Relay-contact scheme

relay-contact network, relay-contact circuit

A mathematical model of electrical devices consisting of contacts and intermediate relays that function at discrete moments of time. The relay-contact circuit is one of the first classes of control systems (cf. Control system) considered from the mathematical point of view, as well as one of the first versions of the concept of a finite automaton (cf. Automaton, finite). Relay-contact circuits were first described in 1938–1944 (see [1]–[3]).

Mathematically, a relay-contact circuit is a finite graph in which all edges and some distinguished vertices are ascribed symbols of alphabets

$$ x = \{ x _ {1} , \overline{x} _ {1} \dots x _ {n} , \overline{x} _ {n} \} ,\ \ Y = \{ Y _ {1} \dots Y _ {m} \} , $$

$$ y = \{ y _ {1} , \overline{y} _ {1} \dots y _ {m} , \overline{y} _ {m} \} ,\ a = \{ a _ {+} , a _ {-} , a _ {1} \dots a _ {s} \} $$

in the following way. Each distinguished vertex of the graph (these vertices are called poles) is assigned a symbol from the alphabet $ a $ such that to different poles are assigned different symbols. The set of edges of the graph is divided into three non-intersecting subsets $ R, K _ {1} , K _ {2} $. The edges from each of these subsets are ascribed symbols from $ Y $, $ x $ and $ y $ in the following way: Each edge from $ R $ is assigned a symbol from $ Y $, all edges from $ R $ being assigned different symbols and all symbols from $ Y $ being assigned to edges from $ R $. With each edge of the set $ K _ {1} $ ($ K _ {2} $, respectively) is assigned a symbol from $ x $ ($ y $, respectively) such that each symbol may be assigned to several edges and some symbols may not be assigned to any edge. If the set $ Y $ is empty, then the set $ K _ {2} $ is also empty. In this case (if only $ K _ {1} $ is non-empty) the relay-contact circuit is a contact scheme. Two relay-contact circuits are called isomorphic if their graphs are isomorphic and if corresponding edges and poles are assigned identical symbols.

The pole $ a _ {+} $ is called the input pole and the poles $ a _ {1} \dots a _ {s} $ are called output poles. The pole $ a _ {-} $ may also sometimes be an output pole. The edges to which are assigned symbols from $ x $ are known as contacts of basic relays (or basic contacts); the set of all edges marked by the symbols $ Y _ {i} , y _ {i} , \overline{y}\; _ {i} $ is called the $ i $-th intermediate relay, $ i = 1 \dots m $; edges marked by symbols from $ y $ are called contacts of intermediate relays; edges marked by symbols from $ Y $ are called windings. A sequence of contacts and windings between several vertices of a relay-contact circuit corresponding to a simple chain of the graph is called a chain.

A relay-contact circuit functions at discrete moments of time $ 1 \dots t \dots $ and the functioning is considered in terms of conductivities that at each moment are (in the model under consideration) Boolean functions. The conductivity of the contacts $ x _ {j} , \overline{x} _ {j} $, $ j = 1 \dots n $, at any moment $ t $ is equal to the value of the corresponding variable ( $ x _ {j} $ or $ \overline{x} _ {j} $). The conductivity of a contact $ y _ {i} $, $ i = 1 \dots m $, of an intermediate relay at the moment 1 is equal to zero, the conductivity of the contact $ \overline{y} _ {i} $ at the moment 1 being equal to one; the conductivity of a winding is always equal to one. Every winding at a moment $ t $ can be in various states. The state of the winding at a moment $ t $ (in the two-valued model in question, 1 and 0 are "excited" and "non-excited" ) can be defined in various ways. For example, a winding $ Y _ {i} $ is in state 1 at the moment $ t $ if and only if either: a) there is a chain $ \sigma $ between the poles $ a _ {+} $ and $ a _ {-} $ that passes through $ Y _ {i} $ and that has conductivity 1 at the moment $ t $; or b) condition a) is fulfilled but there is no chain consisting only of contacts having conductivity 1 at the moment $ t $, and joining two vertices of the chain $ \sigma $ situated on different sides of $ Y _ {i} $.

The conductivity of an intermediate contact $ y _ {i} $ $ ( \overline{y} _ {i} ) $ at a moment $ t $ depends on the state of the winding $ Y _ {i} $ at the moment $ t- 1 $; more precisely, the conductivity of $ y _ {i} $ coincides with it and the conductivity of $ \overline{y} _ {i} $ is opposite to it. The conductivity of a chain of a relay-contact circuit between two poles at a moment $ t $ is equal to the conjunction of the conductivities at the moment $ t $ formed by its contacts and windings. Thus, the state of the windings at a moment $ t $ is, generally speaking, dependent on the sequence of sets of values of the variables $ x _ {1} \dots x _ {n} $ at the preceding moments of time. If at successive moments of time the variables $ x _ {1} \dots x _ {n} $ are supplied with sequences of choices of values between a pair of poles $ a _ {+} $ and $ a _ {k} $, $ k = 1 \dots s $, and in a particular case also between $ a _ {+} $ and $ a _ {-} $, a finite-state sequential function is defined. Isomorphic relay-contact circuits realize the same finite-state sequential function. If the relay-contact circuit is such that by supplying the variables $ x _ {1} \dots x _ {n} $ with the same set of values $ ( \sigma _ {1} \dots \sigma _ {n} ) $, starting from a specific moment $ t $, the windings of the intermediate relays do not change their state (and this for each set of values of the variables $ x _ {1} \dots x _ {n} $), then the state of the windings is said to be "steady" , and the relay-contact circuit realizes Boolean functions. If stabilization of the windings from a moment $ t $ starts immediately for all choices of values of the variables $ x _ {1} \dots x _ {n} $, then the relay-contact circuit is called a one-cycle; if stabilization starts at a moment $ t + \tau $, then the relay-contact scheme is called a $ \tau $-cycle.

The complexity of a relay-contact circuit is defined as the sum of the weights (or of the complexity indices) of all basic contacts, intermediate contacts and windings of the circuit. An asymptotic expression for the complexity of the simplest relay-contact circuit realizing the most complexly-realizable Boolean function of $ n $ variables possible takes the form $ \rho 2 ^ {n} /n $, where $ \rho $ is a constant depending on the chosen manner of functioning, the topology of the circuit and the complexity of the relay contacts and windings.

References

[1]	C. Shannon, "A symbolic analysis of relay and switching circuits" AIEE Trans. , 57 (1938) pp. 713–723
[2]	M.A. Gavrilov, "Relaisschalttechnik" , Deutsch. Verlag Wissenschaft. (1953) (Translated from Russian)
[3]	V.I. Shestakov, "On a logical calculus applicable to the theory of relay-contact circuits" Uchen. Zap. Moskov. Gosudarstv. Univ. Mat. , 73 : 5 (1944) pp. 45–48 (In Russian)
[4]	O.B. Lupanov, "Complexity of relay-contact circuits realization by functions of the algebra of logic" Probl. Kibernetiki , 11 (1964) pp. 25–47 (In Russian)