Periodic generalized automata over the reals

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Periodic generalized automata over the reals Klaus Meer ∗ , Ameen Naif 1 Computer Science Institute, Brandenburg University of Technology Cottbus-Senftenberg, Platz der Deutschen Einheit 1, D-03046 Cottbus, Germany

a r t i c l e

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Article history: Received 9 June 2016 Received in revised form 5 May 2017 Accepted 4 February 2018 Available online xxxx Keywords: Unconventional models of computation Generalized finite automata Real number computations Decision problems for automata over uncountable structures

a b s t r a c t Gandhi, Khoussainov, and Liu introduced a generalized model of finite automata able to work over arbitrary structures. The model mimics the finite automata over finite structures but has an additional ability to perform in a restricted way operations attached to the structure under consideration. As one relevant area of investigations for this model the above mentioned authors identified studying the new automata over uncountable structures such as the real numbers. This research was started in 2015 by the present authors. However, there it turned out that many elementary properties known from classical finite automata are lost. The intrinsic reason for this is that the computational abilities of the new model turn out to be too strong. We propose a restricted periodic version of the model. The new model still has certain computational abilities which, however, are restricted in that computed information is deleted again after a fixed period in time. We show that this limitation regains a lot of classical properties including validity of a pumping lemma, structural as well as decidability results. Thus the new model seems to reflect more adequately what might be considered as finite automata over the reals and similar structures. © 2019 Elsevier Inc. All rights reserved.

1. Introduction In 2012, Gandhi, Khoussainov, and Liu [4] introduced a generalized model of finite automata called (S , k, )-automata, see as well [7] for a more recent refinement of the approach. It is able to work over an arbitrary structure S , and here in particular over infinite alphabets like the real numbers. A structure is characterized by an alphabet (also called universe) together with a finite number of binary functions and relations over that alphabet. Intuitively the model processes words over the underlying alphabet componentwise. Each single step is composed of finitely many test operations relying on the fixed relations as well as finitely many computational operations relying on the fixed functions. For performing the latter an (S , k, )-automaton can use a finite number k of registers together with  constants from the underlying structure. It moves between finitely many states and finally accepts or rejects an input. The motivation to study such generalizations is manifold. In [4] the authors discuss different previous approaches to design finite automata over infinite alphabets and their role in program verification and database theory. One goal is to look

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Corresponding author. E-mail addresses: [email protected] (K. Meer), [email protected] (A. Naif). 1 A. Naif was partially supported by a ‘Promotionsstipendium nach Graduiertenförderungsverordnung des Landes Brandenburg GradV’. The support is gratefully acknowledged. https://doi.org/10.1016/j.ic.2019.104452 0890-5401/© 2019 Elsevier Inc. All rights reserved.

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for a generalized framework that is able to homogenize at least some of these approaches. As the authors remark, many classical automata models like pushdown automata, Petri nets, visibly pushdown automata can be simulated by the new model. Another major motivation results from work on algebraic models of computation over structures like the real and complex numbers. Here, the authors suggested their model as a finite automata variant of the Blum-Shub-Smale BSS model [2,1]. They then ask to analyze such automata over structures like real or algebraically closed fields. This is similar in spirit to the study of quantum finite automata as finite automata restriction of another unconventional model of computation, see for example [8,13]. For some decidability results in this framework similar to the ones studied in the present paper we refer to [14,9]. For the BSS model this line of research has been started by the present authors in [10]. Given the tremendous impact finite automata have in classical computability theory it looks promising to introduce and study a similar concept as restriction of the real computational model introduced by Blum, Shub, and Smale. However, the main lesson from [10] is that the general automata model by Gandhi et al. turns out to be too strong when applied to the real or complex numbers. Almost all basic automata problems turn out to be undecidable in the BSS framework for these two uncountable structures. As another consequence, only weak structural properties can be derived for real languages accepted by such an automaton. The intrinsic reason for this is the automata’s ability to store intermediate results during an entire computation, something obviously not possible in the finite automata world. We therefore suggest a restricted version of the Ghandi-Khoussainov-Liu (for short: GKL) model. The basic idea is to force the automaton to periodically forget after a constant number of steps its intermediate results. This still enables the automaton to perform operations present in the given structure, but in a limited way. For the resulting restricted periodic version of real GKL automata we shall prove both structural as well as several decidability results. Since the automata can perform basic arithmetic operations over R, not surprisingly semi-algebraic sets show up; computability and decidability results then naturally rely on quantifier elimination algorithms for the first order theory of the reals. Our results hopefully indicate that the restricted model is a meaningful alternative giving back many of the features finite automata have, but non-trivially related to the uncountable structure under consideration. Some of the further questions arising are discussed at the end. In the next section we recall the automaton model by Gandhi et al., equipped with the additional restriction of periodicity. The main results are proved in Sections 3 and 4, which collect both decidability results for several questions about our periodic automata and structural properties of the languages accepted by them. We close with a collection of further questions. The paper intends to be a further step towards the development of a generalized model of finite automata. It might be promising to analyse our variant as well for the many further scenarios treated in [4]. 2. Periodic GKL automata We suppose the reader to be familiar with the basics of the Blum-Shub-Smale model of computation and complexity over R. Very roughly, algorithms in this model work over finite strings of real numbers. The operations either can be computational, in which case addition, subtraction, and multiplication are allowed; without much loss of generality we do not consider divisions in this paper to avoid technical inconveniences. Or an algorithm can branch depending on the result of a binary test operation. The latter will be inequality tests of form ‘is x ≥ y?’ The size of a finite string of real numbers is the number of components it has, the cost of an algorithm is the number of operations it performs until it halts. For more details see [1]. The generalized finite automata introduced in [4] work over structures. A structure S consists of a universe D together with finite sets of (binary) functions and relations over the universe. An automaton informally works as follows. It reads a word from the universe, i.e., a finite string of components from D and processes each component once. Reading a component the automaton can perform some tests using the relations in the structure. The tests might involve a fixed set of constants from the universe D that the automaton can use. It can as well perform in a limited way computations on the current component. Towards this aim, there is a fixed number k of registers that can store elements from D. Those registers can be changed using their current value, the current input and the functions related to S . The new aspect we include here is that such computations on the registers cannot be performed unlimited; after at most a fixed number of steps – called the period – have been performed all register values are reset to the initial assignment 0, thus forgetting intermediate results. After having read the entire input word the automaton accepts or rejects it depending on the state in which the computation stops. These automata can both be deterministic and non-deterministic. The approach by Gandhi et al. in particular can be used in order to define generalized finite automata over structures like R and C . We shall in the rest of the paper focus on the real numbers, though all our results hold as well in their corresponding variant for the complex numbers. Statements about computability and decidability refer to the real BSS model of computation. We consider exclusively the structure SR := (R, +, −, ×, null, pr1 , pr2 , ≥, =) of reals as ring with order. As in the original work [4] we include the binary projection operators pr1 , pr2 which give back the first and the second component of a tuple, respectively. The operation null stands for assigning 0 to a register. In order to avoid technicalities for the subtraction operation we allow both orders of the involved arguments, i.e., applying − to two values x, v can mean x − v or v − x.

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Similarly, the order test can be performed both as x ≤ v? and v ≤ x? We do not include division as an operation. This will not significantly change our results. The following definition makes these ideas precise. We alternatively call the resulting automata real periodic (SR , k, , T )-automata or (a bit less clumsy) real periodic GKL automata, where GKL refers to the initials of the authors of [4]. The definition below adapts the general automata definition in [4] and its real number version from [10]. Definition 1 (Periodic GKL automata over R). Let k, T ,  ∈ N be fixed. a) A deterministic periodic (SR , k, , T )-automaton A, also called real periodic GKL automaton, consists of the following objects: – a finite state space Q and an initial state q0 ∈ Q , – a set F ⊆ Q of final (accepting) states, – a set of  registers which contain fixed given constants c 1 , . . . , c  ∈ R, – a set of k registers which can store real numbers denoted by v 1 , . . . , v k , – a counter containing a number t ∈ {0, 1, . . . , T − 1}; we call T the periodicity of the automaton, – a transition function δ : Q × {0, 1}k+ × {0, 1, . . . , T − 1} → Q × {◦1 , . . . , ◦k } × {0, 1, . . . , T − 1}. Rn , i.e., words of finite length with real components. For such an b) The automaton processes elements of R∗ :=



n ≥1

(x1 , . . . , xn ) ∈ Rn it works as follows. The computation starts in q0 with initial assignment 0 ∈ R for the values v 1 , . . . , v k ∈ R. The automaton has a counter which stores an integer t ∈ {0, 1, . . . , T − 1}. At the beginning of a computation its value is 0. A reads the input components step by step. Suppose a value x is read in state q ∈ Q with counter value t. The next state together with an update of the values v i and t is computed as follows: A performs k +  comparisons x1 v 1 ?, x2 v 2 ?, . . . , xk v k ?, xk+1 c 1 ?, . . . , xk+ c  ?, where i ∈ {≥, >, =, <, ≤}. This gives a vector b ∈ {0, 1}k+ , where a component 0 indicates that the comparison that was tested is violated whereas 1 codes that it is valid; depending on state q and b the automaton moves to a state q ∈ Q (which could again be q). The allowed operations are: – an operation reset can be performed; it causes that both the register values v i and the counter value t are reset to 0 and a new period starts. A reset is possible if t < T − 1. If the current counter value is T − 1 a reset must be performed. – In all other cases, the counter value is increased by 1 and the values of all v i are updated applying one of the operations in the structure: v i ← x ◦i v i . Here, ◦i ∈ {+, −, ×, null, pr1 , pr2 }, 1 ≤ i ≤ k depends on q and b only. When the last component of an input is read A performs the tests for this component and moves to its last state without any further computation. It accepts the input if this last state belongs to F , otherwise A rejects. c) Non-deterministic (SR , k, , T )-automata are defined similarly with the only difference that δ becomes a relation in the following sense: If in state q the tests result in b ∈ {0, 1}k+ the automaton can non-deterministically choose a next state and as update operations either a reset or one among finitely many tuples (q , ◦1 , . . . , ◦k ) ∈ Q × {+, −, ×, null, pr1 , pr2 }k . As usual, a non-deterministic automaton accepts an input if there is at least one accepting computation. d) The language of finite strings accepted by A is denoted by L (A) ⊆ R∗ . e) A configuration of A is a tuple from Q × Rk × {0, . . . , T − 1} specifying the current data during a computation. Some remarks are appropriate before we start working with the model. Remark 2. a) There are several reasonable variations of the above model. For example, in the non-deterministic framework one can as well consider λ-transitions. A λ-transition does not read an input component, increases the counter by 1 and moves to a potentially new state without changing register values. If the period is reached also a λ-transition causes a reset of register and counter values to 0. It is easy to see that λ-transitions are equivalent to using the reset operation in the above non-deterministic model. A second variant uses step counters separately for each register. This seems to result in a less stable model with respect to decidability issues because register values could be transferred from one register to another if both have different reset points in time. Thirdly, a full-period model can be considered, in which each period has to be finished without resetting the counter earlier. This will be reasonable in some situations below, see Section 4.1. b) One could allow several initial states for non-deterministic automata without increasing the computational power. Since this would not simplify any of our proofs we refrain from doing so. c) One might wonder whether moves of an automaton should also depend on the counter value. Since the period T is fixed such a dependence can easily be simulated by using for each state q copies q(t ), 0 ≤ t ≤ T − 1 and replacing a transition from p to q by an according one from p (t ) to q(t + 1). Below this is helpful several times. d) It is well known that classical finite automata become Turing-universal if they are equipped with two counters [12]. The situation with GKL automata over the reals is different. In the original (i.e., non-periodic) GKL model over R an automaton can use each of its registers as counter and thus allows an arbitrary number of counters. However, the structural results in [10] show that this model is not BSS-universal. Therefore, undecidability results do not follow from (non-existent) universality.

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e) In [4] constants are considered as part of the structure. Since here we deal exclusively with SR only we think it is more natural to attach only the number of constants to an automaton. Then the same A might be used with different real constants. This is in accordance with the role constant play in the BSS model. Example 3. a) Every classically regular language L ⊆ {0, 1}∗ , when considered as a language in R∗ , can be accepted by a real periodic GKL automaton. This can be seen easily by interpreting a corresponding finite automaton for L as a GKL automaton which does not use its registers. b) An easy example of a language not being acceptable by a periodic automaton is L := {x ∈ R∗ |x = (x1 , . . . , xn ) for some n ∈ N and x1 = xn }. Once n is large enough compared to the automaton’s period it will have no ability to store the value x1 for a final comparison with xn . Note that L is easily acceptable in the non-periodic GKL model. c) For every semi-algebraic set S ⊆ Rn , n ∈ N there is an m ∈ N and a real periodic GKL automaton A such that S is the projection of the set L (A) ∩ Rm onto its first n components. This was shown in [10] for the real GKL automata model, but the proof applies as well in the periodic case; this is true because the main point is to give a bound on the number of steps needed to evaluate the polynomial conditions defining S. The periodicity of the new automaton then should be larger than this bound. (n) (n) d) Consider the fixed dimensional real Knapsack problem K S R : Let n ∈ N be fixed; then define K S R := {(x1 , . . . , xn , 1)| xi ∈ (0, 1), ∃ S ⊂ {1, . . . , n} such that xi = 1}. Here, in view of a later result we exclude instances for which S = {1, . . . , n} (n)

i∈ S

is a solution. Then K S R can be accepted by a non-deterministic periodic (SR , 1, 1, n + 1)-automaton. The automaton uses 1 as its only constant and a single register v 1 . When reading a new component xi of the input the automaton nondeterministically chooses whether xi should participate in the final sum or not. If it should, then xi is added to v 1 , otherwise v 1 remains unchanged. When (x1 , . . . , xn ) has been processed the automaton finally checks whether the last input component xn+1 equals 1. If not A rejects, if yes (i.e., xn+1 = 1) it is also compared with v 1 . The automaton accepts iff v 1 = 1. This automaton needs a period T > n. Below we shall see that the general real Knapsack problem, i.e., with varying input dimension n, is an example of a real language that cannot be accepted by a periodic automaton but satisfies the pumping lemma. 2 Below in Section 4 we outline a more systematic description of acceptable languages which resembles the structure of regular sets over finite alphabets. 3. Decidability Let us start with the study of some typical decision problems for periodic GKL automata. To each periodic (SR , k, , T )-automaton A we attach a directed graph G A . Additionally, the edges are labeled by certain semi-algebraic sets. Both the graph and those sets turn out to be crucial for solving many of the fundamental decision problems about periodic GKL automata. Before being more precise let us describe the intuition behind those definitions. Since A has periodicity T we are naturally led to consider the following two questions: Starting in a state q with counter and register values 0, which other states are reachable within precisely a number of t steps from q, where 1 ≤ t ≤ T ? And when processing which inputs from Rt does this happen? For the further analysis it is valuable to partition sequences of computations into several groups. One collects the partial computations that finish either with a reset or with a full period. The others gather computations that last t < T ‘pure’ computational steps, i.e., steps which do not apply the reset operation. Definition 4. Let A be a deterministic (SR , k, , T )-automaton with state set Q , initial state q0 and final states F ⊆ Q . a) The directed graph G A = ( V , E ) attached to A is defined as follows: For each q ∈ Q the set V contains a vertex q and vertices q(t ) for 1 ≤ t < T . G A has an edge ( p , q) iff there is a computation of A that when starting in p with register values 0 and performing at most T steps reaches q either after a reset or after exactly T steps. G A has an edge ( p , q(t )) for an 1 ≤ t < T iff there is a computation of A that when starting in p with register values 0 reaches q after t steps none of which is a reset. No other edges are present in G A . b) For edges ( p , q) and ( p , q(t )) of G A , respectively, define S ( p , q) ⊆

T

Ri and S ( p, q(t )) ⊆ Rt as the set of those vectors i =1

x for which A moves from p to q or to q(t ), respectively, when reading x according to the conditions under a).

Intuitively, vertices named by p ∈ Q are used for dealing with sequences of at most T computational steps of A which finish with a reset or the full period, whereas the copies q(t ), 1 ≤ t < T are used to reflect the final t steps (without a reset) of a computation. Therefore, there are no directed edges of form (q(t ), p ). Theorem 5. Let A be a deterministic (SR , k, , T )-automaton with attached directed graph G A = ( V , E ), S (u , v ) be defined as above for vertices u , v ∈ V . Then the following holds:

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a) All S ( p , q) are finite unions of semi-algebraic sets S (i ) ( p , q) ⊆ Ri for 1 ≤ i ≤ T , all S ( p , q(t )) are semi-algebraic in Rt .2 b) The edge relation of G A is BSS-computable, i.e., for given u , v ∈ V one can decide by a BSS algorithm whether (u , v ) ∈ E. c) For each q ∈ Q , 1 ≤ t ≤ T − 1 the set V (q, t ) ⊆ Rk of register values v ∈ Rk that occur as valid entries during a computation of A which reaches q such that the counter value is t is semi-algebraic (and thus BSS decidable). Proof. We are in this paper not interested in efficiency results and thus only argue how the questions under consideration lead to certain quantifier elimination tasks in the first order theory of the reals. Since this theory is BSS decidable, see [15] for more on the history of respective algorithms, the claimed results then follow. Ad a) Let p , q ∈ Q be fixed. We first deal with the edge ( p , q). Suppose A uses  constants and is in state p with all register values equal to 0. Without loss of generality we consider an entire period and describe the semi-algebraic set S ( T ) ( p , q) of inputs that transfer A from p to q in T steps. If a reset occurs earlier after i < T steps, the arguments how to obtain S (i ) ( p , q) remain the same. We enumerate all sequences P := ( p 0 , p 1 , p 2 , . . . , p T ) of states p i ∈ Q , where p 0 = p and p T = q, together with sequences β := (b1 , . . . , b T ), b i ∈ {0, 1}k+ of decision vectors such that for A it is possible to move from state p i to p i +1 if the actual input component xi +1 yields the test results coded by the components of b i . We now construct for each such pair (P , β) a first order formula (P ,β) (x) expressing for which inputs x ∈ R T the path P is representing A’s computation on x. Towards this aim we must record the changes of the register values as well as (i ) (i ) (i ) check the related test results. For each 1 ≤ i ≤ T let h1 , . . . , hk : Ri → R be polynomials such that h j (x1 , . . . , xi ) is the (i )

entry in register v j if the computation follows (P , β) on input x1 , . . . , xi starting from register values 0. The h j clearly are polynomials of degree at most i given the way A can compute. (i −1) (x1 , . . . , xi −1 ), . . . , Next, first order formulas ϕi (x1 , . . . , xi ) express that if A is in state p i −1 with register values v 1 = h1 (i −1)

v k = hk

(x1 , . . . , xi −1 ) and reads xi , then all performed tests give a result according to bi , A moves to state p i , and the (i )

(i )

new register values are v 1 = h1 (x1 , . . . , xi ), . . . , v k = hk (x1 , . . . , xi ). Once again, all above conditions can be expressed in first order logic over R due to the form of the tests that can be performed. (P ,β) (x) now is the conjunction of all these  formulas for all 1 ≤ i ≤ T . It follows that S ( T ) ( p , q) = {x ∈ R T | (P ,β) (x)}, where contains all suitable sequences (P ,β)∈

leading in T steps from p to q and respecting the constraints described above. Since there are only finitely many suitable pairs and for each of it the set of x satisfying (P ,β) (x) is semi-algebraic, the same holds for S ( T ) ( p , q). Taking possible earlier resets into account it follows that S ( p , q) =

T

S (i) ( p, q) is the union of semi-algebraic sets S (i) ( p, q) ⊂ Ri . i =1

The argument for edges of the form ( p , q(t )) and semi-algebraicity of S ( p , q(t )) is the same. Ad b) Given the result in a) for each pair (u , v ) of vertices of the graph G A it is decidable whether S (u , v ) = ∅ by (one of) the well known algorithms for quantifier elimination over real closed fields [15]. Such an algorithm is applied to each of the finitely many semi-algebraic sets constituting S (u , v ). Therefore, the edge relation of G A is BSS computable. Ad c) The argument is similar to that under a). First, consider a computation from the starting state q0 that reaches state q such that the counter has value t. Any such computation can be decomposed into a sequence of steps that ends either with a reset or the full period, followed by t final steps. In order to determine the realizable register assignments we have to figure out for which states p ∈ Q automaton A can reach q in t steps when starting with register values 0. Among those states we are only interested in the ones reachable in a computation that finishes with a reset, i.e., those p for which there is a path in G A from q0 to p. The latter can be checked by a usual search algorithm on directed graphs once G A has been computed according to b). Let H (q, t ) be the set of states p reachable in the above sense from q0 and such that (t ) (t ) S ( p , q(t )) = ∅. Then a v ∈ Rk is in V (q, t ) iff there is a p ∈ H (q, t ) and an x ∈ S ( p , q(t )) with v = (h1 (x), . . . , hk (x)), where (t )

h j are defined as in the proof of part a). Since this set is a projection of a finite union of semi-algebraic sets, by Tarski’s theorem it is again semi-algebraic. 2 The above theorem implies several decidability results of fundamental questions about real periodic GKL automata. Decidability here refers to the real number BSS model. Decidability in all cases follows relatively straightforwardly from the proof of Theorem 5. Note that we do not currently know about a full state minimization algorithm, thus the idea behind the classical algorithm for deciding equivalence of deterministic finite automata using minimal ones is not applicable. We present a partial state minimization algorithm in the proposition following the next theorem. Note also that given the significantly extended computability features of the original definition of real GKL automata in [4], none of the problems listed below is decidable in the more general non-periodic model. This was shown in [10]. Theorem 6. The problems below are decidable in the real BSS model:

2

A semi-algebraic set in Rn is a set that can be defined as a finite union of solution sets of polynomial equalities and inequalities.

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a) Emptiness: Given an (SR , k, , T )-automaton A, is L (A) = ∅? b) Reachability I: Given A as in a) with state set Q together with a state q ∈ Q , is there a computation starting in A’s initial state with register entries 0 that reaches q? c) Reachability II: Given an automaton A, a state q, a counter value t ∈ {0, . . . , T − 1}, and a v ∈ Rk , is there a computation starting in A’s initial state with register entries 0 that reaches q such that the counter’s value is t and the register values equal v? d) Reachability III: Similar to Reachability II, but without t being specified? Proof. Decidability of the questions under consideration is a relative immediate consequence of (the proof of) Theorem 5. We prove the statement for deterministic automata only. For non-deterministic ones a suitable re-definition of the semialgebraic sets used in Theorem 5 gives a similar proof. Let F denote the set of final states of the considered automaton. Ad a) For emptiness we compute a set H of states reachable by A from its starting configuration performing a sequence of steps whose final one is a reset. The computation of H can be done using the arguments from the proof of parts a) and b) in Theorem 5. If H ∩ F = ∅ it follows L (A) = ∅. Otherwise, for each p ∈ H , 1 ≤ t < T , and q ∈ Q we compute a description of the semi-algebraic set S ( p , q(t )) and check whether it is empty or not using quantifier elimination. It follows that now L (A) = ∅ iff all those S ( p , q(t )) are empty. Ad b) Using G A and the corresponding sets S (u , v ) it is easy to check whether a state q or one of its copies q(t ), 1 ≤ t < T are reachable in G A from the starting state. This can be done, for example, by a breadth-first search. Ad c) Check whether q (if t = 0) or q(t ) (if t > 0) are reachable in G A . In case it is we analyze the set of attainable register values V (q, t ) as in the proof of Theorem 5. Ad d) As in c), but instead of checking one fixed t the algorithm has to be performed for all 0 ≤ t ≤ T − 1. 2 4. Structural results In this section the focus is on elaborating elementary structural results for languages acceptable by real periodic GKL automata. They extend the corresponding ones for discrete finite automata and include a pumping lemma, closure properties like the equivalence of non-deterministic and deterministic periodic GKL automata when the automata always use full periods, and a kind of generalization of regular expressions to our setting. Note that in its usual form the pumping lemma does not hold for general GKL automata over R and no variant is known to be true; similarly, non-deterministic real GKL automata are strictly more powerful than deterministic ones, see [10] for both issues and [4] for similar results over other structures. Thus, periodicity significantly reduces the computational power of GKL automata and brings them closer to what one might expect from a real version of finite automata. We start with the easy to prove Pumping Lemma. Lemma 7 (Pumping Lemma for real periodic GKL automata). Let L ⊆ R∗ be accepted by a deterministic real periodic GKL automaton A with periodicity T , k registers and s states. Then for all w ∈ L of algebraic size | w | ≥ sT there exist x, y , z ∈ R∗ such that w = xyz, | y | ≥ T , |xy | ≤ sT , and ∀i ∈ N0 xy i z ∈ L. Proof. Periodicity of A implies that for all r ∈ N after at most rT steps of any computation the register values and the counter have been reset to 0 at least r many times. The actual configuration after a reset only depends on the current state. Thus, for inputs w of length at least sT the automaton attains at least one state twice among the states that are attained immediately after one of the at least s many situations when all the register values and the counter start anew from 0. The rest of the proof is as usual: If p is a state occurring twice, x the prefix of w processed by A until p is reached for first time, y the part processed until p occurs for the second time, and z the rest, then w = xyz by definition and xy i z ∈ L for all i ∈ N0 because A’s configuration is ( p , 0, 0) both when it starts to process y and when it has finished. The remaining conditions obviously are satisfied. 2 Example 8. We apply the Pumping Lemma to give some examples of languages that are not acceptable by periodic automata. i) The first one is the canonical discrete example language {0n 1n |n ∈ N}. As in the finite automata world this language is neither acceptable by a periodic real GKL automaton. ii) Our second example is a language related to the rational numbers. More precisely, define L ⊆ R∗ as

L := {(r , x1 , x2 , . . . , xn , 0, t , s)|n, t , s ∈ N, xi ∈ {−1, 1}, s=



i , x i =1

xi , t =



i ,xi =−1

|xi |, r = ts }.

Note that the projection of L to the first component gives the (positive) rationals. It was shown in [10] that L can be accepted by a non-periodic deterministic (SR , 3, 3)-automaton. This result in [10] was a key ingredient for several undecidability results based on undecidability of Q in R in the BSS model. On the other side, the above lemma immediately implies that L cannot be accepted by a periodic automaton. As soon as we have a long enough word w ∈ L which is pumped

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at a subword not including the final three components 0, t, and s, the pumped word will destroy the equality requirements for s and t. Obviously, such a w ∈ L exists. iii) We give a language that satisfies the Pumping Lemma without being acceptable by a periodic automaton. Recall the real version of Knapsack from Example 3, part e); we can define it as K S R = {(x1 , . . . , xn , 1)|n ∈ N, xi ∈ (0, 1), ∃ S ⊂  {1, . . . , n} xi = 1}. For the example to work above we exclude instances for which as only solution all components have i∈ S

to be added, i.e., for which S = {1, . . . , n}. Now, obviously, every x ∈ K S R can be pumped at some place because if a subset of components sums up to 1 we can extend such an x by arbitrarily adding components. And since only proper subsets are considered as solutions one remains within K S R by removing a component that does not belong to a solution. On the other hand K S R cannot be accepted by a periodic automaton. Suppose otherwise and let A denote such an automaton. Consider the first part (x1 , . . . , xs ) of an input such that all xi sum up to a result < 1 and such that A either has done a reset or has finished a full period when reading the input component xs . There are uncountably many such vectors (x1 , . . . , xs ). Then A’s next step on the next input component xs+1 does at most depend on finitely many choices for xs+1 , namely those occurring as constants. But for s large enough there are remaining choices for xs+1 such that the resulting vector belongs to K S R and others for which it does not. So A cannot correctly distinguish these inputs. 2 The Knapsack example of course raises the question for a characterization of acceptable languages. We give one such below that is similar to the study of regular expressions. We do not know a characterization in the spirit of the classical Myhill-Nerode theorem. Another interesting question might be to sharpen the statement of the Pumping Lemma by not only considering loops having a length being an integer multiple of the periodicity; this, however, would require to control the evolvement of the register entries, something that seemed to hinder a sharper structural result in [10]. 4.1. Closure properties For the following results we consider full periods only, i.e., we disallow resets earlier than after the full period T . Note that all previous results are true by the same arguments in this full-period version of the model. Without requiring periodicity non-deterministic real GKL automata are more powerful than deterministic ones [10]. This difference vanishes if we include the periodicity requirement into the model. The proof is based on a powerset construction as its classical counterpart. However, due to the ability to perform computations it is more complex. Theorem 9. The classes of languages over R∗ acceptable by non-deterministic and by deterministic periodic real GKL automata using full periods only are the same. Proof. Let A be a periodic non-deterministic (SR , k, , T )-automaton with state set Q and final states F ⊆ Q . The construction of an equivalent deterministic automaton A in principle uses the classical powerset idea. It is nevertheless more complicated because of the automaton’s ability to compute. An equivalent deterministic automaton has not only to record states that can be reached non-deterministically, but also to store both the corresponding register entries. In particular, it might be possible that A in a state has several choices how to continue to the same successor state but with different computations performed on the registers. Therefore, instead of (unordered) subsets of Q we are lead to consider (ordered) tuples of elements in Q as states of the new automaton A . The main idea for controlling these effects is to figure out how many different configurations can occur at most at one point in time. Let M ∈ N be the maximal number of non-deterministic transitions A can choose from for any of its states. We first want to argue how much working space (organized in form of register blocks of length k) a new deterministic automaton needs in order to record all possible configurations of A that can occur in parallel during a computation. Here, a configuration represents information about the current state, the counter value and the register entries. Since we are not interested in complexity issues below we overestimate this number. The decisive point is that it can be bounded by a constant only depending on A. In order to find such an estimate we first consider the evolvement of a single register v i during a non-deterministic computation. If A starts a period and v i = 0, then within r steps the register value can attain at most M r different assignments as long as r < T . After T steps the register value is reset to 0. So at any point in time of a computation a single register can attain at most M T −1 many different values. Since there are k registers, | Q | states and the period is T the automaton can attain at each fixed time step at most m := | Q | M k( T −1) different configurations. The above leads to the following construction. The deterministic automaton we look for each state q ∈ Q and each counter value 0 ≤ t ≤ T − 1 uses a number M k( T −1) of register blocks of length k each. Using these registers it will be able to record all potential calculations of A from a state q on with counter value t and register assignment 0. In particular it can also bookkeep all reachable configurations during the computation on a given input starting from q0 with t = 0. The deterministic automaton then has mk registers. For each pair (q, t ) ∈ Q × {0, 1, . . . , T − 1} it reserves some fixed ‘working space’ among the register blocks as indicated above. Its set of constants will be chosen to be the same as those used by the given non-deterministic automaton, its period again is T .

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Next we define the state set Q  of A : Q  will be used to code the multi-set of states of Q that can occur with potentially different register values at a certain point of a computation. In addition, for each original state q we consider new states q(t ), 0 ≤ t ≤ T − 1 and interpret them below as the given non-deterministic automaton being in state q with counter value t. Since not necessarily m different configurations are realizable let q∗ ∈ / Q denote a new dummy symbol and define Q  as a subset of Q := ({q(t )|q ∈ Q , 0 ≤ t ≤ T − 1} ∪ {q∗ })m ; a state in Q  lists with the respective cardinalities the set of states in Q reachable non-deterministically when any fixed part of the input has been read. If that way less than m different configurations are reachable the lacking components are filled with q∗ .3 The starting state of A is chosen as follows. There must be one area of the working space corresponding to q0 (0) with register assignments 0. The starting state q0 ∈ Q  codes this using the corresponding components. All other components are set to q∗ . Final states in Q  are those which contain at least one state q(t ) with q ∈ F as a component. The transition function can now be defined straightforwardly; in order not to overload the presentation notationally we just describe its functioning informally. In particular, we do not work out the bookkeeping of the working space. This is tedious, but it should be clear from what has been said before that there is no principle difficulty to do it. If A after t steps on input x can reach a configuration ( p , v 1 , . . . , v k , t ), then A is in a state p  which has p (t ) as one of its components; attached to each component is one block of k registers - and the one attached to the component containing the above p (t ) has ( v 1 , . . . , v k ) as its register values. Next, each of the at most M many non-deterministic transitions A can perform is recorded via changing the corresponding components of p  and their attached register blocks accordingly. If A in state p has less than M many choices, in p  the lacking components are set to q∗ and the corresponding register values simply remain unchanged, unless a period is filled. Finally, A has an accepting computation on x ending in a state q f ∈ F if and only if A reaches at the end a state that has q f (t ) for a t ∈ {0, 1, . . . , T − 1} as one of its components, i.e., A accepts x. 2 An immediate consequence given the equivalence between non-deterministic and deterministic periodic automata using full periods are the following closure properties of the set of languages accepted by periodic automata. Corollary 10. The set of languages in R∗ being acceptable by periodic GKL automata with full periods is closed under union, complementation, and intersection. The equivalence problem asking whether two such automata accept the same language is decidable. Proof. For the union of two acceptable languages the proof is standard; the only minor technical problem is to combine the periods T 1 , T 2 of the two given automata. Define T := T 1 · T 2 as period of a non-deterministic automaton A for L 1 ∪ L 2 . A begins in a new starting state. Its first transition non-deterministically simulates either the first transition of A1 or of A2 from their respective starting states. Afterwards, A deterministically simulates the automaton it has chosen in the first step. If Ai is simulated A resets the register values to 0 in addition after each sequence of T i steps. The choice of T guarantees that a reset performed by A due to its periodicity does not collide with the simulations it performs. Since the class of languages accepted by periodic deterministic automata trivially is closed under complementation it follows as well closeness under intersection for both automata models by Theorem 9. Decidability of the equivalence problem for two automata A1 , A2 now follows from Theorem 6, a) checking emptiness of both L (A1 ) ∩ L (A2 ) and of L (A1 ) ∩ L (A2 ). 2 The above decision algorithm for equivalence of periodic real automata using full periods implies that, at least under certain conditions and in contrast to the non-periodic model [10], state minimization of periodic automata can be performed. The algorithm below, however, is not completely satisfying compared to the situation for finite automata. We comment on it afterwards. Suppose a periodic automaton A accepting an L ⊆ R∗ is given. A has several integer parameters which might influence the definition of a state minimal automaton for L. Beside the number of states of A there is the period T , the number k of registers, and the number  of constants. If we are just interested in minimizing the number of states of an automaton for L it is not obvious to us how to do that if the remaining parameters T , k,  are allowed to be arbitrarily large. Applying the proof given below in such a general framework results in existentially quantifying those parameters. Since they are integral, in general we cannot eliminate the corresponding quantifiers. The situation is different if we look for an equivalent automaton for L such that its parameter values are somehow computationally bounded as a function in the parameters of A. Another problem might result from allowing a minimal automaton to use different real constants. Then it is not clear whether such constants in principle can be computed by a BSS algorithm. For example, the original automaton, using a constant 2, might check for three√input components x1 , x2 , x3 whether x1 = x2 ∧ x3 = 2 ∧ x1 · x2 = x3 . Now another automaton could do the same using 2 as a constant which possibly could not be computed from the data specifying the first automaton. It is unclear whether such an effect might help to reduce the necessary number of states. We therefore restrict the set of automata among which we search for a minimal state automaton in two ways. We only allow the same constants as those the given automaton uses. And we require a computable bound on the remaining

3 In order to make the coding unique we could order the components of any tuple in Q  according to an order on the set {q(t ), |q ∈ Q , 1 ≤ t ≤ T − 1} ∪ {q∗ }, but we refrain from elaborating on this because it will likely not increase understandability.

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parameters of the automaton. Under this additional requirements we can use quantifier elimination algorithms to find a minimal state automaton for L. Proposition 11. Let L ⊆ R∗ be acceptable by a deterministic periodic real GKL automaton A using full period, having s states, period T , k registers and  constants {c 1 , . . . , c  }. Let f : N 3 → N be a BSS computable function which is monotonically strictly increasing in each component. Then there is a BSS algorithm computing another automaton A with parameters s , T  , k ,  such that i) ii) iii) iv)

A uses the same constants as A, A also accepts L, f (s , T  , k ) ≤ f (s, T , k) and

the number s of states is minimal for all automata satisfying conditions i)-iii).

Proof. Given A first compute f (s, T , k). Since f is strictly increasing in each component the set P := {(s , T  , k ) ∈ N 3 | f (s , T  , k ) ≤ f (s, T , k)} is finite; due to computability of f it can also be enumerated. The decision algorithm enumerates all parameter tuples belonging to set P . Since the same constants have to be used for each fixed tuple of values there are only finitely many different automata with the given parameters using those constants. We enumerate them. For each automaton in the enumeration apply the algorithm presented in the proof of the previous theorem to check equivalence between the respective automaton and A. Once all parameter patterns have been checked that way it is clear what the minimal number of states of an automaton accepting L (A) is. Moreover, a corresponding automaton has been computed by the above algorithm. 2 If we are only interested in knowing the minimal number of states we can avoid the requirement to use the same set of constants. Then it would be sufficient to add in the formula for automata equivalence quantifiers requiring existence of suitable constants and include the additional parameter  as argument in the above function f . Though the above theorem gives a minimization procedure the construction does not really depend much on A. It would of course be more interesting to design an algorithm which, like the classical minimization algorithm, stepwise changes A by detecting identifiable parts. However, it is unclear whether this is possible at all. So far we have not been able to set up an approach like the classical one by Myhill-Nerode defining suitable equivalence classes on states and on words. This seems an interesting open problem to us. Presumably, it will also lead to certain quantifier elimination tasks when defining and computing equivalent states. Maybe in order to be successful one has to limit the considered class of allowed equivalent automata, for example, by requiring them to have the same period as the given automaton. Furthermore, the equivalence problem might be interesting from the complexity point of view. There are not too many NPR -complete or PR -complete problems, respectively, from different areas known in the BSS model; it would be interesting to find out more, for example, related to questions about finite automata in this area. The proof of Proposition 11 does not put much focus on the special structure of the minimization problem by reducing everything to a quantifier elimination task. So what is the precise BSS complexity of state minimization of periodic real automata? 4.2. Regular real sets The results especially of Theorem 5 resemble a strong similarity between the structures of real languages acceptable by periodic GKL automata and of regular languages. In this subsection, we want to make this intuition precise. Again, here we consider computations with full periods only. It has been shown in Section 3 that computation cycles of length the periodicity T play a crucial role in the analysis of periodic automata. If such an automaton can move in T steps from state p to q assuming all registers have been initialized to 0, then the (non-empty) semi-algebraic sets S ( p , q) introduced in Definition 4 constitute building blocks of words being processed by the automaton; similarly at the end of a computation with sets of form S ( p , q(t )). There is as well a natural way to define a Kleene-∗ operation on those sets. Starting from the automaton’s side, each cycle in the directed graph G A indicates that a corresponding sequence of operations can be performed by A arbitrarily many times, each time processing a word from a corresponding semi-algebraic set. This indicates that we might obtain a recursive construction of expressions being built on base of certain semi-algebraic sets by using the typical operations for regular expressions. We shall first define what kind of semi-algebraic sets can be taken as building blocks. These sets then will be used to define a family of real sets that we call regular. The main theorem below establishes the equivalence between the latter and the class of languages acceptable by real periodic GKL automata. The restricted way in which automata can read the input, namely once from left to right, determines the polynomial systems that are building blocks of our regular sets. This is reflected in the sequential structure in which variables occur in those polynomials and the degree of each variable being at most 1. Obviously, this structure reproduces the way the automaton can use its registers for computations. Definition 12. a) Let T ∈ N . A sequential polynomial family in real variables x1 , . . . , x T is a set { p i |1 ≤ i ≤ T } of polynomials of the following structure: Each p i depends (at most) on variables x1 , . . . , xi ; p 1 is one of the polynomials {±x1 , 0}, and

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p i is taken from the set {± p i −1 , ±xi , ± p i −1 ◦ ±xi , 0}. Here ± indicates that optionally one of the signs can be chosen, and

◦ ∈ {+, −, ×}.

b) A basic regular real set S is a set in some R T , T ∈ N that can be defined as follows. There is a finite set c 1 , . . . , c  of real constants, a sequential polynomial family { p i }1≤i ≤ T and relation symbols i , i j ∈ {≥, >, =, <, ≤} for all 0 ≤ i ≤ T , 1 ≤ j ≤  such that S is the solution set of the polynomial inequality system given as

x1 1 0

∧ xi i p i −1 (x1 , . . . xi −1 ) for 2 ≤ i ≤ T ∧ xi i j c j for 1 ≤ i ≤ T , 1 ≤ j ≤ . c) A regular real set is a set S ⊆ R∗ that can be generated in finitely many steps from finitely many basic regular real sets in some R T using the following operations: concatenation, union, intersection, and the Kleene star. As final construction  step, such a set can be concatenated from right with a basic regular real set in some (potentially other) R T . Note that above it is allowed to change  and the c j for each new basic regular real set. This is justified by the closure properties. Intersection is only included for simulating the use of k > 1 computation registers. One could similarly define the basic real sets as intersections of the solution sets of the particular sequential polynomial systems and then remove intersection from the list of operations. Example 13. A basic regular set L is the set of vectors of roots of unity for a given order s ∈ N in the following sense: L := {(1, x, x2 , . . . , xs )|xs = 1} ⊆ Rs+1 . A suitable description via a polynomial system of equations is obtained by taking all polynomials p i = xi × p i −1 and using 1 as (only) constant. Theorem 14. The sets accepted by periodic real GKL automata with full periods are precisely the regular real sets. Proof. Let S ⊆ R∗ be regular real. The proof that S is acceptable can be done easily by induction following the recursive structure of S using Corollary 10. Basic regular real sets obviously are acceptable by periodic automata. The building rules for sequential polynomials and the tests allowed precisely reflect the computation of an automaton for T steps. Due to the closure properties of acceptable languages S is acceptable. Note that for concatenating basic regular sets defined with respect to the same T the corollary can easily be extended and the same holds for the Kleene star. Conversely, consider a periodic real automaton A with period T , k registers and constants c 1 , . . . , c  . Employing the notation of Definition 4 and of Theorem 5 we first note that the sets S ( p , q) and S ( p , q(t )) are regular real for all states p , q and all 1 ≤ t ≤ T . For example, S ( p , q) results from a finite union of intersections of the solution sets of k inequality systems, each having the structure given in Definition 12, b). Each such system corresponds to A’s behavior on one of its registers. Finite unions then are necessary because the computation might depend on test results with respect to all registers. The above argument remains the same if S ( p , q(t )) is considered. Now these sets are the building blocks of the inputs accepted by A. More precisely, any accepting computation of A corresponds to a walk in the automaton’s graph G A from node q0 representing the starting state of A to a node q f (t ), where q f is a final state of A and 0 ≤ t ≤ T − 1. All such walks can be decomposed into finitely many building parts. Consider a walk that reaches p r from p 0 in a sequence of steps passing through states p 0 → p 1 → . . . → p r with all p i different. The definition of G A implies that each p i was reached after T steps from p i −1 . Then the set of points in RrT following this computation is the concatenation S ( p 0 , p 1 ) × S ( p 1 , p 2 ) × . . . × S ( p r −1 , p r ) and thus real regular. If p r = p 0 the walk is a loop and can be passed through arbitrarily often yielding the real regular Kleene star of the latter set as part of the input branched along the loop states. And if the walk ends with an edge ( p , q f (t )) this corresponds to the real regular set S ( p , q f (t )). It follows that the set L (A) of inputs from R∗ being accepted can be decomposed into a finite combination of concatenations and Kleene stars starting from basic real regular sets. Thus L (A) is a regular real set. 2 5. Conclusion and further questions In this paper we have modified the automata model introduced by Ghandi, Khoussainov, and Liu in order to obtain a reasonable finite automata model over the structure of real numbers with basic arithmetic operations. The main restriction in our periodic model is the limitation of time during which an automaton can store register values. This seems to correspond well to the restricted memory classical finite automata over finite structures have. Our results show that the periodic automata over R share many of the basic properties classical automata have, of course adapted accordingly. There are several further problems that might be interesting to study. First, one could continue the line of research in this paper and analyze further properties of real languages acceptable by periodic GKL automata. For example, as indicated already we would be interested in a more practical algorithm to minimize the number of states of an automaton, maybe restricting the class of equivalent automata over which the minimization is considered. Next, above we completely disregarded complexity issues and only applied elimination results in their general form. In many of our problems requiring quantifier elimination the formulas obtained have a very specific structure because the automata process an input component only once and it is only influencing T steps of a computation. Therefore, one might ask which of the problems treated above

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are efficiently solvable, which are NPR -complete in the BSS framework? Especially finding new NPR -complete problems is interesting since the list of known such problems still is relatively limited. A third area of further research is considering other underlying structures than the reals. One major motivation of [4] was to have a generalized automata model for many different structures which also homogenizes existing ones. So the question is in how far our restricted version of GKL automata also is meaningful for further structures like those considered in [4]? Finally, it seems promising to analyze the new automata model as well with respect to questions occurring in formal verification and model checking, a typical area of applications for classical finite automata. This leads to several questions that have to be treated first. In the discrete framework finite automata on infinite words play a crucial role here. The classical theory initiated by Büchi, see [16] for a longer introduction into the topic, established a close relation between such infinite computations of finite automata and certain logics in which (at least some of) the corresponding verification tasks then can be formalized. So in a first step one can generalize periodic automata to performing infinite computations over countably infinite sequences of reals. For working with infinite structures meta-finite model theory was developed in [5] and applied to BSS computability theory in [6,3]. Now an important task to start with is to investigate potential links between the latter and a kind of periodic real Büchi automata. In particular, it would be important to figure out whether computations of such an extended automata model can be described by suitable logics that are natural for the metafinite-model theory approach to complexity theory in the BSS model. This work was recently started in [11]. 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