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Unifiability in extensions of K4

C. Gencer, D. De Jongh
2009 Logic Journal of the IGPL  
We extend and generalize the work on unifiability of [8] . We give a semantic characterization for unifiability and non-unifiability in the extensions of K4.  ...  We apply this in particular to extensions of KD4, GL and K4.3 to obtain a syntactic characterization and give a concrete decision procedure for unifiability for those logics.  ...  We owe considerable gratitude to an unknown referee who pointed out many small inaccuracies, and one more important one in the proof of Theorem 18.  ... 
doi:10.1093/jigpal/jzn032 fatcat:o4hsy6cr4jg3zahbps4qw4awmi

Admissible Rules of Modal Logics

Emil Jeřábek
2005 Journal of Logic and Computation  
We construct explicit bases of admissible rules for a representative class of normal modal logics (including the systems K4, GL, S4, Grz, and GL.3), by extending the methods of S. Ghilardi and R.  ...  We also investigate the notion of admissible multiple conclusion rules.  ...  Let L be a normal extension of K4 with FMP.  ... 
doi:10.1093/logcom/exi029 fatcat:tksfg63qvrchvaplums5ea4fb4

Rules with parameters in modal logic I

Emil Jeřábek
2015 Annals of Pure and Applied Logic  
We study admissibility of inference rules and unification with parameters in transitive modal logics (extensions of K4), in particular we generalize various results on parameter-free admissibility and  ...  In the sequel, we will use the same machinery to investigate the computational complexity of admissibility and unification with parameters in cluster-extensible logics, and we will adapt the results to  ...  Moreover, the class of clx logics is closed under joins in the lattice of normal extensions of K4.  ... 
doi:10.1016/j.apal.2015.04.004 fatcat:rfihurobc5genaqt7fllbwt5bu

Unification and Passive Inference Rules for Modal Logics

V. V. Rybakov, M. Terziler, C. Gencer
2000 Journal of Applied Non-Classical Logics  
A complete syntactic description of all formulas which are non-unifiable in wide classes of modal logics is given.  ...  We 1 study unification of formulas in modal logics and consider logics which are equivalent w.r.t. unification of formulas.  ...  Thus we can say that formulas 1≤i≤n 3p i ∧3¬p i form an anti-basis for formulas which are not unifiable in extensions of S4 Indeed, as long as a formula α is not unifiable there is a formula from those  ... 
doi:10.1080/11663081.2000.10511004 fatcat:thbnfkezmvacddnmjvt33hqi2e

Rules with parameters in modal logic II [article]

Emil Jeřábek
2020 arXiv   pre-print
The class of cluster-extensible (clx) logics was introduced in the first part of this series of papers.  ...  We analyze the computational complexity of admissibility and unifiability with parameters in transitive modal logics.  ...  The Institute of Mathematics of the Czech Academy of Sciences is supported by RVO: 67985840.  ... 
arXiv:1905.13157v2 fatcat:e7yxsy5cebf6vebruezshk4eum

Unification in modal and description logics

F. Baader, S. Ghilardi
2010 Logic Journal of the IGPL  
In this article, we give a survey of the results on unification obtained in two closely related, yet different, application areas of unification: description logics and modal logics.  ...  Unification was originally introduced in automated deduction and term rewriting, but has recently also found applications in other fields.  ...  Many examples (ranging from small to medium size) are worked out in full details in [33] . The extension of this calculus to the modal systems K4 and S4 has been presented in [75] .  ... 
doi:10.1093/jigpal/jzq008 fatcat:hrew2tewyrfopf2a5egnmhm23q

Undecidability of the unification and admissibility problems for modal and description logics

Frank Wolter, Michael Zakharyaschev
2008 ACM Transactions on Computational Logic  
More precisely, suppose that for a unifiable formula ϕ in L one can compute a finite complete set S of unifiers in the sense that each unifier s for ϕ in L is less general than some s ∈ S (i.e., there  ...  sets of unifiers, we refer the reader to [].  ...  Let K u ⊆ L ⊆ K4 u . Then P : a → b iff ψ(b) is unifiable in L. Proof of Lemma. (⇐) Suppose that P : a → b.  ... 
doi:10.1145/1380572.1380574 fatcat:ts3c3wnptnboravpvrl3nzspli

Proof theory for admissible rules

Rosalie Iemhoff, George Metcalfe
2009 Annals of Pure and Applied Logic  
Proof systems are defined here for admissible rules of classes of modal logics, including K4, S4, and GL, and also Intuitionistic Logic IPC.  ...  In this paper, a Gentzen-style framework is introduced for analytic proof systems that derive admissible rules of non-classical logics.  ...  Acknowledgements We are very grateful for the constructive criticism and useful suggestions of an anonymous referee which have led to many improvements in the presentation of the paper, and also for the  ... 
doi:10.1016/j.apal.2008.10.011 fatcat:7z7trrc4gjg3vo2z76ppxksd2e

Hypersequent Systems for the Admissible Rules of Modal and Intermediate Logics [chapter]

Rosalie Iemhoff, George Metcalfe
2008 Lecture Notes in Computer Science  
In a previous paper by the authors, formal systems for deriving the admissible rules of Intuitionistic Logic and a class of modal logics were defined in a proof-theoretic framework where the basic objects  ...  Here, the framework is extended to cover derivability of the admissible rules of intermediate logics and a wider class of modal logics, in this case, by taking hypersequent rules as the basic objects.  ...  of "extensible" modal logics including K4, S4, and GL.  ... 
doi:10.1007/978-3-540-92687-0_16 fatcat:gowxk7nr4vgx7e6s6bct6466qa

Page 164 of American Journal of Mathematics Vol. 58, Issue 1 [page]

1936 American Journal of Mathematics  
It has recently been proved in a paper’ which will be referred to as I and which is based on an extension of a method of M.  ...  The purpose of the present note is to carry out this extension.  ... 

Page 5290 of Mathematical Reviews Vol. , Issue 2000h [page]

2000 Mathematical Reviews  
The chapters in order are: Extensions of K4, Logics of bounded alternativity, and Dynamic logic. There is an index with separate lists of symbols, names, and terms, and an extensive bibliography. J.  ...  A complete syntactic description of all formulas which are non-unifiable in wide classes of modal logics is given.  ... 

On the Momentum Problem for Distribution Functions in More Than One Dimension. II

E. K. Haviland
1936 American Journal of Mathematics  
It has recently been proved in a paper’ which will be referred to as I and which is based on an extension of a method of M.  ...  The purpose of the present note is to carry out this extension.  ... 
doi:10.2307/2371063 fatcat:qbcztg7e75dfxgqafqxfuhdzb4

Undecidability of the unification and admissibility problems for modal and description logics [article]

Frank Wolter, Michael Zakharyaschev
2006 arXiv   pre-print
We show that the unification problem 'is there a substitution instance of a given formula that is provable in a given logic?'  ...  We also prove undecidability of the unification and admissibility problems for K and K4 with at least two modal operators and nominals (instead of the universal modality), thereby showing that these problems  ...  Let K u ⊆ L ⊆ K4 u . Then P : a → b iff ψ(b) is unifiable in L. • 7 Proof of Lemma. (⇐) Suppose that P : a → b.  ... 
arXiv:cs/0609052v1 fatcat:6jwhm2e5mvgnbmgbxtak2m3tq4

On Rules

Rosalie Iemhoff
2015 Journal of Philosophical Logic  
This paper contains a brief overview of the area of admissible rules with an emphasis on results about intermediate and modal propositional logics.  ...  A set of maximal unifiers of A is complete if every unifier of A is less general than a unifier in the set and if no two unifiers in the set are comparable.  ...  A substitution σ is a maximal unifier of A if among the unifiers of A it is maximal in the order . It is a most general unifier (mgu) of A if all unifiers of A are less general than it.  ... 
doi:10.1007/s10992-015-9354-x fatcat:rgq6jjp5prh2ze7umdzr4nacie

Page 3560 of Mathematical Reviews Vol. , Issue 92g [page]

1992 Mathematical Reviews  
Summary: “The concept of unified theory is defined in logical and abstract semantic terms, and employed in the analysis of relations between empirical scientific theories.  ...  The analysis shows in which sense it can, and in which it cannot, be said that this theory unifies Wien’s law and the Rayleigh-Jeans law of black-body radiation.”  ... 
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