Any standard proof system such as natural deduction contains infinitely many statements whose simplest proofs are of length at least exponential relative to the size of the statement.
(what feasibility constraints limit in the statement)
The number of steps or amount of logical work needed to demonstrate that something is true in mathematics.
Size of the statement(in logic)
How long or complex a statement is, usually measured by counting the number of symbols or words needed to write it out.
proof system(Propositional proof complexity)
A definition of the derivability symbol ⊢_P which characterizes what it means for a formula φ to be derivable from a given set of axioms and rules; equivalently, a mapping P: {0,1}* → VALID whose domain consists of all binary strings and whose range is the class of all valid formulas
[53] And from this it follows that such orders cannot be reached from below by a sorties-like sequence of feasible orders of growth \(O(1) \prec O(n) \prec O(n^2) \prec O(n^3) \ldots\) When analyzed according to the Cobham-Edmonds thesis, it hence appears that the ‘naive’ notion of feasible computability does not suffer from the sort of instability which Dummett takes to plague the notion of a feasibly constructible number. These observations point to another theme within the writings of some st