Uniqueness Of Normal Forms Research Articles

Complementary Proof Nets for Classical Logic

A complementary system for a given logic is a proof system whose theorems are exactly the formulas that are not valid according to the logic in question. This article is a contribution to the complementary proof theory of classical propositional logic. In particular, we present a complementary proof-net system, textsf{CPN}, that is sound and complete with respect to the set of all classically invalid (one-side) sequents. We also show that cut elimination in textsf{CPN} enjoys strong normalization along with strong confluence (and, hence, uniqueness of normal forms).

A model for analyzing phenomena in multicellular organisms with multivariable polynomials: Polynomial life

Most of life maintains itself through turnover, namely cell proliferation, movement and elimination. Hydra’s cells, for example, disappear continuously from the ends of tentacles, but these cells are replenished by cell proliferation within the body. Inspired by such a biological fact, and together with various operations of polynomials, I here propose polynomial-life model toward analysis of some phenomena in multicellular organisms. Polynomial life consists of multicells that are expressed as multivariable polynomials. A cell is expressed as a term of polynomial, in which point [Formula: see text] is described as a term [Formula: see text] and the condition is described as its coefficient. Starting with a single term and following reductions by set of polynomials, I simulate the development from a cell to a multicell. In order to confirm uniqueness of the eventual multicell-pattern, Gröbner base can be used, which has been conventionally used to ensure uniqueness of normal form in the mathematical context. In this framework, I present various patterns through the polynomial-life model and discuss patterns maintained through turnover. Cell elimination seems to play an important role in turnover, which may shed some light on cancer or regenerative medicine.

New Undecidability Results for Properties of Term Rewrite Systems

This paper is on several basic properties of term rewrite systems: reachability, joinability, uniqueness of normal forms, unique normalization, confluence, and existence of normal forms, for subclasses of rewrite systems defined by syntactic restrictions on variables. All these properties are known to be undecidable for the general class and decidable for ground (variable-free) systems. Recently, there has been impressive progress on efficient algorithms or decidability results for many of these properties. The aim of this paper is to present new results and organize existing ones to clarify further the boundary between decidability and undecidability for these properties. Another goal is to spur research towards a complete classification of these properties for subclasses defined by syntactic restrictions on variables. The proofs of the presented results may be intrinsically interesting as well due to their economy, which is partly based on improved reductions between some of the properties.

Sufficiency Conditions for Bokut' Normal Forms

We provide sufficient conditions for the existence and uniqueness of normal forms of sequences of HNN extensions defined by Bokut′. Furthermore, we show that under an assumption, which holds for various applications, such normal forms always exist (but might not be unique). The conditions are amenable to be used in automatic theorem provers. We discuss also how to obtain a Gröbner–Shirshov basis from the rewrite rules of Bokut′ normal forms under certain assumptions. Finally, we provide an application drawn from a paper of Aanderaa and Cohen to illustrate the sufficiency conditions.

A polynomial algorithm for uniqueness of normal forms of linear shallow term rewrite systems

Term rewrite systems are useful in many areas of computer science. Two especially important areas are decision procedures for the word problem of some algebraic systems and rule-based programming. One of the most studied properties of rewrite systems is confluence, and one of the primary benefits of having a confluent rewrite system is that the system also has uniqueness of normal forms. However, uniqueness of normal forms is an interesting property in its own right and well studied. Also, confluence can be too strong a requirement for some applications. In this paper, we study the decidability of uniqueness of normal forms. Uniqueness of normal forms is decidable for ground rewrite systems, but is undecidable in general. This paper shows that the uniqueness of normal forms problem is decidable for the class of linear shallow term rewrite systems, and gives a decision procedure that is polynomial as long as the arities of the function symbols are bounded or the signature is fixed.

FURTHER REDUCTION OF NORMAL FORMS AND UNIQUE NORMAL FORMS OF SMOOTH MAPS

Further reduction for classical normal forms of smooth maps is considered in this paper. Firstly, based on the idea of computation of simplest normal forms for vector fields [Yu & Yuan, 2003], we compute the transformed map of a given smooth map under a near identity formal transformation, and give a recursive formula for the homogeneous terms of the transformed map, which is a powerful tool for further reduction of classical normal forms of smooth maps. Secondly, by using the recursive formula, the idea of [Chen & Della Dora, 1999] for further reduction of normal forms for maps and the method introduced by Kokubu et al. [1996] for further reduction of normal forms of vector fields, we develop the concepts of Nth order normal forms and infinite order normal forms of smooth maps, and give some sufficient conditions for uniqueness of normal forms of smooth maps. As an application, we show the occurrence of the flip–Neimark–Sacker bifurcation in a financial model.

Free poset algebras and combinatorics of cones

In Free poset Boolean algebra F(P ), uniqueness of normal form of non-zero elements is proved and the notion of support of a non-zero element is, therefore, well defined. An Inclusion---Exclusion-like formula is given by defining, for each non-zero element x, $\overline{\mu}(x)$ using support of x ?F(P ) in a very natural way.

Characterizing polytime through higher type recursion

It is well-known that by a single use of higher type recursion on notation one can define proper Kalmar-elementary functions. This is due to a certain non-linear use of the ‘previous function’ in the step term of the recursion.It is shown how to restrict recursion on notation in all finite types so as to characterise the polynomial time computable functions. The restrictions are obtained by enriching the type structure with ! s, and by adding linear concepts to the lambda calculus.For this purpose, a simply typed term system RNA in the style of Gödel's T is presented, where RNA indicates the essential restrictions to the formation of terms: Relevance, Necessitation, and Affinability. RNA-terms are built from variables and the ‘binary’ constants 0, s0, s1, p by restricted application accounting for R, by a simplification of Prawitz's rule for introducing ‘!’ in his formulation of S4 accounting for N, by elimination of ‘!’, by restricted lambda abstraction accounting for A, and by rules for introducing case analysis and recursion on notation in all finite safe types. Safe types are those types which do not contain ‘!’. Similar to Gödel's T, terms are interpreted over the set-theoretical function spaces.Affinability formalises the underlying concept of linearity which is central to the whole system. It is designed such that it is decidable whether or not a given general term is in RNA.It is shown that RNA is closed under reduction. Strong normalisation for RNA and uniqueness of normal forms then follow. In particular, every closed RNA term of ground type reduces to a numeral denoting the value of the term. Thus, RNA can be considered a simple functional programming language where computation is normalisation.As for the expressive power of RNA, a kind of sharing concept implemented within RNA is employed to prove that every RNA program computing a number theoretical function can be evaluated in time polynomial in the lengths of the given inputs. That all polynomial time computable functions are definable in RNA simply holds by embedding the BC functions. (Joint work with S.Bellantoni and H.Schwichtenberg).

Towards restructuring and normalization of types in databases

The equivalence-preserving transformation and normalization of types in object-oriented databases are discussed. Specifically, a normal form of types based on set-theoretic equivalence is proposed, rewrite rules which transform types into normal forms are presented, and the uniqueness of normal form and the completeness of rewrite rules are proved. The emphasis of this work is on normal forms and corresponding rewrite rules. It provides a new formal approach for the study of restructuring of database schema and other manipulations in object-oriented databases.

Residual theory in λ-calculus: a formal development

AbstractWe present the complete development, in Gallina, of the residual theory of β-reduction in pure λ-calculus. The main result is the Prism Theorem, and its corollary Lévy's Cube Lemma, a strong form of the parallel-moves lemma, itself a key step towards the confluence theorem and its usual corollaries (Church-Rosser, uniqueness of normal forms). Gallina is the specification language of the Coq Proof Assistant (Dowek et al., 1991; Huet 1992b). It is a specific concrete syntax for its abstract framework, the Calculus of Inductive Constructions (Paulin-Mohring, 1993). It may be thought of as a smooth mixture of higher-order predicate calculus with recursive definitions, inductively defined data types and inductive predicate definitions reminiscent of logic programming. The development presented here was fully checked in the current distribution version Coq V5.8. We just state the lemmas in the order in which they are proved, omitting the proof justifications. The full transcript is available as a standard library in the distribution of Coq.

The uniqueness of normal forms via Lie transforms and its applications to Hamiltonian systems

A necessary and sufficient condition is given for the unique normal forms about critical elements-equilibrium points and periodic orbits. With some applications to the Birkhoff's normal form along with its generalized form by K.R. Meyer, the restricted problem of three bodies near L4, the Birkohoff's normalization procedure, and the singular perturbation, of Hamiltonian systems are discussed.