Sequential calculus for proving the properties of regular programs

2005 ◽  
pp. 370-381
Author(s):  
Aida Pliuškevičienė
Keyword(s):  
Sequential Calculus

On the proof theory of the modal logic for arithmetic provability

1981 ◽  
Vol 46 (3) ◽  
pp. 531-538 ◽  
Author(s):  
Daniel Leivant
Keyword(s):  
Modal Logic ◽  
Proof Theory ◽  
Additional Insight ◽  
Semantical Analysis ◽  
Constructive Proofs ◽  
Definite Improvement ◽  
Original Presentation ◽  
Sequential Calculus ◽  
Insight Into

The modal logic GL has been found by Solovay [13] to formalize the provable propositional properties of the provability-predicate for Peano's Arithmetic PA (cf. §1 below). We give several sequential calculi for GL, compare their merits, and use one calculus to syntactically derive several metamathematical results about GL.Some of our results have been proved model theoretically, and similar proofs are fairly straightforward for several of the remaining ones (G. Boolos and the referee have provided such proofs for 4.1, 4.3 and 5.1 below). However, our syntactic techniques often yield more concise and obviously constructive proofs, they offer additional insight into the nature of the systems considered, and are easily adaptable to systems for which semantical analysis is problematic.I am indebted to G. Boolos and to the referee for their valuable advice. The referee has suggested the rule GL of §3 below as an axiomatization of GL; the resulting sequential calculus has allowed a definite improvement of our original presentation.

Sequential calculus

1995 ◽  
Vol 53 (3) ◽  
pp. 123-130 ◽  
Author(s):  
Burghard von Karger ◽  
C.A.R. Hoare
Keyword(s):  
Sequential Calculus

On modal systems having arithmetical interpretations

1984 ◽  
Vol 49 (3) ◽  
pp. 935-942 ◽  
Author(s):  
Arnon Avron
Keyword(s):  
Fixed Point ◽  
Modal Logics ◽  
Cut Elimination ◽  
Kripke Models ◽  
Disjunction Property ◽  
Sequential Calculus ◽  
The Subject ◽  
Modal Systems ◽  
Arithmetical Interpretation ◽  
Fixed Point Techniques

We deal here with two modal logics, GL and Grz, that are known to have interesting arithmetical interpretations connected with the notion of provability. GL is the extensiom of K (or K4) by the schema □(□ A → A) → □ A, and Grz is the extension of S4 by □(□(A → □A) →A) → □A. GL is also known to be sound and complete with respect to the class of all Kripke models that are transitive, irreflexive and well founded. Grz bears the same relation to the corresponding reflexive models. We refer the reader to [1] for a full exposition of the subject. (See also [4], [2], [6].)In §I we develop a sequential calculus for both GL and Grz and give a semantical proof that both systems admit cut-elimination. (Incidentally, this provides an easy proof of the semantical completeness of the two systems.) With respect to GL this yields a correction of an error in [2].In §II we show that cut-elimination fails for QGL (the extension of GL to a language with quantifiers). We further show that, despite this failure, QGL still has some of GL's interesting properties (e.g., the disjunction property). We also show, using fixed-point techniques, that similar properties obtain if we take as semantics for QGL the arithmetical interpretation extended in the obvious way.We want to thank Professor H. Gaifman for his help while working on the subject.

Observability, Connectivity, and Replay in a Sequential Calculus of Classes

2005 ◽  
pp. 296-316 ◽  
Author(s):  
Erika Ábrahám ◽  
Marcello M. Bonsangue ◽  
Frank S. de Boer ◽  
Andreas Grüner ◽  
Martin Steffen
Keyword(s):  
Sequential Calculus
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