Structural rule A famous structural rule is known as cut. Considerable effort is spent by proof theorists in showing that cut rules are superfluous in various logics. More precisely, what is shown is that cut is only (in a sense) a tool for abbreviating proofs, and does not add to the theorems that can be proved. The successful 'removal' of cut rules, known as "cut elimination", is directly related to the philosophy of "computation as normalization" (see Curry–Howard correspondence); it often gives a good indication of the complexity of deciding a given logic. Structural rule In proof theory, a structural rule is
Rule 110 resulted in a legal affair based on a non-disclosure agreement with Wolfram Research. Wolfram Research blocked publication of Cook's proof for several years. Among the 88 possible unique elementary cellular automata, Rule 110 is the only one for which Turing completeness has been proven, although proofs for several similar rules should follow as simple corollaries (e.g. Rule 124, which is the horizontal reflection of Rule 110). Rule 110 is arguably the simplest known Turing complete system. Rule 110, like the Game of Life, exhibits what Wolfram calls "Class 4 behavior", which is neither completely stable nor completely chaotic. Localized structures