tilansiirtymien

Tilansiirtymien (state transitions) describe how a system moves from one state to another in response to inputs or to the passage of time. The term is used in mathematics, computer science, and systems theory to analyze how complex models evolve.

Formally, a discrete system consists of a set of states S and an input alphabet I. A transition function δ determines, for each state s ∈ S and input a ∈ I, the next state δ(s, a) ∈ S. If δ maps to a single state for every pair (s, a) the model is deterministic; if δ(s, a) yields multiple possible next states, it is nondeterministic.

State transitions are commonly depicted with state diagrams, directed graphs in which nodes represent states and labelled edges indicate the input that causes the transition and the resulting state. Beyond automata theory, tilansiirtymien underpin software state machines, digital circuit design, protocol specifications, and model checking in formal verification.

Example: a deterministic finite automaton over the alphabet {a,b} recognizing strings that end with the letter a can be described with two states: q0 (start, non-accepting) and q1 (accepting). Transitions: δ(q0,a)=q1, δ(q0,b)=q0, δ(q1,a)=q1, δ(q1,b)=q0. The input string is accepted when, after processing all symbols, the current state is q1.

Etymology and usage: tilansiirtymien is formed from tila (state) and siirtymä (transition). The form tilansiirtymien is the genitive plural used in Finnish texts to refer to the collection of state transitions. In English, the corresponding term is "state transitions." See also finite automaton, transition function, state machine, state diagram, Markov chain, and dynamical system.