Polynomial-time Martin-Lof type theory

Fragments of extensional Martin-Lof type theory without universes, $ML\sb0,$ are introduced that conservatively extend S. A. Cook and A. Urquhart's $IPV\sp\omega.$ A model for these restricted theories is obtained by interpretation in Feferman's theory APP of operators, a natural model of which is the class of partial recursive functions. In conclusion, an example in group theory is considered.$IPV\sp\omega$ is a higher-order arithmetic that conservatively extends Cook's equational system PV. PV formalizes the notion of feasibly (i.e., polynomial-time verifiably) constructive proof. $IPV\sp\omega$ in turn captures a basic notion of polynomial-time computability for functionals of finite (linear) type as well. However, while $IPV\sp\omega$ formalizes feasibly constructive first-order number theory, it does not naturally apply to higher mathematics, e.g., in general, it lacks the relation of equality among objects of the same type. This motivates the study of extensions of $IPV\sp\omega$ in more expressive formalisms.$ML\sb0$ is a predicative intuitionistic type theory based on the propositions-as-types paradigm which identifies a proposition with the set of its proofs. The fragments of $ML\sb0$ obtained extend the above notions for the polymorphic Martin-Lof type structure and exhibit functionals encoding proofs of theorems. Because of their richness, they are well-suited for the development of feasibly constructive mathematics, particularly in the synthesis of feasible algorithms from proofs.U of I OnlyETDs are only available to UIUC Users without author permissio