This article continues the development of the basic terminology

Abstract not found

this paper. Let X 1 , X 2 be non empty sets, let Y 1 be a non empty subset of X 1 , let Y 2 be a non empty subset of X 2 , and let x be an element of [:Y 1 , Y 2 :]. Then x 1 is an element of Y 1 . Then x 2 is an element of Y 2 . In the sequel n is a natural number. Let us consider n. The functor Fib(n) yielding a natural number is defined by the condition (Def. 1). (Def. 1) There exists a function f 1 from N into [: N, N :] such that (i) Fib(n) = f 1 (n) 1 , (ii) f 1 (0) = ##0, 1##, and (iii) for every natural number n and for every element x of [: N, N :] such that x = f 1 (n) holds f 1 (n +1) = ##x 2 , x 1 + x 2 ##

this paper. 1. Preliminaries Let s be a state of SCM FSA and let P be an initial finite partial state of SCM FSA . We say that

Summary. We define and prove some simple facts on Cartesian categories and its duals co-Cartesian categories. The Cartesian category is defined as a category with the fixed terminal object, the fixed projections, and the binary products. Category C has finite products if and only if C has a terminal object and for every pair a,b of objects of C the product a × b exists. We say that a category C has a finite product if every finite family of objects of C has a product. Our work is based on ideas of [10], where the algebraic properties of the proof theory are investigated. The terminal object of a Cartesian category C is denoted by 1C. The binary product of a and b is written as a × b. The projections of the product are written as pr1(a,b) and as pr2(a,b). We define the products f × g of arrows f: a → a ′ and g: b → b ′ as < f · pr1,g · pr2>: a × b → a ′ × b ′.

this paper. 1. Preliminaries We use the following convention: k, m are natural numbers, x, X are sets, and N is a set with non empty elements

this paper. For simplicity, we adopt the following convention: m denotes a natural number, I denotes a macro instruction, s, s 1 , s 2 denote states of SCMFSA , a denotes an integer location, and

Summary. The aim of this work is to provide a bridge between the theory of contextfree grammars developed in [10], [6] and universally free manysorted algebras([14]. The third scheme proved in the article allows to prove that two homomorphisms equal on the set of free generators are equal. The first scheme is a slight modification of the scheme in [6] and the second is rather technical, but since it was useful for me, perhaps it might be useful for somebody else. The concept of flattening of a many sorted function F between two manysorted sets A and B (with common set of indices I) is introduced for A with mutually disjoint components (pairwise disjoint function – the concept introduced in [13]). This is a function on the union of A, that is equal to F on every component of A. A trivial many sorted algebra over a signature S is defined with sorts being singletons of corresponding sort symbols. It has mutually disjoint sorts.

this paper. For simplicity, we adopt the following rules: a, b denote integer locations, f denotes a finite sequence location, i 1 , i 2 , i 3 denote instruction-locations of SCMFSA , T denotes an instruction type of SCMFSA , and k denotes a natural number. Next we state two propositions: (1) For every function f and for all sets a, A, b, B, c, C such that a 6= b and a 6= c holds (f+\Delta(a7\Gamma! . A)+\Delta(b7\Gamma! . B)+\Delta(c7\Gamma! . C))(a) = A: (2) For all sets a, b holds hai +\Delta (1; b) = hbi: Let l 1 , l 2 be integer locations and let a, b be integers. Then [l 1 7\Gamma! a; l 2 7\Gamma! b] is a finite partial state of SCMFSA . The following propositions are true: (3) a = 2 the instruction locations of SCMFSA . (4) f = 2 the instruction locations of SCMFSA . (5) Data-LocSCMFSA 6= the instruction locations of SCMFSA . (6) Data

This paper is based on a previous work of the first author [15] in which a mathematical model of the computer has been presented. The model deals with random access memory, such as RASP of C. C. Elgot and A. Robinson [13], however, it allows for a more realistic modeling of real computers. This new model of computers has been named by the author (Y. Nakamura, [15]) Architecture Model for Instructions (AMI). It is more developed than previous models, both in the description of hardware (e.g., the concept of the program counter, the structure of memory) as well as in the description of instructions (instruction codes, addresses). The structure of AMI over an arbitrary collection of mathematical domains N consists of: - a non-empty set of objects, - the instruction counter, - a non-empty set of objects called instruction locations, - a non-empty set of instruction codes, - an instruction code for halting, - a set of instructions that are ordered pairs with the first element being an instruction code and the second a finite sequence in which members are either objects of the AMI or elements of one of the domains included in N, - a function that assigns to every object of AMI its kind that is either an instruction