29 found
Order:
  1.  59
    (1 other version)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.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   29 citations  
  2. (1 other version)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 (...)
    Direct download (9 more)  
     
    Export citation  
     
    Bookmark   14 citations  
  3.  89
    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 (...)
    Direct download (10 more)  
     
    Export citation  
     
    Bookmark   12 citations  
  4.  17
    The neat embedding problem for algebras other than cylindric algebras and for infinite dimensions.Robin Hirsch & Tarek Sayed Ahmed - 2014 - Journal of Symbolic Logic 79 (1):208-222.
  5.  35
    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 (...))
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   11 citations  
  6. (1 other version)Relation Algebras by Games.Robin Hirsch & Ian Hodkinson - 2003 - Bulletin of Symbolic Logic 9 (4):515-520.
     
    Export citation  
     
    Bookmark   10 citations  
  7.  35
    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 (...)
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   10 citations  
  8.  53
    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.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   6 citations  
  9.  76
    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 (...)
    Direct download (10 more)  
     
    Export citation  
     
    Bookmark   4 citations  
  10.  17
    Relation algebras of intervals.Robin Hirsch - 1996 - Artificial Intelligence 83 (2):267-295.
  11.  23
    Ordered domain algebras.Robin Hirsch & Szabolcs Mikulás - 2013 - Journal of Applied Logic 11 (3):266-271.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  12.  46
    Relation Algebra Reducts of Cylindric Algebras and Complete Representations.Robin Hirsch - 2007 - Journal of Symbolic Logic 72 (2):673 - 703.
    We show, for any ordinal γ ≥ 3, that the class RaCAγ is pseudo-elementary and has a recursively enumerable elementary theory. ScK denotes the class of strong subalgebras of members of the class K. We devise games, Fⁿ (3 ≤ n ≤ ω), G, H, and show, for an atomic relation algebra A with countably many atoms, that Ǝ has a winning strategy in Fω(At(A)) ⇔ A ∈ ScRaCAω, Ǝ has a winning strategy in Fⁿ(At(A)) ⇐ A ∈ ScRaCAn, Ǝ (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  13.  26
    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.
    Direct download  
     
    Export citation  
     
    Bookmark   3 citations  
  14.  87
    A finite relation algebra with undecidable network satisfaction problem.Robin Hirsch - 1999 - Logic Journal of the IGPL 7 (4):547-554.
    We define a finite relation algebra and show that the network satisfaction problem is undecidable for this algebra.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  15. Positive fragments of relevance logic and algebras of binary relations.Robin Hirsch & Szabolcs Mikulás - 2011 - Review of Symbolic Logic 4 (1):81-105.
    We prove that algebras of binary relations whose similarity type includes intersection, union, and one of the residuals of relation composition form a nonfinitely axiomatizable quasivariety and that the equational theory is not finitely based. We apply this result to the problem of the completeness of the positive fragment of relevance logic with respect to binary relations.
    Direct download (7 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  16.  26
    First-Order Axiomatisations of Representable Relation Algebras Need Formulas of Unbounded Quantifier Depth.Rob Egrot & Robin Hirsch - 2022 - Journal of Symbolic Logic 87 (3):1283-1300.
    Using a variation of the rainbow construction and various pebble and colouring games, we prove that RRA, the class of all representable relation algebras, cannot be axiomatised by any first-order relation algebra theory of bounded quantifier depth. We also prove that the class At(RRA) of atom structures of representable, atomic relation algebras cannot be defined by any set of sentences in the language of RA atom structures that uses only a finite number of variables.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  17.  17
    Algebra of Non-deterministic Programs: Demonic Operations, Orders and Axioms.Robin Hirsch, Szabolcs Mikulás & Tim Stokes - 2022 - Logic Journal of the IGPL 30 (5):886-906.
    Demonic composition, demonic refinement and demonic union are alternatives to the usual ‘angelic’ composition, angelic refinement (inclusion) and angelic (usual) union defined on binary relations. We first motivate both the angelic and the demonic via an analysis of the behaviour of non-deterministic programs, with the angelic associated with partial correctness and demonic with total correctness, both cases emerging from a richer algebraic model of non-deterministic programs incorporating both aspects. Zareckiĭ has shown that the isomorphism class of algebras of binary relations (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  18.  48
    Undecidability of representability as binary relations.Robin Hirsch & Marcel Jackson - 2012 - Journal of Symbolic Logic 77 (4):1211-1244.
    In this article we establish the undecidability of representability and of finite representability as algebras of binary relations in a wide range of signatures. In particular, representability and finite representability are undecidable for Boolean monoids and lattice ordered monoids, while representability is undecidable for Jónsson's relation algebra. We also establish a number of undecidability results for representability as algebras of injective functions.
    Direct download (7 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  19.  35
    Meet-completions and ordered domain algebras.R. Egrot & Robin Hirsch - 2015 - Logic Journal of the IGPL 23 (4):584-600.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  20.  44
    Corrigendum to:“Relation algebra reducts of cylindric algebras and complete representations”.Robin Hirsch - 2013 - Journal of Symbolic Logic 78 (4):1345-1346.
  21.  36
    From points to intervals.Robin Hirsch - 1994 - Journal of Applied Non-Classical Logics 4 (1):7-27.
  22.  78
    Meet-completions and representations of ordered domain algebras.Robin Hirsch & R. Egrot - unknown
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  23. 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.
     
    Export citation  
     
    Bookmark  
  24. 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.
     
    Export citation  
     
    Bookmark  
  25. (1 other version)On the number of variables required for proofs.Robin Hirsch, I. Hodkinson & Roger Maddux - 2002 - Journal of Symbolic Logic 67 (1):197-213.
     
    Export citation  
     
    Bookmark   1 citation  
  26.  16
    Tractable approximations for temporal constraint handling.Robin Hirsch - 2000 - Artificial Intelligence 116 (1-2):287-295.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  27.  27
    The temporal logic of two dimensional Minkowski spacetime is decidable.Robin Hirsch & Mark Reynolds - 2018 - Journal of Symbolic Logic 83 (3):829-867.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  28.  9
    Undecidability of Algebras of Binary Relations.Robin Hirsch, Ian Hodkinson & Marcel Jackson - 2021 - In Judit Madarász & Gergely Székely (eds.), Hajnal Andréka and István Németi on Unity of Science: From Computing to Relativity Theory Through Algebraic Logic. Springer. pp. 267-287.
    Let S be a signature of operations and relations definable in relation algebra, let R be the class of all S-structures isomorphic to concrete algebras of binary relations with concrete interpretations for symbols in S, and let F be the class of S-structures isomorphic to concrete algebras of binary relations over a finite base. To prove that membership of R or F for finite S-structures is undecidable, we reduce from a known undecidable problem—here we use the tiling problem, the partial (...)
    No categories
    Direct download  
     
    Export citation  
     
    Bookmark  
  29.  80
    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.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark