Filters








299,010 Hits in 6.1 sec

Krull Dimension and Classical Krull Dimension of Modules

Jaime Castro Pérez, José Ríos Montes
2014 Communications in Algebra  
We study a class of first-order theories whose complete quantifier-free types with one free variable either have a trivial positive part or are isolated by a positive quantifier-free formula-plus a few  ...  We also prove that the length of any increasing sequence of prime types is bounded, so every formula has finite Krull dimension.  ...  Definition 1 We say that the theory T is Krull-minimal if for every tuple a, D0 the trivial type x = x is prime over a; D1 for every consistent positive quantifier-free formula ζ(a, x) there is a quantifier-free  ... 
doi:10.1080/00927872.2013.781611 fatcat:nd6hkq7llzecre4p5x2rdt3pby

The complexity of explicit definitions

Harvey Friedman
1976 Advances in Mathematics  
We begin with a careful statement of what we will call the (X, Y, Z)definability theorem. A relational type is a set X of constant, relation, and function symbols.  ...  An X-structure consists of a nonempty domain D together with interpretations of the symbols in X on D. If 6I!  ...  There is no bound on the complexity of an explicit definition even if the implicit definition is assumed to consist of four universalized equations. definability theorem.  ... 
doi:10.1016/0001-8708(76)90167-5 fatcat:af75a7uedvcy5pljbap7ojopmm

Mechanising Mondex with Z/Eves

Leo Freitas, Jim Woodcock
2007 Formal Aspects of Computing  
We describe our experiences in mechanising the specification, refinement, and proof of the Mondex Electronic Purse using the Z/Eves theorem prover.  ...  The work contributes to the Repository for the Verified Software Grand Challenge.  ...  We would like to thank the other members of the Mondex club, especially those who have been working directly on the verification effort. This includes Michael  ... 
doi:10.1007/s00165-007-0059-y fatcat:rlikrbxz7bgjrlpoqo3kmdhy5q

Arithmetic Translations of Axiom Systems

Hao Wang
1951 Transactions of the American Mathematical Society  
As the theorem reveals the strength of the assertion Con(S) for each given system S, it can be applied in considerations regarding problems of relative consistency.  ...  With the adoption of a suitable definition for the notion of translatability, the theorem can be rephrased as saying that every system S is translatable into the system obtained from number theory by adding  ...  Thus, for example, each set theory which contains the number theory Z as a part and in which Con(Z) is a theorem is stronger than Z (that is, not translatable into Z).  ... 
doi:10.2307/1990692 fatcat:doviuts7xnblvoy7g24tz5rnda

Arithmetic translations of axiom systems

Hao Wang
1951 Transactions of the American Mathematical Society  
As the theorem reveals the strength of the assertion Con(S) for each given system S, it can be applied in considerations regarding problems of relative consistency.  ...  With the adoption of a suitable definition for the notion of translatability, the theorem can be rephrased as saying that every system S is translatable into the system obtained from number theory by adding  ...  Thus, for example, each set theory which contains the number theory Z as a part and in which Con(Z) is a theorem is stronger than Z (that is, not translatable into Z).  ... 
doi:10.1090/s0002-9947-1951-0043046-1 fatcat:27m42ud6sngnpk6gf2v7rjxyzm

On formal specification of a proof tool [chapter]

R. D. Arthan
1991 Lecture Notes in Computer Science  
Acknowledgments The FST project (IED project 1563) is jointly funded by International Computers Limited and by the Information Engineering Directorate of the UK Department of Trade and Industry. [  ...  For example, definitions by recursion over a free type or definitions of a set like the definition of wff in section 2.1 above may usefully be handled in this way.  ...  The forms of Z paragraph such as free type definitions which do not come in boxes are high-lighted by a bar in the left margin. Defining occurrences of global variables are shown in bold type.  ... 
doi:10.1007/3-540-54834-3_22 fatcat:rgng53mshvgjxjte5m6oovl2ey

On the proof theory of Coquand's calculus of constructions

Jonathan P. Seldin
1997 Annals of Pure and Applied Logic  
Strong normalization is proved for deductions. This strong normalization result implies the consistency of the underlying system, but it is still possible to make contradictory assumptions.  ...  The calculus of constructions is formulated as a natural deduction system in which deductions follow the constructions of the terms to which types are assigned.  ...  This is a minor variation on the environment r of Theorem 20: Theorem 21. Zf r be a strongly consistent environment, then T,c : Peanol is consistent. : Peanol,z : PropkM : 2, ( consider 5B2(z).  ... 
doi:10.1016/s0168-0072(96)00008-5 fatcat:2olfkhtdovgn3pobdsu6tutpay

The stability of the family of $A_2$-type arrangements

Takuro Abe
2006 Journal of Mathematics of Kyoto University  
We give a necessary and sufficient condition for the stability and the freeness of the family of A 2 -type arrangements.  ...  Moreover, we determine explicitly when the normalization of the sheafification of its module of reduced logarithmic vector fields is isomorphic to T P 2 (−2).  ...  The author is so grateful to Masahiko Yoshinaga for his help, advice, and encouragement to this article.  ... 
doi:10.1215/kjm/1250281752 fatcat:7njarznccbhnjbnmwclb3kdlde

A general class of noncontext free grammars generating context free languages

Sarwan K. Aggarwal, James A. Heinen
1979 Information and Control  
It is also shown that the property of SVMT-boundedness can sometimes be used to conclude that a given language is context free while certain previous results cannot be used for this purpose.  ...  The concept of SVMT-bounded grammars is introduced.  ...  The following definition introduces a particular type of strict partial ordering defined on the elements of V k) Z* (all SVMT-strings): DEFINITION 3.  ... 
doi:10.1016/s0019-9958(79)90675-2 fatcat:f5ppirxvrjd27he2r7j2kivnni

The Theory behind Theory Mine

Alan Bundy, Flaminia Cavallo, Lucas Dixon, Moa Johansson, Roy McCasland
2015 IEEE Intelligent Systems  
We hope it will be possible to make it available in the medium term. Meanwhile, the current paper goes some way to fill the gap left by its non-publication.  ...  for commercial reasons.  ...  An example of a non-free type would be the integers, defined as Z ::= 0 | s(Z) | p(Z) where p is the predecessor function, since s(p(x)) = p(s(x).  ... 
doi:10.1109/mis.2015.42 fatcat:5diehwtgs5ennfnjxm66kswdoa

Passive induction and a solution to a Paris–Wilkie open question

Dan E. Willard
2007 Annals of Pure and Applied Logic  
Our prior work showed that the semantic tableaux version of the Second Incompleteness Theorem did generalize for the most common definition of IΣ 0 appearing in the standard textbooks.  ...  However, there was an alternate interesting definition of this axiom system in the Wilkie-Paris article in the Annals of Pure and Applied Logic 35 (1987) pp. 261-302 which we did not examine in our year  ...  (Here ¬ Ψ is a passive formula because it has no free variables): ∀x ∀z { { ¬Ψ ∧ ∀y ≤ z [ ¬Ψ =⇒ ¬Ψ ] } =⇒ ∀y ≤ z ¬Ψ } (24) Definition 1 had indicated that the root of a semantic tableaux proof of the theorem  ... 
doi:10.1016/j.apal.2007.01.003 fatcat:w6aiei5cffbozglziczfna3hc4

GSOS Formalized in Coq

Ken Madlener, Sjaak Smetsers
2013 2013 International Symposium on Theoretical Aspects of Software Engineering  
The present paper formalizes the underlying theory in the theorem prover COQ, and proves the consistency property, also known as the adequacy theorem.  ...  The inspiration for our work has been drawn from the field of bialgebraic semantics.  ...  , in B is the constructor of the streams Z B , and has type B Z B → Z B .  ... 
doi:10.1109/tase.2013.34 dblp:conf/tase/MadlenerS13 fatcat:muig25u2hvaqzanq4h6sx7hwfi

Correction to: "Peano models with many generic classes"

James Schmerl
1981 Pacific Journal of Mathematics  
Notice that if i e I and r t Q Γ consists of those formulas in Γ which have v t as the only free variable, then Γ is a minimal type ( [1] ).  ...  More generally, for J Q I let Γj £ Γ consist of those formulas in Γ which have v 5 (j e J) as their only free variables.  ... 
doi:10.2140/pjm.1981.92.195 fatcat:i6y3422qxvhejpmxwl64bg56ce

Spectral theory of Sturm–Liouville difference operators

Guoliang Shi, Hongyou Wu
2009 Linear Algebra and its Applications  
These examples are obtained using a general procedure for constructing difference operators realizing discrete Sturm-Liouville problems, and the minimum conditions for such difference operators to be self-adjoint  ...  In particular, several eigenvalue comparison results are proved.  ...  The first named author was supported by the National Nature Science Foundation of China (Grant Number 10671141).  ... 
doi:10.1016/j.laa.2008.09.035 fatcat:txgvn4ffcvdltfyk5fkrrjkwtu

WKL0 and induction principles in model theory

David R. Belanger
2015 Annals of Pure and Applied Logic  
The Arithmetic Comprehension Axiom is axiom scheme: For each arithmetical formula φ(x) is an arithmetical formula in the language of second-order arithmetic with a free first-order variable x and an arbitrary  ...  In this subsection we define and give some alternate characterizations of each of these principles. The uninterested reader may skip it and refer back as needed. Definition 1.2.  ...  We say X has the pairwise free amalgamation property if for every pair p(x), q(ȳ) of types in X, there is a type r(x,ȳ,z) in X extending both q i and q j . 4 .  ... 
doi:10.1016/j.apal.2015.04.001 fatcat:el5bqgiyffdobcpxc3he53yxi4
« Previous Showing results 1 — 15 out of 299,010 results