A new look at the interpolation problem

1975 ◽  
Vol 40 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Jacques Stern
Keyword(s):  
Model Theory ◽  
Interpolation Problem ◽  
Classical Case ◽  
Interpolation Theorem ◽  
Alternative Proof ◽  
Interesting Application ◽  
Interesting Discussion ◽  
New Look

The original aim of this paper was to show that forcing provided a useful and unifying tool in the model theory of finite and admissible languages. Roughly speaking, any result obtained by a Henkin proof, by Makkai's method or by an omitting type argument can be given an alternative proof via forcing. Finally, instead of giving new proofs of a number of known theorems, we have chosen to focus on the interpolation theorem for many-sorted languages. The main result of this paper is thus a generalization of Feferman's theorems ([2], [3]) with a completely different proof; this was announced in [8] and [9].For more on forcing techniques in model theory the reader should consult [4] as well as [5]. He should also compare forcing and boolean-valued models developed in [6] for the classical case and in [7] for admissible languages. Throughout the paper familiarity with the standard concepts of model theory is assumed.The author wishes to express his gratitude to J. L. Krivine and K. McAloon for supervising his work. Thanks are also due to J. P. Ressayre for helpful suggestions and to R. L. Vaught for an interesting discussion on forcing in model theory. Finally, an interesting application has been pointed out by S. Feferman and has been included with his permission.

scholarly journals About the Noether’s theorem for fractional Lagrangian systems and a generalization of the classical Jost method of proof

2019 ◽  
Vol 22 (4) ◽  
pp. 871-898 ◽  
Author(s):  
Jacky Cresson ◽  
Anna Szafrańska
Keyword(s):  
Local Group ◽  
Type Theorem ◽  
Classical Case ◽  
Alternative Proof ◽  
Lagrangian Systems ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Definition Of ◽  
Variational Symmetries ◽  
Discrete Setting

Abstract Recently, the fractional Noether’s theorem derived by G. Frederico and D.F.M. Torres in [10] was proved to be wrong by R.A.C. Ferreira and A.B. Malinowska in (see [7]) using a counterexample and doubts are stated about the validity of other Noether’s type Theorem, in particular ([9], Theorem 32). However, the counterexample does not explain why and where the proof given in [10] does not work. In this paper, we make a detailed analysis of the proof proposed by G. Frederico and D.F.M. Torres in [9] which is based on a fractional generalization of a method proposed by J. Jost and X.Li-Jost in the classical case. This method is also used in [10]. We first detail this method and then its fractional version. Several points leading to difficulties are put in evidence, in particular the definition of variational symmetries and some properties of local group of transformations in the fractional case. These difficulties arise in several generalization of the Jost’s method, in particular in the discrete setting. We then derive a fractional Noether’s Theorem following this strategy, correcting the initial statement of Frederico and Torres in [9] and obtaining an alternative proof of the main result of Atanackovic and al. [3].

Linear reasoning in modal logic

1984 ◽  
Vol 49 (4) ◽  
pp. 1363-1378 ◽  
Author(s):  
Melvin Fitting
Keyword(s):  
Classical Case ◽  
Interpolation Formula ◽  
Companion Paper ◽  
Interpolation Theorem ◽  
Modal Logics ◽  
Reasoning System ◽  
Gentzen Systems ◽  
Theoretic Argument ◽  
Proof Procedure ◽  
Consistency Properties

In [1] Craig introduced a proof procedure for first order classical logic called linear reasoning. In it, a proof of P ⊃ Q consists of a sequence of formulas, each of which implies the next, beginning with P and ending with Q. And one of the formulas in the sequence will be an interpolation formula for P ⊃ Q. Indeed, this was the first proof of the Craig interpolation theorem, some of whose important consequences were demonstrated in a companion paper [2]. In this paper we present systems of linear reasoning for several standard modal logics: K, T, K4, S4, D, D4, and GL. Similar systems can be constructed for several regular, nonnormal modal logics too, though we do not do so here. And just as in the classical case, interpolation theorems are easy consequences. Such theorems are well known for the logics considered here. There is a model theoretic argument in [6], an argument using Gentzen systems in [8], an argument using consistency properties in [4] and [5], and an argument using symmetric Gentzen systems in [5]. This paper presents what seems to be the first modal proof that follows Craig's original methods. We note that if the modal rules given here are dropped, a classical linear reasoning system results, which is related to, but not the same as those in [1] and [10].Since the basic linear reasoning ideas are fully illustrated by the propositional case, we present that first, to keep the clutter down. Later we show how the techniques can generally be extended to encompass quantifiers. We do not follow [1] in making heavy use of prenex form, since it is not generally available in modal logics. Fortunately, it plays no essential role.

Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic

1997 ◽  
Vol 62 (2) ◽  
pp. 457-486 ◽  
Author(s):  
Jan Krajíček
Keyword(s):  
Lower Bounds ◽  
Cutting Planes ◽  
Communication Complexity ◽  
Interpolation Theorem ◽  
Alternative Proof ◽  
Bounded Arithmetic ◽  
Proof Systems ◽  
Restricted Form ◽  
Arithmetic Theory ◽  
Equational Calculus

AbstractA proof of the (propositional) Craig interpolation theorem for cut-free sequent calculus yields that a sequent with a cut-free proof (or with a proof with cut-formulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuit-size is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: (1)Feasible interpolation theorems for the following proof systems:(a)resolution(b)a subsystem of LK corresponding to the bounded arithmetic theory (α)(c)linear equational calculus(d)cutting planes.(2)New proofs of the exponential lower bounds (for new formulas)(a)for resolution ([15])(b)for the cutting planes proof system with coefficients written in unary ([4]).(3)An alternative proof of the independence result of [43] concerning the provability of circuit-size lower bounds in the bounded arithmetic theory (α).In the other direction we show that a depth 2 subsystem of LK does not admit feasible monotone interpolation theorem (the so called Lyndon theorem), and that a feasible monotone interpolation theorem for the depth 1 subsystem of LK would yield new exponential lower bounds for resolution proofs of the weak pigeonhole principle.

Finite Conformal Hypergraph Covers and Gaifman Cliques in Finite Structures

2003 ◽  
Vol 9 (3) ◽  
pp. 387-405 ◽  
Author(s):  
Ian Hodkinson ◽  
Martin Otto
Keyword(s):  
Model Theory ◽  
Extension Property ◽  
Alternative Proof ◽  
Finite Model ◽  
Finite Model Property ◽  
Finite Model Theory ◽  
Relational Structures ◽  
Finite Structures ◽  
Guarded Fragment ◽  
Open Question

AbstractWe provide a canonical construction of conformal covers for finite hypergraphs and present two immediate applications to the finite model theory of relational structures. In the setting of relational structures, conformal covers serve to construct guarded bisimilar companion structures that avoid all incidental Gaifman cliques—thus serving as a partial analogue in finite model theory for the usually infinite guarded unravellings. In hypergraph theoretic terms, we show that every finite hypergraph admits a bisimilar cover by a finite conformal hypergraph. In terms of relational structures, we show that every finite relational structure admits a guarded bisimilar cover by a finite structure whose Gaifman cliques are guarded. One of our applications answers an open question about a clique constrained strengthening of the extension property for partial automorphisms (EPPA) of Hrushovski, Herwig and Lascar. A second application provides an alternative proof of the finite model property (FMP) for the clique guarded fragment of first-order logic CGF, by reducing (finite) satisfiability in CGF to (finite) satisfiability in the guarded fragment, GF.

Well-behaved modal logics

1984 ◽  
Vol 49 (4) ◽  
pp. 1393-1402
Author(s):  
Harold T. Hodes
Keyword(s):  
Modal Logic ◽  
Model Theory ◽  
Classical Model ◽  
Classical Case ◽  
Recursion Theory ◽  
Modal Logics ◽  
Modal Model ◽  
The Right ◽  
Domain Property ◽  
The Way

Much of the literature on the model theory of modal logics suffers from two weaknesses. Firstly, there is a lack of generality; theorems are proved piecemeal about this or that modal logic, or at best small classes of logics. Much of the literature, e.g. [1], [2], and [3], confines attention to structures with the expanding domain property (i.e., if wRu then Ā(w) ⊆ Ā(u)); the syntactic counterpart of this restriction is assumption of the converse Barcan scheme, a move which offers (in Russell's phrase) “all the advantages of theft over honest toil”. Secondly, I think there has been a failure to hit on the best ways of extending classical model theoretic notions to modal contexts. This weakness makes the literature boring, since a large part of the interest of modal model theory resides in the way in which classical model theoretic notions extend, and in some cases divide, in the modal setting. (The relation between α-recursion theory and classical recursion theory is analogous to that between modal model theory and classical model theory. Much of the work in α-recursion theory involved finding the right definitions (e.g., of recursive-in) and separating concepts which collapse in the classical case (e.g. of finiteness and boundedness).)The notion of a well-behaved modal logic is introduced in §3 to make possible rather general results; of course our attention will not be restricted to structures with the expanding domain property. Rather than prove piecemeal that familiar modal logics are well-behaved, in §4 we shall consider a class of “special” modal logics, which obviously includes many familiar logics and which is included in the class of well-behaved modal logics.

scholarly journals Model Theory in a Paraconsistent Environment

2021 ◽  
Vol 27 (2) ◽  
pp. 216-216
Author(s):  
Bruno Costa Coscarelli
Keyword(s):  
Model Theory ◽  
Paraconsistent Logic ◽  
Interpolation Theorem ◽  
Point Of View ◽  
The Third ◽  
Reasoning System ◽  
Paraconsistent Reasoning ◽  
E Mail ◽  
Craig’S Interpolation Theorem ◽  
Completeness Theorems

AbstractThe purpose of this thesis is to develop a paraconsistent Model Theory. The basis for such a theory was launched by Walter Carnielli, Marcelo Esteban Coniglio, Rodrigo Podiack, and Tarcísio Rodrigues in the article ‘On the Way to a Wider Model Theory: Completeness Theorems for First-Order Logics of Formal Inconsistency’ [The Review of Symbolic Logic, vol. 7 (2014)].Naturally, a complete theory cannot be fully developed in a single work. Indeed, the goal of this work is to show that a paraconsistent Model Theory is a sound and worthy possibility. The pursuit of this goal is divided in three tasks: The first one is to give the theory a philosophical meaning. The second one is to transpose as many results from the classical theory to the new one as possible. The third one is to show an application of the theory to practical science.The response to the first task is a Paraconsistent Reasoning System. The start point is that paraconsistency is an epistemological concept. The pursuit of a deeper understanding of the phenomenon of paraconsistency from this point of view leads to a reasoning system based on the Logics of Formal Inconsistency. Models are regarded as states of knowledge and the concept of isomorphism is reformulated so as to give raise to a new concept that preserves a portion of the whole knowledge of each state. Based on this, a notion of refinement is created which may occur from inside or from outside the state.In order to respond to the second task, two important classical results, namely the Omitting Types Theorem and Craig’s Interpolation Theorem are shown to hold in the new system and it is also shown that, if classical results in general are to hold in a paraconsistent system, then such a system should be in essence how it was developed here.Finally, the response to the third task is a proposal of what a Paraconsistent Logic Programming may be. For that, the basis for a paraconsistent PROLOG is settled in the light of the ideas developed so far.Abstract prepared by Bruno Costa Coscarelli.E-mail: [email protected]: http://repositorio.unicamp.br/jspui/handle/REPOSIP/331697

scholarly journals A model-theoretic approach to descriptive general frames: the van Benthem characterization theorem

2020 ◽  
Vol 30 (7) ◽  
pp. 1331-1355
Author(s):  
Nick Bezhanishvili ◽  
Tim Henke
Keyword(s):  
Modal Logic ◽  
Model Theory ◽  
Characterization Theorem ◽  
Interpolation Theorem ◽  
Theoretic Approach ◽  
Basic Model ◽  
Finite Structures ◽  
Beth Definability ◽  
Skolem Theorem ◽  
General Frames

Abstract The celebrated van Benthem characterization theorem states that on Kripke structures modal logic is the bisimulation-invariant fragment of first-order logic. In this paper, we prove an analogue of the van Benthem characterization theorem for models based on descriptive general frames. This is an important class of general frames for which every modal logic is complete. These frames can be represented as Stone spaces equipped with a ‘continuous’ binary relation. The proof of our theorem generalizes Rosen’s proof of the van Benthem theorem for finite frames and uses as an essential technique a new notion of descriptive unravelling. We also develop a basic model theory for descriptive general frames and show that in many ways it behaves like the model theory of finite structures. In particular, we prove the failure of the compactness theorem, of the Beth definability theorem, of the Craig interpolation theorem and of the upward Löwenheim–Skolem theorem.1

PEDIATRIC NEWS Unveils a New Look for the New Year

2010 ◽  
Vol 44 (1) ◽  
pp. 2
Author(s):  
Catherine Cooper Nellist ◽  
Mary Jo Dales
Keyword(s):  
New Look
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