8 research outputs found
Clones and Genoids in Lambda Calculus and First Order Logic
A genoid is a category of two objects such that one is the product of itself
with the other. A genoid may be viewed as an abstract substitution algebra. It
is a remarkable fact that such a simple concept can be applied to present a
unified algebraic approach to lambda calculus and first order logic
Clone Theory: Its Syntax and Semantics, Applications to Universal Algebra, Lambda Calculus and Algebraic Logic
The primary goal of this paper is to present a unified way to transform the
syntax of a logic system into certain initial algebraic structure so that it
can be studied algebraically. The algebraic structures which one may choose for
this purpose are various clones over a full subcategory of a category. We show
that the syntax of equational logic, lambda calculus and first order logic can
be represented as clones or right algebras of clones over the set of positive
integers. The semantics is then represented by structures derived from left
algebras of these clones
Universal Algebra and Mathematical Logic
In this paper, first-order logic is interpreted in the framework of universal
algebra, using the clone theory developed in three previous papers. We first
define the free clone T(L, C) of terms of a first order language L over a set C
of parameters in a standard way. The free right algebra F(L, C) of formulas
over T(L, C) is then generated by atomic formulas. Structures for L over C are
represented as perfect valuations of F(L, C), and theories of L are represented
as filters of F(L). Finally Godel's completeness theorem and first
incompleteness theorem are stated as expected
Clone Theory and Algebraic Logic
The concept of a clone is central to many branches of mathematics, such as
universal algebra, algebraic logic, and lambda calculus. Abstractly a clone is
a category with two objects such that one is a countably infinite power of the
other. Left and right algebras over a clone are covariant and contravariant
functors from the category to that of sets respectively. In this paper we show
that first-order logic can be studied effectively using the notions of right
and left algebras over a clone. It is easy to translate the classical treatment
of logic into our setting and prove all the fundamental theorems of first-order
theory algebraically
A New Approach to Abstract Machines - Introduction to the Theory of Configuration Machines
An abstract machine is a theoretical model designed to perform a rigorous
study of computation. Such a model usually consists of configurations,
instructions, programs, inputs and outputs for the machine. In this paper we
formalize these notions as a very simple algebraic system, called a
configuration machine. If an abstract machine is defined as a configuration
machine consisting of primitive recursive functions then the functions computed
by the machine are always recursive. The theory of configuration machines
provides a useful tool to study universal machines