Church's Theory Research Articles

Kripke semantics for higher-order type theory applied to constraint logic programming languages

We define a Kripke semantics for Intuitionistic Higher-Order Logic with constraints formulated within Church's Theory of Types via the addition of a new constraint base type. We then define an executable fragment, hoHH(C), of the logic: a higher-order logic programming language with typed λ-abstraction, implication and universal quantification in goals and constraints, and give a modified model theory for this fragment. Both formal systems are shown sound and complete for their respective semantics.We also solve the impredicativity problem in λProlog semantics, namely how to give a definition of truth without appealing to induction on subformula structure. In the last section we give a simple semantics-based conservative extension proof that the language hoHH(C) satisfies a uniformity property along the lines of [39].

Spatio-temporal dynamics of an encapsulated gas bubble in an ultrasound field

Coupled equations describing the radial and translational dynamics of an encapsulated gas bubble in an ultrasound field are derived by using the Lagrangian formalism. The equations generalize Church's theory by allowing for the translation motion of the bubble and radiation losses due to the compressibility of the surrounding liquid. The expression given by Church for the inner bubble radius corresponding to the unstrained state of the bubble shell is also refined, assuming that the shell can be of arbitrary thickness and impermeable to gas. Comparative linear analysis of the radial equation is carried out relative to Church's theory. It is shown that there are substantial departures from predictions of Church's theory. The proposed model is applied to evaluate radiation forces exerted on encapsulated bubbles and their translational displacements. It is shown that in the range of relatively high frequencies encapsulated bubbles are able to translate more efficiently than free bubbles of the equivalent size.

Completeness and Cut-elimination in the Intuitionistic Theory of Types

In this paper we define a model theory and give a semantic proof of cut-elimination for ICTT, an intuitionistic formulation of Church's theory of types defined by Miller et al. and the basis for the λProlog programming language. Our approach, extending techniques of Takahashi and Andrews and tableaux machinery of Fitting, Smullyan, Nerode and Shore, is to prove a completeness theorem for the cut-free fragment and show semantically that cut is a derived rule. This allows us to generalize a result of Takahashi and Schutte on extending partial truth valuations in impredicative systems. We extend Andrews' notion of Hintikka sets to intuitionistic higher-order logic in a way that also defines tableau-provability for intuitionistic type theory. In addition to giving a completeness theorem without using cut we then show, using cut, how to establish completeness of more conventional term models. These models give a declarative semantics for the logic underlying the λProlog programming language.

Embedding Display Calculi into Logical Frameworks:: Comparing Twelf and Isabelle

Logical frameworks are computer systems which allow a user to formalise mathematics using specially designed languages based upon mathematical logic and Church's theory of types. They can be used to derive programs from logical specifications, thereby guaranteeing the correctness of the resulting programs. They can also be used to formalise rigorous proofs about logical systems. We compare several methods of implementing the display (sequent) calculus δRA for relation algebra in the logical frameworks Isabelle and Twelf. We aim for an implementation enabling us to formalise, within the logical framework, proof-theoretic results such as the cut-elimination theorem for δRA, and any associated increase in proof length. We discuss issues arising from this requirement.

The Liberalization of The Scholastic Theory of Socio-Economic Policy

It has been ten years since the publication of Pope John XXIII's Mater et Magistra. Once a subject of controversy, interest in the encyclical has declined in recent years, preempted, perhaps, by the com? plexity and urgency of other problems facing the Roman Catholic Church. Nevertheless, John's encyclical, a major turning point in the development of scholastic or Roman Catholic socio-economic thought, still raises many questions concerning future developments in the Catho? lic church's theory of economic policy.1 In a thoughtful article concerning the implications of Mater et Magistra, A. F. McKee states that the Roman Catholic church now,

M. H. A. Newman and A. M. Turing. A formal theorem in Church's theory of types. The journal of symbolic logic, vol. 7 (1942), pp. 28–33.

M. H. A. Newman and A. M. Turing. A formal theorem in Church's theory of types. The journal of symbolic logic, vol. 7 (1942), pp. 28–33.

A formal theorem in Church's theory of types

This note is concerned with the logical formalism with types recently introduced by Church [1] (and called (C) in this note) It was shewn in his paper (Theorem 26α) that if Yα stands for(a form of the “axiom of infinity” for the type α), Yα can be proved formally, from Yι and the axioms 1 to 7, for all types α of the forms ι′, ι″, …. For other types the question was left open, but for the purposes of an intrinsic characterisation of the Church type-stratification given by one of us, it is desirable to have the remaining cases cleared up. A formal proof of Yα is now given for all types α containing ι, but the proof uses, in addition to Axioms 1 to 7 and Yι, also Axiom 9 (in connection with Def. 4), and Axiom 10 (in Theorem 9).