scholarly journals An inductive definition of the class of 3-connected quadrangulations of the plane

1989 ◽  
Vol 78 (1-2) ◽  
pp. 45-53 ◽  
Author(s):  
Vladimir Batagelj
Keyword(s):  
Inductive Definition ◽  
Definition Of

A TYPE-FREE THEORY OF HALF-MONOTONE INDUCTIVE DEFINITIONS

1995 ◽  
Vol 06 (03) ◽  
pp. 203-234 ◽  
Author(s):  
YUKIYOSHI KAMEYAMA
Keyword(s):  
Type Theory ◽  
Characteristic Point ◽  
Logical Theory ◽  
Inductive Definition ◽  
Program Extraction ◽  
First Order ◽  
Inductive Definitions ◽  
Free Theory ◽  
Program Proof ◽  
Definition Of

This paper studies an extension of inductive definitions in the context of a type-free theory. It is a kind of simultaneous inductive definition of two predicates where the defining formulas are monotone with respect to the first predicate, but not monotone with respect to the second predicate. We call this inductive definition half-monotone in analogy of Allen’s term half-positive. We can regard this definition as a variant of monotone inductive definitions by introducing a refined order between tuples of predicates. We give a general theory for half-monotone inductive definitions in a type-free first-order logic. We then give a realizability interpretation to our theory, and prove its soundness by extending Tatsuta’s technique. The mechanism of half-monotone inductive definitions is shown to be useful in interpreting many theories, including the Logical Theory of Constructions, and Martin-Löf’s Type Theory. We can also formalize the provability relation “a term p is a proof of a proposition P” naturally. As an application of this formalization, several techniques of program/proof-improvement can be formalized in our theory, and we can make use of this fact to develop programs in the paradigm of Constructive Programming. A characteristic point of our approach is that we can extract an optimization program since our theory enjoys the program extraction theorem.

A proof of the cut-elimination theorem in simple type theory

1973 ◽  
Vol 38 (2) ◽  
pp. 215-226
Author(s):  
Satoko Titani
Keyword(s):  
Boolean Algebra ◽  
Type Theory ◽  
Arbitrary Number ◽  
Formal System ◽  
Cut Elimination ◽  
Simple Type ◽  
Inductive Definition ◽  
Simple Type Theory ◽  
Maximal Ideals ◽  
Definition Of

In [4], I introduced a quasi-Boolean algebra, and showed that in a formal system of simple type theory, from which the cut rule is omitted, wffs form a quasi-Boolean algebra, and that the cut-elimination theorem can be formulated in algebraic language. In this paper we use the result of [4] to prove the cut-elimination theorem in simple type theory. The theorem was proved by M. Takahashi [2] in 1967 by using the concept of Schütte's semivaluation. We use maximal ideals of a quasi-Boolean algebra instead of semivaluations.The logical system we are concerned with is a modification of Schütte's formal system of simple type theory in [1] into Gentzen style.Inductive definition of types.0 and 1 are types.If τ1, …, τn are types, then (τ1, …, τn) is a type.Basic symbols.a1τ, a2τ, … for free variables of type τ.x1τ, x2τ, … for bound variables of type τ.An arbitrary number of constants of certain types.An arbitrary number of function symbols with certain argument places.

Monotone inductive definitions over the continuum

1976 ◽  
Vol 41 (1) ◽  
pp. 188-198 ◽  
Author(s):  
Douglas Cenzer
Keyword(s):  
Mathematical Logic ◽  
Recursion Theory ◽  
Axiom System ◽  
Baire Space ◽  
Natural Numbers ◽  
Inductive Definition ◽  
Inductive Definitions ◽  
Definition Of ◽  
The Continuum

Monotone inductive definitions occur frequently throughout mathematical logic. The set of formulas in a given language and the set of consequences of a given axiom system are examples of (monotone) inductively defined sets. The class of Borel subsets of the continuum can be given by a monotone inductive definition. Kleene's inductive definition of recursion in a higher type functional (see [6]) is fundamental to modern recursion theory; we make use of it in §2.Inductive definitions over the natural numbers have been studied extensively, beginning with Spector [11]. We list some of the results of that study in §1 for comparison with our new results on inductive definitions over the continuum. Note that for our purposes the continuum is identified with the Baire space ωω.It is possible to obtain simple inductive definitions over the continuum by introducing real parameters into inductive definitions over N—as in the definition of recursion in [5]. This is itself an interesting concept and is discussed further in [4]. These parametric inductive definitions, however, are in general weaker than the unrestricted set of inductive definitions, as is indicated below.In this paper we outline, for several classes of monotone inductive definitions over the continuum, solutions to the following characterization problems:(1) What is the class of sets which may be given by such inductive definitions ?(2) What is the class of ordinals which are the lengths of such inductive definitions ?These questions are made more precise below. Most of the results of this paper were announced in [2].

Motivating the Rules of Sequent Calculus

2017 ◽  
Author(s):  
Neil Tennant
Keyword(s):  
Natural Deduction ◽  
Sequent Calculus ◽  
Natural Isomorphism ◽  
Inductive Definition ◽  
Proof Search ◽  
Definition Of ◽  
Automated Proof

Parallelized elimination rules in natural deduction correspond to Left rules in the sequent calculus; and introduction rules correspond to Right rules. These rules may be construed as inductive clauses in the inductive definition of the notion of sequent proof. There is a natural isomorphism between natural deductions in Core Logic and the sequent proofs that correspond to them. We examine the relations, between sequents, of concentration and dilution; and describe what it is for one sequent to strengthen another. We examine some possible global restrictions on proof-formation, designed to prevent proofs from proving dilutions of sequents already proved by a subproof. We establish the important result that the sequent rules of Core Logic maintain concentration, and we explain its importance for automated proof-search.

On a partial syntactical criterion for the left distributivity and the idempotency

2010 ◽  
Vol 60 (2) ◽  
Author(s):  
Přemysl Jedlička
Keyword(s):  
Inductive Definition ◽  
Definition Of

AbstractWe study here so called cuts of terms and their classes modulo the identities of the left distributivity and the idempotency. We give an inductive definition of such classes and this gives us a criterion that decides in some cases whether two terms are equivalent modulo both identities.

Foldings of star manifolds

1993 ◽  
Vol 123 (6) ◽  
pp. 1011-1016
Author(s):  
H. R. Farran ◽  
E. El-Kholy ◽  
S. A. Robertson
Keyword(s):  
Smooth Manifold ◽  
General Setting ◽  
Smooth Manifolds ◽  
Inductive Definition ◽  
Metrical Structure ◽  
Totally Geodesic Submanifold ◽  
Geodesic Submanifold ◽  
Crucial Feature ◽  
Definition Of

SynopsisThis paper is a sequel to [4]. Its purpose is to show that the concept of isometric foldings of Riemannian manifolds can be extended to a much wider class of manifolds without losing the main structure theorem. We present here what we believe to be a definitive form of the folding concept for smooth manifolds.The theory discussed here is based on the idea of a 1-spread [2], where the role of geodesies on a Riemannian manifold is assumed by smooth, unoriented and unparametrised curves on a smooth manifold. The absence of metrical structure forces a fresh approach to the basic definitions. A crucial feature of the Riemannian theory does survive, however, in this general setting: a 1-spread on a sufficiently smooth manifold M induces a 1-spread on sufficiently small spheres surrounding any point of M. With the help of this fact, we are able to construct an inductive definition of “star folding” f:M → N between smooth manifolds M and N, and to retain the theorem that the manifold M is stratified by the “folds”, each of which has the character of a “totally geodesic” submanifold with respect to the above-mentioned curves.

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