Arithmetic based on the Church numerals in illative combinatory logic

Studia Logica ◽  
1988 ◽  
Vol 47 (2) ◽  
pp. 129-143
Author(s):  
M. W. Bunder
Keyword(s):  
Combinatory Logic ◽  
The Church

Strong reduction and normal form in combinatory logic

1967 ◽  
Vol 32 (2) ◽  
pp. 213-223 ◽  
Author(s):  
Bruce Lercher
Keyword(s):  
Normal Form ◽  
Combinatory Logic ◽  
Strong Reduction ◽  
The Church

The notion of strong reduction is introduced in Curry and Feys' book Combinatory logic [1] as an analogue, in the theory of combinatore, to reduction (more exactly, βη-reduction) in the theory of λ-conversion. The existence of an analogue and its possible importance are suggested by an equivalence between the theory of combinatore and λ-conversion, and the Church-Rosser theorem in λ-conversion. This theorem implies that if a formula X is convertible to a formula X* which cannot be further reduced—is irreducible, or in normal form—then X is convertible to X* by a reduction alone. Moreover, the reduction may be performed in a certain prescribed order.

The Church-Rosser property in symmetric combinatory logic

2005 ◽  
Vol 70 (2) ◽  
pp. 536-556 ◽  
Author(s):  
Katalin Bimbó
Keyword(s):  
Fixed Point ◽  
Place Identity ◽  
Combinatory Logic ◽  
Left And Right ◽  
The One ◽  
The Church

AbstractSymmetic combinatory logic with the symmetric analogue of a combinatorially complete base (in the form of symmetric λ-calculus) is known to lack the Church-Rosser property. We prove a much stronger theorem that no symmetric combinatory logic that contains at least two proper symmetric combinatory has the Church-Rosser property. Although the statement of the result looks similar to an earlier one concerning dual combinatory logic, the proof is different because symmetric combinators may form redexes in both left and right associated terms. Perhaps surprisingly, we are also able to show that certain symmetric combinatory logics that include just one particular constant are not confluent. This result (beyond other differences) clearly sets apart symmetric combinatory logic from dual combinatory logic, since all dual combinatory systems with a single combinator or a single dual combinator are Church-Rosser. Lastly, we prove that a symmetric combinatory logic that contains the fixed point and the one-place identity combinator has the Church-Rosser property.

A Formal Model of Universal Algorithmic Assembly and Molecular Computation

2010 ◽  
Vol 2 (3) ◽  
pp. 55-68 ◽  
Author(s):  
Bruce MacLennan
Keyword(s):  
Formal Model ◽  
Combinatory Logic ◽  
Network Graph ◽  
Molecular Computation ◽  
Supramolecular Networks ◽  
Autonomous Molecular Computation ◽  
And Control ◽  
The Church ◽  
Theoretical Foundations ◽  
Simple Network

In this paper, the author describes a systematic and general approach to nanostructure synthesis and control through autonomous molecular combinatory computing. Combinatory computing is based on simple network (graph) substitution operations, deriving from combinatory logic (Curry, Feys, & Craig, 1958), which are sufficient for any computation. When these operations are implemented by autonomous molecular processes, they provide a means for computing within supramolecular networks, which may be used to assemble these networks or control their behavior. Further, the Church-Rosser Theorem (Curry, Feys, & Craig, 1958) proves that substitutions may be performed in any order or in parallel without affecting the computational result; this is a very advantageous property for autonomous molecular computation. In addition to the theoretical foundations of molecular combinatory computing, the author discusses possible molecular implementations as well as accomplishments in the (simulated) synthesis of membranes, channels, nanotubes, and other nanostructures.

The Church-Rosser property in dual combinatory logic

2003 ◽  
Vol 68 (1) ◽  
pp. 132-152 ◽  
Author(s):  
Katalin Bimbó
Keyword(s):  
General Theorem ◽  
Combinatory Logic ◽  
Unique Decomposition ◽  
Per Se ◽  
Structurally Free Logics ◽  
The Church

AbstractDual combinators emerge from the aim of assigning formulas containing ← as types to combinators. This paper investigates formally some of the properties of combinatory systems that include both combinators and dual combinators. Although the addition of dual combinators to a combinatory system does not affect the unique decomposition of terms, it turns out that some terms might be redexes in two ways (with a combinator as its head, and with a dual combinator as its head). We prove a general theorem stating that no dual combinatory system possesses the Church-Rosser property. Although the lack of confluence might be problematic in some cases, it is not a problem per se. In particular, we show that no damage is inflicted upon the structurally free logics, the system in which dual combinators first appeared.

Spiritual Healing, Sick Kids and the Law: Inequities in the American Healthcare System

2003 ◽  
Vol 29 (2-3) ◽  
pp. 269-299
Author(s):  
Janna C. Merrick
Keyword(s):  
Medical Science ◽  
Spiritual Healing ◽  
Best Interests ◽  
Legal Rights ◽  
High Tech ◽  
Divine Love ◽  
Religious Minority ◽  
Human Need ◽  
East End ◽  
The Church

Main Street in Sarasota, Florida. A high-tech medical arts building rises from the east end, the county's historic three-story courthouse is two blocks to the west and sandwiched in between is the First Church of Christ, Scientist. A verse inscribed on the wall behind the pulpit of the church reads: “Divine Love Always Has Met and Always Will Meet Every Human Need.” This is the church where William and Christine Hermanson worshipped. It is just a few steps away from the courthouse where they were convicted of child abuse and third-degree murder for failing to provide conventional medical care for their seven-year-old daughter.This Article is about the intersection of “divine love” and “the best interests of the child.” It is about a pluralistic society where the dominant culture reveres medical science, but where a religious minority shuns and perhaps fears that same medical science. It is also about the struggle among different religious interests to define the legal rights of the citizenry.

Ideas and Leaders in Contentious Politics: One Parish Priest in El Salvador's Popular Movement

2014 ◽  
Vol 38 (01) ◽  
pp. 76-101
Author(s):  
PETER M. SANCHEZ
Keyword(s):  
Collective Action ◽  
El Salvador ◽  
Liberation Theology ◽  
Contentious Politics ◽  
Religious Leaders ◽  
Political Movement ◽  
Parish Priest ◽  
Historical Moment ◽  
Resource Mobilisation ◽  
The Church

AbstractThis paper examines the actions of one Salvadorean priest – Padre David Rodríguez – in one parish – Tecoluca – to underscore the importance of religious leadership in the rise of El Salvador's contentious political movement that began in the early 1970s, when the guerrilla organisations were only just beginning to develop. Catholic leaders became engaged in promoting contentious politics, however, only after the Church had experienced an ideological conversion, commonly referred to as liberation theology. A focus on one priest, in one parish, allows for generalisation, since scores of priests, nuns and lay workers in El Salvador followed the same injustice frame and tactics that generated extensive political mobilisation throughout the country. While structural conditions, collective action and resource mobilisation are undoubtedly necessary, the case of religious leaders in El Salvador suggests that ideas and leadership are of vital importance for the rise of contentious politics at a particular historical moment.

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