The Algebraic Theory of Semigroups, Vol. I. By A. H. Clifford and G. B. Preston. Pp. xv + 224. $10.60. 1961. (American Mathematical Society, Providence.)

Automata theory based on quantum logic (abbr l-valued automata theory) may be viewed as a logical approach of quantum computation In this paper, we characterize some fundamental properties of l-valued automata theory, and discover that some properties of the truth-value lattices of the underlying logic are equivalent to certain properties of automata More specifically (i) the transition relations of l-valued automata are extended to describe the transitions enabled by strings of input symbols, and particularly, these extensions depend on the distributivity of the truth-value lattices (Proposition 31); (ii) some properties of the l-valued successor and source operators and l-valued subautomata are demonstrated to be equivalent to a property of the truth-value lattices which is exactly equivalent to the distributive law (Proposition 43 and Corollary 44) This is a new characterization of Boolean algebras in the framework of l-valued automata theory; (iii) we verify that the intersection of two l-valued subautomata is still an l-valued subautomaton if and only if the multiplication (&) is distributive over the union in the truth-value lattices (Proposition 45), which is strictly weaker than the usual distributivity; (iv) we show that some topological characterizations in terms of the l-valued successor and source operators also rely on the distributivity of truth-value lattices (Theorem 56) Finally, we address some related topics for further study

Automata theory based on quantum logic: some characterizations

A fuzzy topology associated with a fuzzy finite state machine

Přispěvek si klade za cil vytvařet hypergrupy linearnich parcialnich diferencialnich operatorů prvniho řadu s proximitami, z nichž jedna vytvaři tolerancni pologrupu na potencni množině zminěných diferencialnich operatorů. Konstrukce zkoumaných hypergrup jsou založeny na tzv. “Koncovem lemmatu” aplikovanem na uspořadane grupy diferencialnich operatorů. Navic, je zkonstruovana regularně předuspořadana transpozicni hypergrupa uvažovaných parcialnich diferencialnich operatorů.

On Certain Proximities and Preorderings on the Transposition Hypergroups of Linear First-Order Partial Differential Operators

Let A = ( Q, X, δ ) be an X -automaton, with Q its state set. For a subset B ⊆ Q , the source of B is defined as σB = { q Ȩ Q | δ ( q, x ) Ȩ B for some x Ȩ X }. The source turns out to be a closure operator for Q and defines a topology on Q , vis. B ⊆ Q is closed iff B = σB . It is shown that many automata-theoretic concepts, e.g., separation, connectivity, retrievability, strong connectivity, etc., have standard topological analogs under this topology and many results concerning these concepts are direct consequences of this observation.

A topology for automata: A note

The source of a set of states of an automaton is introduced as the set of all predecessors of members of the set. The source is shown to be a valuable tool in proofs; it is also shown to provide new insights and directions in automata theory. Relationships are explored between the source on the one hand and, on the other, subautomata, primaries, blocks, connectivity, separation, strong connectedness, retrievability, Abelian automata, homomorphisms, and other concepts.

The source as a tool in automata

On the Connectivity of the Product of Automata

Left-discrimination of a semigroup is defined and shown to be a sufficient condition that a semigroup be isomorphic to the input semigroup of its semigroup automaton, a necessary condition if the semigroup is finite. The left-discrimination sequence of an automaton is defined as a sequence of semigroups beginning with the input semigroup of the automaton, each member being the input semigroup of the semigroup automaton of its predecessor. It is related directly to a particular monotonically decreasing sequence of subautomata of the original automaton. This sequence is shown to be preserved by homomorphisms and is extended and used in an algorithm for determining the homomorphisms on one finite automaton to another.

The left-discrimination sequence of an automaton

Publisher Summary This chapter presents several characteristics of connectedness and of separation in automata and discusses their relationships to other connectivity properties and to the basic structure of automata. An automaton is finite only if its set of states is finite. A primary of a nonempty finite automaton is a maximal singly generated subautomaton. The union and the intersection of subautomata of A are themselves subautomata of A. The chapter illustrates a method for finding the minimal separated subautomata of a finite automaton and explains its use in determining the homomorphisms, endomorphisms, isomorphisms, and automorphisms of finite automata. The separated parts of an automaton can be regarded as unrelated automata with the same input alphabet. Every nonempty automaton is made entirely of blocks. Many problems are profitably reducible to the natural building blocks. Two blocks of a nonempty automaton are either identical or disjoint. A nonempty subautomaton C of A is separated only if C is the union of blocks of A.

Connectivity and separation in automata

searched only if we replace the \"fixed constmlt\" by a number congruent to-Q/v2. However, even if this is done, there is still the problem that when Q ~-0, only one table location is examined! To correct these problems, replace steps (3) and (4) of Bell's algorithm with: (3) Initialize A with C, where C is defined below. (4) Increment A by 2Q. For this algorithm, we have a = Q + C, b = Q. We must then choose C so that C ~-QifQ ¢ 0andC ¢-Qif Q ~-0. The algorithm will then search (p + 1)/2 locations if Q ~ 0, and will search all p locations if Q-= 0. The trouble with this algorithm is that it requires testing for Q ~ 0, which means performing an extra division. A seemingly possible way out is to observe that if (p-a)/vb-j, b ¢ 0, then the algorithm searchesj fewer locations before it starts reexamining locations. We can then try to choose C so that we get j to be small, thereby examining nearly half the table before repeating. However, this requires that we make C ~-(j + 1)Q. There does not appear to be any simple algorithm for choosing a C satisfying this congruence for a small j when Q fi 0, and choosing C ¢-Q when Q ~ 0. ~ It seems that the division is necessary. The corrected version of Bell's algorithm still contains a gross inefficiency. For Q ~ 0, it decided that the search is a failure after p tries, instead of the necessary (p q-1)/2 tries. This is easily corrected by changing the criterion for failure. In summary, Bell's algorithm requires a correction which adds an extra division to the initialization procedure. This must be considered in evaluating its efficiency. Bell's table comparing the efficiency of his method with that of Maurer's indicates that this extra initialization cost is justified only if checking a single entry is a relatively time consuming operation.

Zamir Bavel

Papers

The source as a tool in automata

On the Connectivity of the Product of Automata

The left-discrimination sequence of an automaton

Connectivity and separation in automata

On the number of automorphisms of a singly generated automaton