## Works by Robin Hirsch

19 found
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1. Robin Hirsch & Ian Hodkinson (1997). Step by Step-Building Representations in Algebraic Logic. 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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2. Robin Hirsch, Ian Hodkinson & Roger D. Maddux (2002). Relation Algebra Reducts of Cylindric Algebras and an Application to Proof Theory. 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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3. Robin Hirsch, Ian Hodkinson & Roger D. Maddux (2002). Provability with Finitely Many Variables. 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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4. Robin Hirsch & Ian Hodkinson (1997). Complete Representations in Algebraic Logic. 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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5. Robin Hirsch & Ian Hodkinson (2009). Strongly Representable Atom Structures of Cylindric Algebras. 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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6. Robin Hirsch & Ian Hodkinson (2000). Relation Algebras with N-Dimensional Relational Bases. 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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7. Robin Hirsch & Ian Hodkinson (2001). Relation Algebras From Cylindric Algebras, I. 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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8. Robin Hirsch & Ian Hodkinson (1999). Mosaics and Step-by-Step| Remarks onA Modal Logic of Relations' by Venema & Marx. In E. Orłowska (ed.), Logic at Work. Heidelberg
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9. Robin Hirsch (1999). A Finite Relation Algebra with Undecidable Network Satisfaction Problem. 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.

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10. Robin Hirsch & Ian Hodkinson (2001). Relation Algebras From Cylindric Algebras, II. 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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11. Robin Hirsch & Szabolcs Mikulás (2011). Positive Fragments of Relevance Logic and Algebras of Binary Relations. 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.

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12. Robin Hirsch & Ian Hodkinson (1997). Axiomatising Various Classes of Relation and Cylindric Algebras. 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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13. Robin Hirsch & Marcel Jackson (2012). Undecidability of Representability as Binary Relations. 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.

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14. Robin Hirsch, Ian Hodkinson & Roger D. Maddux (2011). Weak Representations of Relation Algebras and Relational Bases. 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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15. Robin Hirsch (1994). From Points to Intervals. Journal of Applied Non-Classical Logics 4 (1):7-27.

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16. Robin Hirsch & Szabolcs Mikulás (2013). Ordered Domain Algebras. Journal of Applied Logic 11 (3):266-271.

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17. Robin Hirsch (2007). Relation Algebra Reducts of Cylindric Algebras and Complete Representations. 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 (...)

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19. Robin Hirsch & Tarek Sayed Ahmed (2014). The Neat Embedding Problem for Algebras Other Than Cylindric Algebras and for Infinite Dimensions. Journal of Symbolic Logic 79 (1):208-222.