49 found
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  1. Temporal Logic Mathematical Foundations and Computational Aspects.Dov M. Gabbay, Ian Hodkinson & Mark Reynolds - 1994
     
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  2. Step by Step-Building Representations in Algebraic Logic.Robin Hirsch & Ian Hodkinson - 1997 - Journal of Symbolic Logic 62 (1):225-279.
    We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterized according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Finite relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether an (...)
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  3.  14
    Decidable Fragments of First-Order Temporal Logics.Ian Hodkinson, Frank Wolter & Michael Zakharyaschev - 2000 - Annals of Pure and Applied Logic 106 (1-3):85-134.
    In this paper, we introduce a new fragment of the first-order temporal language, called the monodic fragment, in which all formulas beginning with a temporal operator have at most one free variable. We show that the satisfiability problem for monodic formulas in various linear time structures can be reduced to the satisfiability problem for a certain fragment of classical first-order logic. This reduction is then used to single out a number of decidable fragments of first-order temporal logics and of two-sorted (...)
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  4. Relation Algebra Reducts of Cylindric Algebras and an Application to Proof Theory.Robin Hirsch, Ian Hodkinson & Roger D. Maddux - 2002 - Journal of Symbolic Logic 67 (1):197-213.
    We confirm a conjecture, about neat embeddings of cylindric algebras, made in 1969 by J. D. Monk, and a later conjecture by Maddux about relation algebras obtained from cylindric algebras. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal α ≥ 3 there is a logically valid sentence X, in a first-order language L with equality and exactly one nonlogical binary relation symbol E, such that X contains only 3 variables (each of which (...)
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  5.  7
    Spatial Logic of Tangled Closure Operators and Modal Mu-Calculus.Robert Goldblatt & Ian Hodkinson - 2017 - Annals of Pure and Applied Logic 168 (5):1032-1090.
  6.  12
    Atom Structures of Cylindric Algebras and Relation Algebras.Ian Hodkinson - 1997 - Annals of Pure and Applied Logic 89 (2):117-148.
    For any finite n 3 there are two atomic n-dimensional cylindric algebras with the same atom structure, with one representable, the other, not.Hence, the complex algebra of the atom structure of a representable atomic cylindric algebra is not always representable, so that the class RCAn of representable n-dimensional cylindric algebras is not closed under completions. Further, it follows by an argument of Venema that RCAn is not axiomatisable by Sahlqvist equations, and hence nor by equations where negation can only occur (...)
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  7.  37
    Provability with Finitely Many Variables.Robin Hirsch, Ian Hodkinson & Roger D. Maddux - 2002 - Bulletin of Symbolic Logic 8 (3):348-379.
    For every finite n ≥ 4 there is a logically valid sentence φ n with the following properties: φ n contains only 3 variables (each of which occurs many times); φ n contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol): φ n has a proof in first-order logic with equality that contains exactly n variables, but no proof containing only n - 1 variables. This result was first proved using the machinery of (...)
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  8.  23
    Complete Representations in Algebraic Logic.Robin Hirsch & Ian Hodkinson - 1997 - Journal of Symbolic Logic 62 (3):816-847.
    A boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension (at least 3) are elementary.
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  9.  34
    Finite Conformal Hypergraph Covers and Gaifman Cliques in Finite Structures.Ian Hodkinson & Martin Otto - 2003 - Bulletin of Symbolic Logic 9 (3):387-405.
    We 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 (...)
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  10. The McKinsey–Lemmon Logic is Barely Canonical.Robert Goldblatt & Ian Hodkinson - 2007 - Australasian Journal of Logic 5:1-19.
    We study a canonical modal logic introduced by Lemmon, and axiomatised by an infinite sequence of axioms generalising McKinsey’s formula. We prove that the class of all frames for this logic is not closed under elementary equivalence, and so is non-elementary. We also show that any axiomatisation of the logic involves infinitely many non-canonical formulas.
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  11.  90
    Loosely Guarded Fragment of First-Order Logic has the Finite Model Property.Ian Hodkinson - 2002 - Studia Logica 70 (2):205 - 240.
    We show that the loosely guarded and packed fragments of first-order logic have the finite model property. We use a construction of Herwig and Hrushovski. We point out some consequences in temporal predicate logic and algebraic logic.
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  12.  3
    Hybrid Formulas and Elementarily Generated Modal Logics.Ian Hodkinson - 2006 - Notre Dame Journal of Formal Logic 47 (4):443-478.
    We characterize the modal logics of elementary classes of Kripke frames as precisely those modal logics that are axiomatized by modal axioms synthesized in a certain effective way from "quasi-positive" sentences of hybrid logic. These are pure positive hybrid sentences with arbitrary existential and relativized universal quantification over nominals. The proof has three steps. The first step is to use the known result that the modal logic of any elementary class of Kripke frames is also the modal logic of the (...)
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  13.  3
    Loosely Guarded Fragment of First-Order Logic has the Finite Model Property.Ian Hodkinson - 2002 - Studia Logica 70 (2):205-240.
    We show that the loosely guarded and packed fragments of first-order logic have the finite model property. We use a construction of Herwig and Hrushovski. We point out some consequences in temporal predicate logic and algebraic logic.
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  14.  52
    Sahlqvist Correspondence for Modal Mu-Calculus.Johan van Benthem, Nick Bezhanishvili & Ian Hodkinson - 2012 - Studia Logica 100 (1-2):31-60.
  15.  7
    Relation Algebras with N-Dimensional Relational Bases.Robin Hirsch & Ian Hodkinson - 2000 - Annals of Pure and Applied Logic 101 (2-3):227-274.
    We study relation algebras with n-dimensional relational bases in the sense of Maddux. Fix n with 3nω. Write Bn for the class of non-associative algebras with an n-dimensional relational basis, and RAn for the variety generated by Bn. We define a notion of relativised representation for algebras in RAn, and use it to give an explicit equational axiomatisation of RAn, and to reprove Maddux's result that RAn is canonical. We show that the algebras in Bn are precisely those that have (...))
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  16.  16
    Strongly Representable Atom Structures of Cylindric Algebras.Robin Hirsch & Ian Hodkinson - 2009 - Journal of Symbolic Logic 74 (3):811-828.
    A cylindric algebra atom structure is said to be strongly representable if all atomic cylindric algebras with that atom structure are representable. This is equivalent to saying that the full complex algebra of the atom structure is a representable cylindric algebra. We show that for any finite n >3, the class of all strongly representable n-dimensional cylindric algebra atom structures is not closed under ultraproducts and is therefore not elementary. Our proof is based on the following construction. From an arbitrary (...)
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  17.  11
    Relation Algebras From Cylindric Algebras, I.Robin Hirsch & Ian Hodkinson - 2001 - Annals of Pure and Applied Logic 112 (2-3):225-266.
    We characterise the class S Ra CA n of subalgebras of relation algebra reducts of n -dimensional cylindric algebras by the notion of a ‘hyperbasis’, analogous to the cylindric basis of Maddux, and by representations. We outline a game–theoretic approximation to the existence of a representation, and how to use it to obtain a recursive axiomatisation of S Ra CA n.
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  18.  7
    Relation Algebras From Cylindric Algebras, II.Robin Hirsch & Ian Hodkinson - 2001 - Annals of Pure and Applied Logic 112 (2-3):267-297.
    We prove, for each 4⩽ n ω , that S Ra CA n+1 cannot be defined, using only finitely many first-order axioms, relative to S Ra CA n . The construction also shows that for 5⩽n S Ra CA n is not finitely axiomatisable over RA n , and that for 3⩽m S Nr m CA n+1 is not finitely axiomatisable over S Nr m CA n . In consequence, for a certain standard n -variable first-order proof system ⊢ m (...)
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  19.  18
    Monodic Packed Fragment with Equality is Decidable.Ian Hodkinson - 2002 - Studia Logica 72 (2):185-197.
    We prove decidability of satisfiability of sentences of the monodic packed fragment of first-order temporal logic with equality and connectives Until and Since, in models with various flows of time and domains of arbitrary cardinality. We also prove decidability over models with finite domains, over flows of time including the real order.
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  20.  14
    Bare Canonicity of Representable Cylindric and Polyadic Algebras.Jannis Bulian & Ian Hodkinson - 2013 - Annals of Pure and Applied Logic 164 (9):884-906.
    We show that for finite n⩾3n⩾3, every first-order axiomatisation of the varieties of representable n-dimensional cylindric algebras, diagonal-free cylindric algebras, polyadic algebras, and polyadic equality algebras contains an infinite number of non-canonical formulas. We also show that the class of structures for each of these varieties is non-elementary. The proofs employ algebras derived from random graphs.
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  21.  15
    Sahlqvist Correspondence for Modal Mu-Calculus.Johan Benthem, Nick Bezhanishvili & Ian Hodkinson - 2012 - Studia Logica 100 (1-2):31-60.
    We define analogues of modal Sahlqvist formulas for the modal mu-calculus, and prove a correspondence theorem for them.
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  22.  21
    Erdős Graphs Resolve Fine's Canonicity Problem.Robert Goldblatt, Ian Hodkinson & Yde Venema - 2004 - Bulletin of Symbolic Logic 10 (2):186-208.
    We show that there exist 2 ℵ 0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any first-order definable class of relational structures. Using a variant of this construction, we resolve a long-standing question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any first-order definable class of Kripke frames (a monomodal example can then be obtained using simulation results of (...)
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  23.  50
    Weak Representations of Relation Algebras and Relational Bases.Robin Hirsch, Ian Hodkinson & Roger D. Maddux - 2011 - Journal of Symbolic Logic 76 (3):870 - 882.
    It is known that for all finite n ≥ 5, there are relation algebras with n-dimensional relational bases but no weak representations. We prove that conversely, there are finite weakly representable relation algebras with no n-dimensional relational bases. In symbols: neither of the classes RA n and wRRA contains the other.
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  24.  20
    All Normal Extensions of S5-Squared Are Finitely Axiomatizable.Nick Bezhanishvili & Ian Hodkinson - 2004 - Studia Logica 78 (3):443-457.
    We prove that every normal extension of the bi-modal system S52 is finitely axiomatizable and that every proper normal extension has NP-complete satisfiability problem.
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  25.  28
    Finite H-Dimension Does Not Imply Expressive Completeness.Ian Hodkinson - 1994 - Journal of Philosophical Logic 23 (5):535 - 573.
    A conjecture of Gabbay (1981) states that any class of flows of time having the property known as finite H-dimension admits a finite set of expressively complete one-dimensional temporal connectives. Here we show that the class of 'circular' structures refutes the generalisation of this conjecture to Kripke frames. We then construct from this class, by a general method, a new class of irreflexive transitive flows of time that refutes the original conjecture. Our paper includes full descriptions of a method for (...)
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  26.  36
    Mosaics and Step-by-Step| Remarks onA Modal Logic of Relations' by Venema & Marx.Robin Hirsch & Ian Hodkinson - 1999 - In E. Orłowska (ed.), Logic at Work. Heidelberg.
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  27.  6
    Complexity of Monodic Guarded Fragments Over Linear and Real Time.Ian Hodkinson - 2006 - Annals of Pure and Applied Logic 138 (1):94-125.
    We show that the satisfiability problem for the monodic guarded, loosely guarded, and packed fragments of first-order temporal logic with equality is 2Exptime-complete for structures with arbitrary first-order domains, over linear time, dense linear time, rational number time, and some other classes of linear flows of time. We then show that for structures with finite first-order domains, these fragments are also 2Exptime-complete over real number time and hence over most of the commonly used linear flows of time, including the natural (...)
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  28.  5
    Axiomatising Various Classes of Relation and Cylindric Algebras.Robin Hirsch & Ian Hodkinson - 1997 - Logic Journal of the IGPL 5 (2):209-229.
    We outline a simple approach to axiomatising the class of representable relation algebras, using games. We discuss generalisations of the method to cylindric algebras, homogeneous and complete representations, and atom structures of relation algebras.
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  29.  18
    On Canonical Modal Logics That Are Not Elementarily Determined.Robert Goldblatt, Ian Hodkinson & Yde Venema - 2003 - Logique Et Analyse 181:77-101.
  30.  6
    The Finite Model Property for Logics with the Tangle Modality.Robert Goldblatt & Ian Hodkinson - 2018 - Studia Logica 106 (1):131-166.
    The tangle modality is a propositional connective that extends basic modal logic to a language that is expressively equivalent over certain classes of finite frames to the bisimulation-invariant fragments of both first-order and monadic second-order logic. This paper axiomatises several logics with tangle, including some that have the universal modality, and shows that they have the finite model property for Kripke frame semantics. The logics are specified by a variety of conditions on their validating frames, including local and global connectedness (...)
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  31. Complete Representations in Algebraic Logic.Robin Hirsch & Ian Hodkinson - 1997 - Journal of Symbolic Logic 62 (3):816-847.
    A boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension are elementary.
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  32. Step by Step-Building Representations in Algebraic Logic.Robin Hirsch & Ian Hodkinson - 1997 - Journal of Symbolic Logic 62 (1):225-279.
    We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterized according to the outcome of certain games. The Lyndon conditions defining representable relation algebras and a similar schema for cylindric algebras are derived. Finite relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is $\omega$-categorical. We (...)
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  33.  26
    The K-Variable Property is Stronger Than H-Dimension K.Ian Hodkinson & András Simon - 1997 - Journal of Philosophical Logic 26 (1):81-101.
    We study the notion of H-dimension and the formally stronger k-variable property, as considered by Gabbay, Immerman and Kozen. We exhibit a class of flows of time that has H-dimension 3, and admits a finite expressively complete set of onedimensional temporal connectives, but does not have the k-variable property for any finite k.
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  34.  12
    Nick Bezhanishvili.Ian Hodkinson - 2004 - Studia Logica 78:443-457.
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  35.  12
    On Canonicity and Completions of Weakly Representable Relation Algebras.Ian Hodkinson & Szabolcs Mikulás - 2012 - Journal of Symbolic Logic 77 (1):245-262.
    We show that the variety of weakly representable relation algebras is neither canonical nor closed under Monk completions.
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  36.  15
    The Modal Logic of Affine Planes is Not Finitely Axiomatisable.Ian Hodkinson & Altaf Hussain - 2008 - Journal of Symbolic Logic 73 (3):940-952.
    We consider a modal language for affine planes, with two sorts of formulas (for points and lines) and three modal boxes. To evaluate formulas, we regard an affine plane as a Kripke frame with two sorts (points and lines) and three modal accessibility relations, namely the point-line and line-point incidence relations and the parallelism relation between lines. We show that the modal logic of affine planes in this language is not finitely axiomatisable.
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  37.  19
    A Bisimulation Characterization Theorem for Hybrid Logic with the Current-State Binder.Ian Hodkinson & Hicham Tahiri - 2010 - Review of Symbolic Logic 3 (2):247-261.
    We prove that every first-order formula that is invariant under quasi-injective bisimulations is equivalent to a formula of the hybrid logic . Our proof uses a variation of the usual unravelling technique. We also briefly survey related results, and show in a standard way that it is undecidable whether a first-order formula is invariant under quasi-injective bisimulations.
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  38.  19
    The Bounded Fragment and Hybrid Logic with Polyadic Modalities.Ian Hodkinson - 2010 - Review of Symbolic Logic 3 (2):279-286.
    We show that the bounded fragment of first-order logic and the hybrid language with and operators are equally expressive even with polyadic modalities, but that their fragments are equally expressive only for unary modalities.
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  39.  7
    L. Csirmaz, D. Gabbay, M. De Rijke, Eds., Logic Colloquium '92, Studies in Logic Language, and Information'.Ian Hodkinson - 1997 - Journal of Logic, Language and Information 6 (4):453-457.
  40.  3
    Languages, Meta-Languages and METATEM, A Discussion Paper.Howard Barringer, Graham Gough, Derek Brough, Dov Gabbay & Ian Hodkinson - 1996 - Logic Journal of the IGPL 4 (2):255-272.
    Meta-languages are vital to the development and usage of formal systems, and yet the nature of meta-languages and associated notions require clarification. Here we attempt to provide a clear definition of the requirements for a language to be a meta-language, together with consideration of issues of proof theory, model theory and interpreters for such a language.
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  41.  3
    Games, Logic, and Constructive Sets, Edited by Mints G. And Muskens R., CSLI Lecture Notes, Vol. 161. CSLI Publications, Stanford, CA, 2003, Xii+ 128 Pp. [REVIEW]Ian Hodkinson - 2005 - Bulletin of Symbolic Logic 11 (3):439-442.
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  42.  1
    Axiomatizing Hybrid Logic Using Modal Logic.Ian Hodkinson & Louis Paternault - 2010 - Journal of Applied Logic 8 (4):386-396.
  43. The Tangled Derivative Logic of the Real Line and Zero-Dimensional Space.Robert Goldblatt & Ian Hodkinson - 2016 - In Lev Beklemishev, Stéphane Demri & András Máté (eds.), Advances in Modal Logic, Volume 11. CSLI Publications. pp. 342-361.
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  44. Mosaics and Step-by-Step. Remarks on “A Modal Logic of Relations”.Robin Hirsch, Ian Hodkinson, Maarten Marx, Szabolsc Mikulás & Mark Reynolds - 1999 - In E. Orłowska (ed.), Logic at Work. Heidelberg.
     
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  45. Relation Algebras by Games.Robin Hirsch & Ian Hodkinson - 2003 - Bulletin of Symbolic Logic 9 (4):515-520.
     
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  46. Games, Logic, and Constructive Sets. [REVIEW]Ian Hodkinson - 2005 - Bulletin of Symbolic Logic 11 (3):439-441.
  47. J. Väänänen, Models and Games.Ian Hodkinson - 2012 - Bulletin of Symbolic Logic 18 (3):406.
  48. Models and Games. [REVIEW]Ian Hodkinson - 2012 - Bulletin of Symbolic Logic 18 (3):406-407.
  49. REVIEWS-Games, Logic, and Constructive Sets.Ian Hodkinson - 2005 - Bulletin of Symbolic Logic 11 (3):439-441.
     
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