Functional completeness and canonical forms in many-valued logics1

1962 ◽  
Vol 27 (4) ◽  
pp. 409-422 ◽  
Author(s):  
William H. Jobe
Keyword(s):  
Functional Completeness ◽  
Canonical Forms ◽  
Infinite Class ◽  
Truth Values ◽  
Binary Operations ◽  
Finite Set

This paper examines the questions of functional completeness and canonical completeness in many-valued logics, offering proofs for several theorems on these topics.A skeletal description of the domain for these theorems is as follows. We are concerned with a proper logic L, containing a denumerably infinite class of propositional symbols, P, Q, R, …, a finite set of unary operations, U1, U2,…, Ub, and a finite set of binary operations, B1, B2, …, Bc. Well-formed formulas in L are recursively defined by the conventional set of rules. With L there is associated an integer, M ≧ 2, and the integers m, where (1 ≦m≦M), are the truth values of L.

scholarly journals Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Song He ◽  
Zhenjie Li ◽  
Prashanth Raman ◽  
Chi Zhang
Keyword(s):  
Large Class ◽  
Configuration Space ◽  
Moduli Space ◽  
Cluster Algebras ◽  
Configuration Spaces ◽  
Minkowski Sum ◽  
Canonical Forms ◽  
Infinite Class ◽  
Special Cases ◽  
Generalized Associahedra

Abstract Stringy canonical forms are a class of integrals that provide α′-deformations of the canonical form of any polytopes. For generalized associahedra of finite-type cluster algebras, there exist completely rigid stringy integrals, whose configuration spaces are the so-called binary geometries, and for classical types are associated with (generalized) scattering of particles and strings. In this paper, we propose a large class of rigid stringy canonical forms for another class of polytopes, generalized permutohedra, which also include associahedra and cyclohedra as special cases (type An and Bn generalized associahedra). Remarkably, we find that the configuration spaces of such integrals are also binary geometries, which were suspected to exist for generalized associahedra only. For any generalized permutohedron that can be written as Minkowski sum of coordinate simplices, we show that its rigid stringy integral factorizes into products of lower integrals for massless poles at finite α′, and the configuration space is binary although the u equations take a more general form than those “perfect” ones for cluster cases. Moreover, we provide an infinite class of examples obtained by degenerations of type An and Bn integrals, which have perfect u equations as well. Our results provide yet another family of generalizations of the usual string integral and moduli space, whose physical interpretations remain to be explored.

On completely recursively enumerable classes and their key arrays

1956 ◽  
Vol 21 (3) ◽  
pp. 304-308 ◽  
Author(s):  
H. G. Rice
Keyword(s):  
Decision Procedure ◽  
Cardinal Number ◽  
Maximum Amount ◽  
Infinite Class ◽  
Finite Sets ◽  
Amount Of Information ◽  
Recursively Enumerable ◽  
Finite Set ◽  
Partial Recursive Functions ◽  
Set Up

The two results of this paper (a theorem and an example) are applications of a device described in section 1. Our notation is that of [4], with which we assume familiarity. It may be worth while to mention in particular the function Φ(n, x) which recursively enumerates the partial recursive functions of one variable, the Cantor enumerating functions J(x, y), K(x), L(x), and the classes F and Q of r.e. (recursively enumerable) and finite sets respectively.It is possible to “give” a finite set in a way which conveys the maximum amount of information; this may be called “giving explicitly”, and it requires that in addition to an effective enumeration or decision procedure for the set we give its cardinal number. It is sometimes desired to enumerate effectively an infinite class of finite sets, each given explicitly (e.g., [4] p. 360, or Dekker [1] p. 497), and we suggest here a device for doing this.We set up an effective one-to-one correspondence between the finite sets of non-negative integers and these integers themselves: the integer , corresponds to the set αi, = {a1, a2, …, an} and inversely. α0 is the empty set. Clearly i can be effectively computed from the elements of αi and its cardinal number.

The M-valued calculus of non-contradiction

1953 ◽  
Vol 18 (3) ◽  
pp. 237-241
Author(s):  
Alan Rose
Keyword(s):  
Functional Completeness ◽  
Truth Values ◽  
Alternative Method ◽  
Truth Value ◽  
Set Up ◽  
Image Position ◽  
Infinite Set

The 2-valued calculus of non-contradiction of Dexter has been extended to 3-valued logic. The methods used were, however, too complicated to be capable of generalisation to m-valued logics. The object of the present paper is to give an alternative method of generalising Dexter's work to m-valued logics with one designated truth-value. The rule of procedure is generalised in the same way as before, but the deductive completeness of the system is proved by applying results of Rosser and Turquette. The system has an infinite set of primitive functions, written n(P1, P2, …, Pr) (r = 1,2, …). With the notation of Post, n(P1, P2, …, Pr) has the same truth-value as ~(P1 & P2 & … & Pr). Thus n(P) is Post's primitive ~P, and we can define & byWe use n2(P1, P2, …, Pr) as an abbreviation for n(n(P1, P2, …, Pr)); similarly for higher powers of n. But if we set up the 1-1 correspondence of truth-values i ↔ m−i+1, then & corresponds to ∨ and ~m−1 corresponds to ~. Hence the functional completeness of our system follows from a theorem of Post.We define the functions N(P), N(P, Q) byThus the truth-value of N(P) is undesignated if and only if the truth-value of P is designated, and the truth-value of N(P, Q) is undesignated if and only if the truth-values of P and Q are both designated.

A remark on functional completeness of binary expansions of Kleene’s strong 3-valued logic

2020 ◽  
Author(s):  
Gemma Robles ◽  
José M Méndez
Keyword(s):  
Classical Result ◽  
General Theorem ◽  
Functional Completeness ◽  
Truth Values ◽  
Element Set

Abstract A classical result by Słupecki states that a logic L is functionally complete for the 3-element set of truth-values THREE if, in addition to functionally including Łukasiewicz’s 3-valued logic Ł3, what he names the ‘$T$-function’ is definable in L. By leaning upon this classical result, we prove a general theorem for defining binary expansions (i.e. expansions with a binary connective) of Kleene’s strong logic that are functionally complete for THREE.

A theorem concerning the composition of functions of several variables ranging over a finite set

1960 ◽  
Vol 25 (3) ◽  
pp. 203-208 ◽  
Author(s):  
Arto Salomaa
Keyword(s):  
Natural Numbers ◽  
Number Range ◽  
Truth Values ◽  
Several Variables ◽  
Finite Set ◽  
Composition Of Functions

Consider functions whose variables, finite in number, range over a fixed finite set N and whose values are elements of N. The elements of N are denoted simply by the natural numbers 1,2, …, n. There are nnm distinct m-place functions. If N is chosen to be the set of n truth-values then the functions considered are obviously truth-functions in n-valued logic.

Are Contexts Semantic Determinants?

1980 ◽  
Vol 6 ◽  
pp. 161-183 ◽  
Author(s):  
Philip P. Hanson
Keyword(s):  
Structural Feature ◽  
Truth Condition ◽  
Truth Conditions ◽  
Truth Values ◽  
The World ◽  
Finite Set

By a ‘semantic determinant’ I will mean“…a structural feature of the world necessary for the determination of truth and falsity …,” or briefly, “Semantic determinants are things that determine valuations.” (Thomason (1972), pp. 301, 302),where such determination is functionaland where valuations are themselves functions from sentences into truth values. A familar example is the ‘domain of individuals’ relative to which truth conditions are given for sentences containing “all,” “some,” and related expressions, where these can be construed as ‘quantifiers.’ Thus, e.g., “Some men are albinos” is true just in case at least one individual in the domain which is a man is also an albino. For this truth condition to be making a definite claim we must suppose that the domain is adefinite(though not necessarily finite) set of individualsfixed in advance.

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