This article presents an example of correct circuit design through interactive transformation. Interactive transformation differs from traditional hardware design transformation frameworks in that it focuses on the issue of finding suitable hardware architecture for the specified system and the issue of architecture correctness. The transformation framework divides every transformation in designs into two steps. The first step is to find a proper architecture implementation. Although the framework does not guarantee existence of such an implementation, nor its discovery, it does provide a characterization of architectural implementation so that the question "is this a correct implementation?" can be answered by equational rewriting. The framework allows a correct architecture implementation to be automatically incorporated with control descriptions to obtain a new system description. The significance of this transformation framework lies in the fact that it requires simpler mechanism o...

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Context-Free Grammars (CFGs) are a simple and intuitively appealing formalism for the description of programming languages, but lack the computational power to describe many common language features. Over the past three decades, numerous extensions of the CFG model have been developed. Most of these extensions retain a CFG kernel, and augment it with a distinct facility with greater computational power. However, in all the most powerful CFG extensions, the clarity of the CFG kernel is undermined by the opacity of the more powerful extending facility. An intuitively appealing strategy for CFG extension is grammar adaptability, the principle that declarations in a program effectively modify the context-free grammar of the programming language. An adaptable grammar is equipped with some formal means for modifying its own CFG kernel. Most previous adaptable grammar formalisms have, unfortunately, failed to realize the potential clarity of this concept. In this thesis, a representative samp...

The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way cylindric and polyadic algebras algebraize the first-order predicate logic. In this paper we prove that the lattice of lambda theories is not modular and that the variety generated by the term algebra of a semi-sensible lambda theory is not congruence modular. Another result of the paper is that the Mal'cev condition for congruence modularity is inconsistent with the lambda theory generated by equating all the unsolvable lambda-terms.

. We present a single identity for the variety of all lattices that is much simpler than those previously known to us. We also show that the variety of weakly associative lattices is one-based, and we present a generalized one-based theorem for subvarieties of weakly associative lattices that can be defined with absorption laws. The automated theorem-proving program Otter was used in a substantial way to obtain the results. 1 Introduction Equational identities are, perhaps, the simplest form of sentences expressing many basic properties of algebras. Several familiar classes of algebras, such as semigroups, groups, rings, lattices, and Boolean algebras, are defined by equational identities. Such a class of algebras is known as an equational Supported by the Office of Scientific Computing, U.S. Department of Energy, under Contract W-31-109-Eng-38. y Supported by an operating grant from NSERC of Canada (#A8215). class of algebras or a variety of algebras (for mathematical properti...

this paper, we show that the equational theory of TBAs is one-based. Our methods for finding a single identity for the theory of TBAs are interesting from two distinct points of view. First, from the algebraic, since TBAs enjoy both permutable and distributive congruences, they admit a single ternary polynomial p(x; y; z), the socalled Pixley polynomial [1, p. 405]. We first find such a polynomial p(x; y; z) and use a technique of R. Padmanabhan and R. W. Quackenbush [7] to construct a single identity for the equational theory in question. This is done in Section 2. Second, from the viewpoint of automated reasoning, we use the program Otter to discover new single identities based upon the results of the algebraic view. Actually we obtain here three new identities--- shorter in length than those obtained by the formal algebraic process of Section 2---each characterizing the equational theory of TBAs. The relevant Otter proofs are also included. 2 The Algebraic View

We study the class of theories for which solving unification problems is equivalent to solving systems of linear equations over a semiring. It encompasses important examples like the theories of Abelian monoids, idempotent Abelian monoids, and Abelian groups. This class has been introduced by the authors independently of each other as "commutative theories " (Baader) and "monoidal theories" (Nutt). We show that commutative theories and monoidal theories indeed define the same class (modulo a translation of the signature), and we prove that it is undecidable whether a given theory belongs to it. In the remainder of the paper we investigate combinations of commutative/monoidal theories with other theories. We show that finitary commutative/monoidal theories always satisfy the requirements for applying general methods developed for the combination of unification algorithms for disjoint equational theories. Then we study the adjunction of monoids of homomorphisms to commutative /monoidal t...

The paper considers some of the main trends of the recent development of mathematical fuzzy logic as an important tool in the toolbox of approximate reasoning techniques. Particularly the focus is on fuzzy logics as systems of formal logic constituted by a formalized language, by a semantics, and by a calculus for the derivation of formulas. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon these theoretical considerations. Key words: mathematical fuzzy logic, algebraic semantics, continuous t-norms, left-continuous t-norms, Pavelka-style fuzzy logic, fuzzy set theory, non-monotonic fuzzy reasoning 1

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A partial field P is an algebraic structure that behaves very much like a field except that addition is a partial binary operation, that is, for some a; b 2 P, a + b may not be defined. We develop a theory of matroid representation over partial fields. It is shown that many important classes of matroids arise as the class of matroids representable over a partial field. The matroids representable over a partial field are closed under standard matroid operations such as the taking of minors, duals, direct sums and 2-sums. Homomorphisms of partial fields are defined. It is shown that if ' : P 1! P 2 is a non-trivial partial field homomorphism, then every matroid representable over P 1 is representable over P 2. The connection with Dowling group geometries is examined. It is shown that if G is a nite abelian group, and r> 2, then there exists a partial field over which the rank{r Dowling group geometry is representable if and only if G has at most one element of order 2, that is, if G is a group in which the identity has at most two square roots.