the initial state of X is populated, the X-machine runs to completion, and the outputs are then observed. In general, a machine may deadlock (be blocked), or livelock (never halt), or perform one or more complete computations. For this reason, more recent research has focused on deterministic X-machines, whose behaviour can be controlled and observed more precisely. Example A compiler with a peep-hole optimizer can be thought of as a machine for optimizing program structure. In this Optimizer-machine, the encoding function α takes source code from the input-type Y (the program source) and loads it into the memory-type X (a parse tree). Suppose that the machine has several states, called FindIncrements, FindSubExprs and Completed. The machine starts in the initial state FindIncrements, which is linked to other states via the transitions: FindIncrements →DoIncrement FindIncrements FindIncrements →SkipIncrement FindSubExprs FindSubExprs →DoSubExpr FindSubExprs FindSubExprs →SkipSubExpr Completed The relation DoIncrement maps a parsed subtree corresponding to "x := x + 1" into the optimized subtree "++x". The relation DoSubExpr maps a parse tree containing multiple occurrences of the same expression "x + y ... x + y" into an optimized version with a local variable to store the repeated computation "z := x + y; ... z ... z". These relations are only enabled if X contains the domain values (subtrees) on which they operate. The remaining relations SkipIncrement and SkipSubExpr are nullops (identity relations) enabled in the complementary cases. So, the Optimizer-machine will run to completion, first converting trivial additions into in-place increments (while in the FindIncrements state), then it will move on to the FindSubExprs state and perform a series of common sub-expression removals, after which it will move to the final state Completed. The decoding function β will then map from the memory-type X (the optimized parse-tree) into the output-type Z (optimized machine code). Convention When referring to Eilenberg's original model, "X-machine" is typically written with a lower-case "m", because the sense is "any machine for processing X". When referring to later specific models, the convention is to use a capital "M" as part of the proper name of that variant. 1980s Interest in the X-machine was revived in the late 1980s by Mike Holcombe,<ref name="Hol88a">M. Holcombe (1988) 'X-machines as a basis for dynamic system specification', Software Engineering Journal 3(2), pp. 69-76.</ref> who noticed that the model was ideal for software formal specification purposes, because it cleanly separates control flow from