Model checking is the process of checking whether a given structure is a model of a given logical formula. The concept is general and applies to all kinds of logics and suitable structures. A simple model-checking problem is testing whether a given formula in the propositional logic is satisfied by a given structure. An important class of model checking methods have been developed to algorithmically verify formal systems. This is achieved by verifying if the structure, often derived from a hardware or software design, satisfies a formal specification, typically a temporal logic formula. Pioneering work in the model checking of temporal logic formulae was done by E. M. Clarke and E. A. Emerson in 1981 and by J. P. Queille and J. Sifakis in 1982. Clarke, Emerson, and Sifakis shared the 2007 Turing Award for their work on model checking. Model checking is most often applied to hardware designs. For software, because of undecidability (see Computability theory) the approach cannot be fully algorithmic; typically it may fail to prove or disprove a given property. The structure is usually given as a source code description in an industrial hardware description language or a special-purpose language. Such a program corresponds to a finite state machine (FSM), i.e., a directed graph consisting of nodes (or vertices) and edges. A set of atomic propositions is associated with each node, typically stating which memory elements are one. The nodes represent states of a system, the edges represent possible transitions which may alter the state, while the atomic propositions represent the basic properties that hold at a point of execution. Formally, the problem can be stated as follows: given a desired property, expressed as a temporal logic formula p, and a structure M with initial state s, decide if . If M is finite, as it is in hardware, model checking reduces to a graph search.