases of the problem of reaching a given convex region, which they call the chamber hitting problem, and which they relate to questions of formal-language theory. Chonev et al. =-=[15]-=- give decidability and complexity results for the problem in case that the “chamber” is a linear subspace of dimension up to 3. Recent progress on this problem is reported in [16]. The case where the ...

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...heory. Chonev et al. [15] give decidability and complexity results for the problem in case that the “chamber” is a linear subspace of dimension up to 3. Recent progress on this problem is reported in =-=[16]-=-. The case where the target is a closed half-space is considered in [45]. See also the survey [44]. In a variant studied by Cortier [20], a point in Nn is subjected to an arbitrary number of iteration...

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...e., f(x) = 0 either on [0, ε] or on [−ε, 0]. For the problem convergence to a fixed point, its complexity in the 1-dimensional, continuous case remains open. 6Recent work by Finkel, Göller and Haase =-=[25]-=- includes a PSPACE algorithm for a more general problem: machines with a single register subject to polynomial updates. 31 region label constraints f(x, y) NW x < 1, y ≥ 2m (x−m, y − 2m) W inr,i 0≤r<1...

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...agarias’ excellent volume shows clearly that this problem is related both to Dynamical System Theory and to Computability Theory. Regarding Computability, Conway [12, 13] and several subsequent works =-=[21, 10, 14, 26, 23]-=- proved undecidability results for generalized Collatz problems by showing how to simulate a counter machine. We shall make essential use of this idea, building on the reductions in [23], which were t...

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...ty status is apparently still open. Its analysis [54] shows that number-theoretic considerations, somewhat in the flavour of the Collatz problem, are involved in understanding such programs. See also =-=[48, 4]-=-. 5.3 Open problems Here is a recap of some open problems—all concern piecewise-affine functions over the integers, and have been described in more detail in previous sections. 1. Is there any constan...

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...al. [?] give decidability and complexity results for the problem in case that the “chamber” is a linear subspace of dimension up to 3. See also the survey [?]. Peter van Emde Boas and Marek Karpinski =-=[38]-=- describe the “mouse in an octant” problem, attributed to Lothar Budach, a problem which has the flavour of a Collatz-like problem (intuitively, closer to our CCP than to Conway’s GCP), and whose deci...

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...on problem called “mortality” which is quite different from our problem; it deals with a set of matrices, and asks whether the zero matrix is a product of some sequence of matrices from this set. See =-=[46, 29, 3]-=-. One may phrase the essence of the difference between the problems in that our mortality problem ask whether all trajectories lead to zero, while their problem asks whether there is a trajectory lead...

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...a larger number of stages f1, f2, . . . , fp is shown to be undecidable once p is beyond some (unspecified) constant. Over Z (in one dimension), decidability for any number of functions is claimed in =-=[27]-=-, while undecidability for Zn, n ≥ 2, follows from results on the matrix mortality problem (see below). A beautiful paper by Bell and Potapov [2] shows how various reachability problems are related to...

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...on problem called “mortality” which is quite different from our problem; it deals with a set of matrices, and asks whether the zero matrix is a product of some sequence of matrices from this set. See =-=[46, 29, 3]-=-. One may phrase the essence of the difference between the problems in that our mortality problem ask whether all trajectories lead to zero, while their problem asks whether there is a trajectory lead...

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...dress this criticism in another way, namely by considering robust constructions that work in dynamical systems subject to some noise. There is research about the computational power of robust systems =-=[11, 32]-=- and about “robust” versions of decision problems about dynamical systems [12, 28, 29]. 42 Some extensions, or variants, of the orbit problem prove harder (in terms of computational complexity, or in ...

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...putation is iterated until an end-condition is met, have gained much interest in program analysis and several heuristic approaches have been proposed (e.g., various constructions of ranking functions =-=[3, 7, 11, 31]-=-). Note that these works concentrate on integer data, and on functions which are linear, piecewise linear, or defined by linear constraints (which is a wider class). In [37], Tiwari draws on inspirati...

Citation Context

...agarias’ excellent volume shows clearly that this problem is related both to Dynamical System Theory and to Computability Theory. Regarding Computability, Conway [12, 13] and several subsequent works =-=[21, 10, 14, 26, 23]-=- proved undecidability results for generalized Collatz problems by showing how to simulate a counter machine. We shall make essential use of this idea, building on the reductions in [23], which were t...

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...alting from all initial states, still diverge if started in a configuration that is not reachable in a proper computation (i.e., from an initial state) 1 . For Turing machines, Hooper [19] and Herman =-=[18]-=- proved the undecidability of mortality (under two different definitions of the state space). Kurtz and Simon proved the following THEOREM 1.5 ([23]). The mortality problem for counter machines (CMs) ...

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...ty status is apparently still open. Its analysis [54] shows that number-theoretic considerations, somewhat in the flavour of the Collatz problem, are involved in understanding such programs. See also =-=[48, 4]-=-. 5.3 Open problems Here is a recap of some open problems—all concern piecewise-affine functions over the integers, and have been described in more detail in previous sections. 1. Is there any constan...

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...putation is iterated until an end-condition is met, have gained much interest in program analysis and several heuristic approaches have been proposed (e.g., various constructions of ranking functions =-=[6, 11, 17, 47]-=-). Note that these works concentrate on integer data, and on functions which are linear, piecewise linear, or defined by linear constraints (which is a wider class). In [53], Tiwari draws on inspirati...

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...at work in dynamical systems subject to some noise. There is research about the computational power of robust systems [11, 32] and about “robust” versions of decision problems about dynamical systems =-=[12, 28, 29]-=-. 42 Some extensions, or variants, of the orbit problem prove harder (in terms of computational complexity, or in terms of establishing results about them). A celebrated open problem is the decidabili...

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...dress this criticism in another way, namely by considering robust constructions that work in dynamical systems subject to some noise. There is research about the computational power of robust systems =-=[11, 32]-=- and about “robust” versions of decision problems about dynamical systems [12, 28, 29]. 42 Some extensions, or variants, of the orbit problem prove harder (in terms of computational complexity, or in ...