The Development and Application of a Natural Duality for Ternary Algebras

This paper examines ternary algebras, their practical applications and how to translate between the natural dual and the restricted Priestley dual of these algebras. We begin by determining the term functions of the standard ternary algebra as an application of the Baker-Pixley Theorem. This is followed by an overview of some of the practical applications of ternary algebras. We establish the ability to transform both the natural dual of an arbitrary ternary algebra into its restricted Priestley dual, and the restricted Priestley dual of an arbitrary ternary algebra into its natural dual. The translation process is formalized and two applications are demonstrated. This discovery is prefaced by using existing natural dualities theory to establish an optimal natural duality and the restricted Priestley duality of ternary algebras. 1. Overview This report aims to encompass the objectives described in the initial project description. These outcomes are: • produce an overview of the many applications of ternary algebras, • apply the theory of natural dualities to study ternary algebras, • investigate the extent to which restricted Priestley duality and natural duality can be used in tandem. Almost all the results in this report are presented without proof. The author and her supervisor will eventually write up all the proofs for publication. 2. Introduction In classical propositional logic every proposition is either true or false. That is, classical logic has exactly two truth-values. The algebraic counterpart of classical logic is Boolean algebra. On the other hand, any form of logic that allows for more than two truth-values belongs to the realm of non-classical logic. Examples of algebras arising from non-classical logic are Kleene algebras and ternary algebras. The standard Boolean algebra is given by 2 = 〈{0, 1};∨,∧,¬, 0, 1〉. ∨ 0 1 0 0 1 1 1 1 ∧ 0 1 0 0 0 1 0 1 ¬ 0 1 1 0 Figure 1. The operations of 2 The research of the author was supported by AMSI. The author would like to thank Professor Brian Davey for imparting his knowledge and for his time and encouragement. 1 2 STACEY P. MENDAN The standard Kleene algebra is given by K = 〈{0, d, 1};∨,∧,¬, 0, 1〉, and the standard ternary algebra is given by 3 = 〈{0, d, 1};∨,∧,¬, 0, d, 1〉. ∨ 0 d 1 0 0 d 1 d d d 1 1 1 1 1 ∧ 0 d 1 0 0 0 0 d 0 d d 1 0 d 1 ¬ 0 1 d d 1 0 Figure 2. The operations of K and 3 A comparison of 2 with K reveals that 2 satisfies the Law of the Excluded Middle p ∨ ¬p ≡ 1 while K fails this law as d ∨ ¬d = d. The only difference between the standard Kleene algebra K and the standard ternary algebra 3 is that d is included in the type of 3 as a nullary operation. We conclude this section with some definitions. Definition 2.1. A term is a “meaningful” expression built from • operation symbols: ∨,∧,¬, 0, d, 1, • variable symbols: x1, x2, x3, . . . , • delimiters: (, ), . . . . For example, ((x1 ∨ x2) ∧ (¬(x1 ∨ x2))) ∨ d is a binary term while ¬x1))∧ is not a “meaningful”expression. A Boolean term refers to a term that does not involve the operation symbol d. Every term yields a corresponding term function on 3 and every Boolean term yields a corresponding term function on both 2 and 3. For example, the binary Boolean term ¬(x ∨ y) ∧ ¬x yields the term functions on 2 and 3 represented as 2-valued and 3-valued truth tables in Figure 3. p q ¬(x ∨ y) ∧ ¬x 0 0 1 0 1 0 1 0 0 1 1 0 p q ¬(x ∨ y) ∧ ¬x 0 0 1 0 d d 0 1 0 d 0 d d d d d 1 0 1 0 0 1 d 0 1 1 0 Figure 3. f : 2 → 2 and g : 3 → 3 The term functions on 3 corresponding to Boolean terms play an important role in applications and are referred to as B-term functions. The term function g in Figure 3 is an example of a B-term function of 3. DUALITY AND TERNARY ALGEBRAS 3 Term functions are used in the analysis of circuits. Terms and term functions are relevant tools in several fields including electrical engineering and computer science. 3. Some applications of ternary algebras We consider some of the applications of ternary algebras. However, the reader should be aware that the author did not possess the necessary expertise in electronics to understand the intricacies of these applications. Ternary algebras were initially applied by Goto in 1949 to analyze the indefinite behaviours of relay circuits as well as to synthesize such circuits [12, 13]. Other pioneers to apply ternary algebras include Moisil and Roginskii. See references 17, 18 and 30 in [4]. Since then the potential of ternary algebras has been further recognized. In particular, ternary algebras may be applicable to a circuit containing ambiguity either at the input stage or the output stage. For example, Muller used ternary algebras to study transient phenomena in switching circuits [17]. While Mukaidono demonstrated that ternary algebras can be used to design fail-safe logic circuits by letting d correspond to a failure state [16]. Mukaidono also showed that ternary algebras can correct input failures [15, 16]. He motivates the correction of input failures by highlighting that a fail-safe logic circuit capable of self-correcting as many input failures as possible during normal operation will have the advantages of improved safety and reliability and a decreasing of inactive states. In the case of CMOS circuits, an ambiguous output is represented by the value d [4]. Let us consider a specific example of an application of ternary algebras. It was shown by Yoeli and Rinon that ternary algebras could be utilized to detect static hazards in combinational switching circuits [19]. In particular they use B-term functions of 3. The use of B-term functions is justified as the overall performance (including transient behaviour) of a binary electronic combinational switching circuit composed of AND-, ORand NOT-logic circuits will be adequately described by the corresponding B-term function [19]. To detect whether a circuit contains a static hazard we simply need to find the corresponding Boolean term, denote all changing inputs by d and determine the resulting value of the B-term function. A circuit containing a hazard will have the output value d [4]. Mukaidono went one step further and demonstrated that various kinds of static hazards contained in combinational switching circuits can be detected and identified by B-term functions [14]. In particular, he derived a method which could algebraically detect all logic hazards contained in the circuits. Mukaidono also pointed out that there were some dynamic hazards which were detectable by Bterm functions. Traditionally, asynchronous circuits have been viewed as difficult to understand and design [5]. Hence many of the digital circuits in use today are synchronous. However, asynchronous circuits have the potential benefits of increased speed and reduced power consumption. In addition to these advantages, the development of several asynchronous design methodologies has made the design of much larger and more complex circuits possible. This is one of the limitations of synchronous circuits; building large complex circuits as synchronous circuits can be challenging. 4 STACEY P. MENDAN An important tool used to detect potential timing errors in asynchronous circuits is based on ternary algebras [3, 4, 6]. This tool is known as ternary simulation and was introduced by Eichelberger [11]. The advantages to ternary simulation are: • unlike binary analysis algorithms which are exponential in the number of state variables, ternary simulation is linear in the number of state variables [4], • ternary simulation provides information that even the most accurate circuit simulator cannot [3], • ternary simulation requires only slightly more computational effort than ordinary logic simulation and hence can be used to check large digital systems operating over long input sequences [3]. Bryant applied ternary simulation to a variety of circuits designed in both nMOS and CMOS [3]. Finally, in addition to the above applications of ternary algebras an example of an application outside the realm of electronics and circuit design is the 3-valued attribute exploration created by Burmeister [7]. A more recent application of ternary algebras was by Negeulescu to a framework for modelling interactive systems known as process spaces [4, 18]. 4. The term functions of 3 We can find the term functions of 3 by applying the following theorem from Baker and Pixley [1] (see also [8]). We begin with the concept of a lattice-based finite algebra. Definition 4.1. A finite algebra 〈M ;F 〉 is lattice-based if there exist binary operations ∨, ∧ ∈ F such that 〈M ;∨,∧〉 is a lattice. Theorem 4.2. If L is any lattice-based finite algebra, then f : L → L is a term function of L if and only if f preserves every binary relation r on L such that r is a subalgebra of L. We will apply this theorem to find the term functions of 3.