We give resource bounded versions of the Completeness Theorem forpropositional and predicate logic. For example, it is well known thatevery computable consistent propositional theory has a computablecomplete consistent extension. We show that, when length is measuredrelative to the binary representation of natural numbers and formulas,every polynomial time decidable propositional theory has anexponential time (EXPTIME) complete consistent extensionwhereas there is a nondeterministic polynomial time (NP)decidable theory which has no polynomial time complete consistentextension when length is measured relative to the binaryrepresentation of natural numbers and formulas. It is well known thata propositional theory is axiomatizable (respectively decidable) ifand only if it may be represented as the set of infinite paths througha computable tree (respectively a computable tree with no deadends). We show that any polynomial time decidable theory may berepresented as the set of paths through a polynomial time decidabletree. On the other hand, the statement that every polynomial timedecidable tree represents the set of complete consistent extensions ofsome theory which is polynomial time decidable, relative to the tallyrepresentation of natural numbers and formulas, is equivalent toP=NP. For predicate logic, we develop a complexity theoreticversion of the Henkin construction to prove a complexity theoreticversion of the Completeness Theorem. Our results imply that that anypolynomial space decidable theory Δ possesses a polynomialspace computable model which is exponential space decidable and thusΔ has an exponential space complete consistent extension.Similar results are obtained for other notions of complexity.
"Complexity, decidability and completeness." J. Symbolic Logic 71 (2) 399 - 424, June 2006. https://doi.org/10.2178/jsl/1146620150