Finite Automata Models of Quantized Systems: Conceptual Status and Outlook

Since Edward Moore, finite automata theory has been inspired by physics, in particular by quantum complementarity. We review automaton complementarity, reversible automata and the connections to generalized urn models. Recent developments in quantum information theory may have appropriate formalizations in the automaton context. 1 Physical connections Physics and computer science share common interests. They may pursue their investigations by different methods and formalisms, but once in a while it is quite obvious that the interrelations are pertinent. Take, for example, the concepts of information and computation. Per definition, any theory of information and computation, in order to be applicable, should refer to physically operationalizable concepts. After all, information and computation is physical [1]. Conversely, concepts of computer science have increasingly influenced physics. Two examples for these developments have been the recent developments in classical continuum theory, well known under the term “deterministic chaos,” as well as quantum information and computation theory. Quantum systems nowadays are often perceived as very specific and delicate (due to decoherence; i.e., the irreversible loss of state information in measurements) reversible computations. Whether or not this correspondence resides in the very foundations of both sciences remains speculative. Nevertheless, one could conjecture a correspondence principle by stating that every feature of a computational model should be reflected by some physical system. Conversely, every physical feature, in particular of a physical theory, should correspond to a feature of an appropriate computational model. This is by no means trivial, as for instance the abundant use of nonconstructive continua in physics indicates. No finitely bounded computation could even in principle store, process and retrieve the nonrecursively enumerable and even algorithmically incompressible random reals, of which the continuum “mostly” exists. But also recent attempts to utilize quantum computations for speedups or even to solve problems which are unsolvable within classical recursion theory [2] emphasize the interplay between physics and computer science. Already quite early, Edward Moore attempted a formalization of quantum complementarity in terms of finite deterministic automata [3]. Quantum complementarity is the feature of certain microphysical systems not to allow the determination of all of its properties with arbitrary precision at once. Moore was interested in the initial state determination problem: given a particular finite automaton which is in an unknown initial state; find that initial state by the analysis of input-output experiments on a single such automaton. Complementarity manifests itself if different inputs yield different properties of the initial automaton state while at the same time steering the automaton into a state which is independent of its initial one. Moore’s considerations have been extended in many ways. Recently, different complementarity classes have been characterized [4] and their likelihood has been investigated [5, 6]. We shall briefly review a calculus of propositions referring to the initial state problem which resembles quantum logic in many ways [7, 8]. Automaton theory can be liked to generalized urn models [9]. In developing the analogy to quantum mechanics further, reversible deterministic finite automata have been introduced [10]. New concepts in quantum mechanics [11] suggest yet different finite automaton models. 2 Automaton partition logics Consider a Mealy automaton 〈S, I,O, δ, λ〉, where S, I,O are the sets of states, input and output symbols, respectively. δ(s, i) = s and λ(s, i) = o, s, s ∈ S, i ∈ I and o ∈ O are the transition and the output functions, respectively. The initial state determination problem can be formalized as follows. Consider a particular automaton and all sequences of input/output symbols which result from all conceivable experiments on it. These experiments induce a state partition in the following natural way. Every distinct set of input/output symbols is associated with a set of initial automaton states which would reproduce that sequence. This set of states may contain one or more states, depending on the ability of the experiment to separate different initial automaton states. A partitioning of the automaton states associated with an input sequence is obtained if one considers the variety of all possible output sequences. Stated differently: given a set of input symbols, the set of automaton states “decays” into disjoint subsets associated with the possible output sequences. This partition can then be identified with a Boolean algebra, with the elements of the partition interpreted as atoms. By pasting the Boolean algebras of the “finest” partitions together, one obtains a calculus of proposition associated with the particular automaton. This calculus of propositions is referred to as automaton partition logic. The converse is true as well: given any partition logic, it is always possible to (nonuniquely) construct a corresponding automaton with the following specifications: associate with every element of the set of partitions a single input symbol. Then take the partition with the highest number of elements and associate a single output symbol with any one element of this partition. The automaton output function can then be defined by associating a single output symbol per element of the partition (corresponding to a particular input symbol). Finally, choose a transition function which completely looses the state information after only one transition; i.e., a transition function which maps all automaton state into a single one. We just mention that another, independent, way to obtain automata from partition logics is by considering the set of two-valued states. In that way, a multitude of worlds can be constructed, many of which feature quantum complementarity. For example, consider the Mealy automaton 〈{1, 2, 3}, {1, 2, 3}, {0, 1}, δ = 1, λ(s, i) = δsi〉 (the Kronecker function δsi = 1 if s = i, and zero otherwise). Its states are partitioned into {{1}, {2, 3}}, {{2}, {1, 3}}, {{3}, {1, 2}}, for the inputs 1, 2, and 3, respectively. Every partition forms a Boolean algebra 2. The partition logic depicted in Fig. 1 is obtained by “pasting” the three algebras together; i.e., by maintaining the order structure and by identifying identical elements; in this case ∅, {3, 1, 2}. It is a modular, nonboolean lattice MO3 of the “chinese lantern” form. A systematic study [8, pp. 38-39] shows that automata reproduce (but are not limited to) all finite subalgebras of Hilbert lattices of finite-dimensional quantum logic.