Martin W. Bunder

Systems of illative combinatory logic complete for first-order propositional and predicate calculus

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators or, in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. The two direct translations turn out to be complete. The paper fulfills the program of Church [1932], [1933] and Curry [1930] to base logic on a consistent system of A-terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent). ?

Linking the Calkin-Wilf and Stern-Brocot trees

Links between the Calkin-Wilf tree and the Stern-Brocot tree are discussed answering the questions: What is thejth vertex in thenth level of the Calkin-Wilf tree? and Where is the vertexrslocated in the Calkin-Wilf tree? A simple mechanism is described for converting the jth vertex in the nth level of the Calkin-Wilf tree into the jth entry in the nth level of the Stern-Brocot tree. We also provide a simple method for evaluating terms in the Hyperbinary sequence thus answering a challenge raised in Quantum in September 1997. We also examine successors and predecessors in both trees.

Completeness of the propositions-as-types interpretation of intuitionistic logic into illative combinatory logic

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative combinatory logic that are sound for first order intuitionistic propositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. In the cited paper we proved completeness of the two direct translations. In the present paper we prove that also the two indirect translations are complete. These proofs are direct whereas in another version, [3], we proved completeness by showing that the two corresponding illative systems are conservative over the two systems for the direct translations. Moreover we shall prove that one of the systems is also complete for predicate calculus with higher type functions. ?

Combinatory abstraction using B, B′ and friends

Abstract In this paper we characterise precisely the sets of terms whose abstractions can be defined using the following partial bases of combinators: { B , B ′, I }, { B , B ′, I , W }, { B , B ′, I , K }, { B , T , I }, { B , T , I , W } and { B , T , I , I }. The reduction axioms for B ′ and T are B ′XYZ Y(XZ) T XYZ YXZ . The first two B ′-bases correspond via type-assignment to two interesting implicational logics. T has the re-ordering property of B ′ but not its bracketing property, and turns out to be strictly stronger than B ′ but strictly weaker than CI whose reduction axiom is CI XY YX .

Locating terms in the Stern-Brocot tree

In this paper we discover an efficient method for answering two related questions involving the Stern-Brocot tree: What is thejth term in thenth level of the tree? and What is the exact position of the fractionrsin the tree?

Implementing the "Fool's model" of combinatory logic

This paper studies ‘Fools models’ of combinatory logic, and relates them to Hindleys ‘D-completeness’ problem. A ‘fools model’ is a family of sets of → formulas, closed under condensed detachment. Alternatively, it is a ‘model’ ofCL in naive set theory. We examine Resolution; and the P-W problem. A sequel shows T→ is D-complete; also, its extensions. We close with an implementation FMO of these ideas.

Some rough consequence logics and their interrelations

This paper considers a number of alternative rough consequence logics which come in a natural way from the logics Lr and LR previously studied by Chakraborty and Banerjee. The systems have been compared to variants of S5, and the logic Triv of Hughes and Cresswell. A comparison has also been made with Lr and LR, and therefore with Jaskowskis discussive logic J, as J is equivalent to LR.

Completeness of two systems of illative combinatory logic for first-order propositional and predicate calculus

Abstract. Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers 4 systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. In a preceding paper, Barendregt, Bunder and Dekkers, 1993, we proved completeness of the two direct translations. In the present paper we prove completeness of the two indirect translations by showing that the corresponding illative systems are conservative over the two systems for the direct translations. In another version, DBB (1997), we shall give a more direct completeness proof. These papers fulfill the program of Church and Curry to base logic on a consistent system of

Security analysis of michael: the IEEE 802.11i message integrity code

\lambda

Proof finding algorithms for implicational logics

-terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent).