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Unifiability in extensions of K4

2009
*
Logic Journal of the IGPL
*

We extend and generalize the work on

doi:10.1093/jigpal/jzn032
fatcat:o4hsy6cr4jg3zahbps4qw4awmi
*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. ...##
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Admissible Rules of Modal Logics

2005
*
Journal of Logic and Computation
*

We construct explicit bases

doi:10.1093/logcom/exi029
fatcat:tksfg63qvrchvaplums5ea4fb4
*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. ...##
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Rules with parameters in modal logic I

2015
*
Annals of Pure and Applied Logic
*

We study admissibility

doi:10.1016/j.apal.2015.04.004
fatcat:rfihurobc5genaqt7fllbwt5bu
*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*. ...##
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Unification and Passive Inference Rules for Modal Logics

2000
*
Journal of Applied Non-Classical Logics
*

A complete syntactic description

doi:10.1080/11663081.2000.10511004
fatcat:thbnfkezmvacddnmjvt33hqi2e
*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 ...##
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Rules with parameters in modal logic II
[article]

2020
*
arXiv
*
pre-print

The class

arXiv:1905.13157v2
fatcat:e7yxsy5cebf6vebruezshk4eum
*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. ...##
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Unification in modal and description logics

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] . ...

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

2008
*
ACM Transactions on Computational Logic
*

More precisely, suppose that for a

doi:10.1145/1380572.1380574
fatcat:ts3c3wnptnboravpvrl3nzspli
*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. ...##
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Proof theory for admissible rules

2009
*
Annals of Pure and Applied Logic
*

Proof systems are defined here for admissible rules

doi:10.1016/j.apal.2008.10.011
fatcat:7z7trrc4gjg3vo2z76ppxksd2e
*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 ...##
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Hypersequent Systems for the Admissible Rules of Modal and Intermediate Logics
[chapter]

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. ...

##
###
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

1936
*
American Journal of Mathematics
*

It has recently been proved

doi:10.2307/2371063
fatcat:qbcztg7e75dfxgqafqxfuhdzb4
*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*. ...##
###
Undecidability of the unification and admissibility problems for modal and description logics
[article]

2006
*
arXiv
*
pre-print

We show that the unification problem 'is there a substitution instance

arXiv:cs/0609052v1
fatcat:6jwhm2e5mvgnbmgbxtak2m3tq4
*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. ...##
###
On Rules

2015
*
Journal of Philosophical Logic
*

This paper contains a brief overview

doi:10.1007/s10992-015-9354-x
fatcat:rgq6jjp5prh2ze7umdzr4nacie
*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. ...##
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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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