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A Uniform Method for Testing Computational Complementarity

Studies of computational complementarity properties in finite state interactive automata may shed light on the nature of both quantum and classical computation. But, complementarity is difficult to test even for small-size automata. This paper introduces the concept of an observation graph of an automaton which is used as the main tool for the design of an algorithm which tests, in a uniform manner, two types of complementarity properties. Implementations have been run on a standard desktop computer examining all 5-state binary automata. 1 Two Computational Complementarity Principles Building on Moore’s “Gedanken” experiments, in [15, 14] complementarity was modeled by means of finite automata. Two new computational complementarity principles have been introduced and studied in [3, 6, 5, 4, 2] using Moore’s automata. To understand Moore’s approach it is enough, at this stage, to say that the machines we are going to consider are finite in the sense that they have a finite number of states, a finite number of input symbols, and a finite number of output symbols. Such a machine has a strictly deterministic behaviour: the current state of the machine depends only on its previous state and previous input; the current output depends only on the present state. A (simple) Moore experiment can be described as follows: a copy of the machine will be experimentally observed, i.e. the experimenter will input a finite sequence of input symbols to the machine and will observe the sequence of output symbols. The correspondence between input and output symbols depends on the particular chosen machine and on its initial state. The experimenter will study the sequences of input and output symbols and will try to conclude that “the machine being experimented on was in state q at the beginning of the experiment”.1 Moore’s experiments have been studied from a mathematical point of view by various researchers, notably by Ginsburg [9], Chaitin [7], Conway [8], and Brauer [1]. A comprehensive survey on testing finite state machines is presented in [11]. In what follows we are going to use two non-equivalent concepts of computational complementarity based upon modeling finite automata (see [3]). Informally, they can This is often referred to as a state identification experiment. be expressed as follows. Consider the class of all elements of reality2 and consider the following properties. A Any two distinct elements of reality can be mutually distinguished by a suitably chosen measurement procedure. B For any element of reality, there exists a measurement which distinguishes between this element and all the others. That is, a distinction between any one of them and all the others is operational. C There exists a measurement which distinguishes between any two elements of reality. That is, a single pre-defined experiment exists to distinguish between an arbitrary pair of elements of reality. (Classical case.) Complementarity corresponds to the following cases: CI Property A but not property B (and therefore not C): The elements of reality can be mutually distinguished by experiments, but one of these elements cannot be distinguished from all the other ones by any single experiment. CII Property B but not property C: Any element of reality can be distinguished from all the other ones by a single experiment, but there does not exist a single experiment which distinguishes between any pair of distinct elements. 2 Moore Automata A finite deterministic automaton consists of a finite set of states and a set of transitions from state to state that occur on input symbols chosen from some fixed alphabet. For each symbol there is exactly one transition out of each state, possible back to the state itself. So, formally, a finite automaton consists of a finite set Q of states, an input alphabet Σ, and a transition function δ : Q× Σ → Q. Sometimes a fixed state, say 1, is considered to be the initial state, and a subset F of Q denotes the final states. A Moore automaton is a finite deterministic automaton having an output function f : Q → O, where O is a finite set of output symbols. At each time the automaton is in a given state q and is continuously emitting the output f(q). The automaton remains in state q until it receives an input signal σ, when it assumes the state δ(q, σ) and starts emitting f(δ(q, σ)). In this paper we are going to concentrate on the case of automata on a binary alphabet Σ = {0, 1} having O = Σ. So, from now on, a Moore automaton will be just a triple M = (Q, δ, f). Let Σ∗ be the set of all finite sequences (words) over the alphabet Σ, including the empty word (the neutral element in the semigroup of string concatenation); by Σ+ we denote Σ∗ \ { }. The transition function δ can be extended to a function δ : Q × Σ∗ → Q, as follows: δ(q, ) = q, δ(q, σw) = δ(δ(q, σ), w),∀q ∈ Q, σ ∈ Σ, w ∈ Σ∗. The output produced by an experiment started in state q with input sequence w ∈ Σ∗ is described by E(q, w), where E is the function E : Q × Σ∗ −→ Σ∗ defined as follows: E(q, ε) = f(q), E(q, σw) = f(q)E(δ(q, σ), w), q ∈ Q, σ ∈ Σ, w ∈ Σ∗, and f : Q −→ O(= Σ) is the output function. Consider, for example, Moore’s automaton, in which Q = {1, 2, 3, 4}, Σ = {0, 1}. The transition is given by the following tables The terms “elements of reality”, “properties”, and “observables” will be used as synonyms.