transitionsconnected

Transitionsconnected is a term used in the study of state-transition systems and automata theory to describe a property of the transition graph. It refers to a condition in which the set of transitions forms a connected subgraph when adjacency is defined by shared states. Concretely, for transitions e1 = (s1, t1) and e2 = (s2, t2), the system is transitionsconnected if there exists a finite sequence of transitions e1, e3, ..., ek = e2 such that consecutive transitions in the sequence share a common state (either a source or a target). This means any two transitions can be linked by a chain of transitions through intermediate states.

The concept is used in modeling, verification, and design of state machines, workflow diagrams, and process graphs to ensure a cohesive transition structure. It is related to, but distinct from, standard graph connectivity concepts such as the connectivity of the underlying state graph or the strongly connected components of states. Transitionsconnectedness emphasizes the connectivity of transitions themselves rather than solely the reachability among states.

Applications include design guidance for ensuring testability and navigability in models, as well as verification checks that aim to avoid isolated transition components. In practice, checking transitionsconnectedness can aid in identifying gaps in coverage and improving the composition of transition sets in complex systems.

Example: A simple automaton with three states A, B, and C and transitions a: A→B, b: B→C, c: C→A is transitionsconnected, since any pair of transitions can be connected through a sequence of transitions via shared states. See also graph theory, automata theory, state machines, and edge connectivity.