@phdthesis{fourman74, author = {Michael Paul Fourman}, title = {Connections between category theory and logic}, school = {Oxford University}, year = {1974}, month = {November}, pages = {v + 102}, note = {D.Phil. thesis}, address = {Linacre College, Oxford} }

@incollection{fourman75, author = {Michael P. Fourman}, title = {Sheaf models for set theory}, booktitle = {Tagungsbericht Oberwolfach}, note = {(abstract)}, month = {July}, year = {1975} }

@article{fourman77a, author = {Michael P. Fourman}, title = {The logic of topoi and sheaves}, note = {(abstract)}, journal = {Journal of Symbolic Logic}, volume = {42}, number = {1}, year = {1977}, pages = {138--139} }

@article{fourman77b, author = {Michael P. Fourman}, title = {Comparaison des r\'eeles d'un topos; structures lisses sur un topos \'{e}l\'{e}mentaire}, journal = {Cah. G\'{e}om. Top. Diff.}, year = {1977}, volume = {16}, number = {3}, pages = {233--239} }

@incollection{fourman77c, author = {Michael P. Fourman}, title = {The logic of topoi}, booktitle = {Handbook of Mathematical Logic}, publisher = {North-Holland}, chapter = {D-6}, year = {1977}, editor = {Jon Barwise}, pages = {1053--1090}, isbn = {0-7204-2285-X} }

@article{fourman77d, author = {Michael P. Fourman and C. Rousseau}, title = {Parameters in complex analysis and the logic of topoi}, note = {(abstract)}, journal = {Journal of Symbolic Logic}, volume = {42}, number = {3}, year = {1977}, pages = {499} }

@article{fourman77e, author = {Michael P. Fourman}, title = {Approximation Sheaves as internal Banach-Spaces}, note = {(abstract) 77T-B67}, pages = {A-224}, journal = {Notices Amer. Math. Soc.}, volume = {24}, number = {2}, year = {1977} }

@article{fourman77f, author = {Michael P. Fourman}, title = {A representation for internal sober spaces}, note = {(abstract) 77T-G48}, pages = {A-301}, journal = {Notices Amer. Math. Soc.}, volume = {24}, number = {3}, year = {1977} }

@incollection{fourman77g, author = {Michael P. Fourman}, title = {Formal Spaces}, year = {1977}, pages = {38--39}, url = {http://homepages.inf.ed.ac.uk/mfourman/research/publications/pdf/fourman77-formal-spaces.pdf}, booktitle = {Tagungsbericht: {K}ategorien, {O}berwolfach, 31.7--6.8.1977} }

@inproceedings{fourman79a, author = {Michael P. Fourman and Dana S. Scott}, title = {Sheaves and Logic}, crossref = {fourman79}, year = {1979}, pages = {302-401}, doi = {10.1007/BFb0061824} }

@inproceedings{fourman79b, author = {Michael P. Fourman and J.M.E. Hyland}, title = {Sheaf Models for Analysis}, crossref = {fourman79}, year = {1979}, pages = {280-301}, url = {http://homepages.inf.ed.ac.uk/mfourman/research/publications/pdf/fourman79-sheaf-models-for-analysis.pdf}, doi = {10.1007/BFb0061823} }

@proceedings{fourman79, booktitle = {Applications of Sheaves}, title = {Proceedings of the Research Symposium on Applications of Sheaf Theory to Logic, Algebra, and Analysis, Durham, July 9–21, 1977}, year = {1979}, editor = {Michael Fourman and Christopher Mulvey and Dana Scott}, volume = {753/1979}, series = {Lecture Notes in Mathematics}, publisher = {Springer}, url = {http://dx.doi.org/10.1007/BFb0061811}, isbn = {978-3-540-09564-4}, organization = {London Mathematical Society} }

@article{fourman80, author = {Michael P. Fourman}, title = {Sheaf Models for Set Theory}, pages = {91-101}, journal = {Journal of Pure and Applied Alegebra}, volume = {19}, year = {1980} }

@article{fourman81, author = {Michael P. Fourman}, title = {Continuous Truth}, note = {(abstract) 81-T-03-135}, journal = {Abstracts Amer. Math. Soc.}, year = {1981} }

@article{fourman82a, author = {Michael P. Fourman and A.Scedrov}, title = {The ``World's Simplest Axiom of Choice'' fails}, pages = {325-332}, journal = {Manuscripta Mathematica}, volume = {38}, year = {1982}, url = {http://homepages.inf.ed.ac.uk/mfourman/research/publications/pdf/fourman82-worlds-simplest-axiom-of-choice-fails.pdf}, abstract = {We use topos-theoretic methods to show that intuitionistic set theory with coutable or dependent choice does not imply that every family, all of whose elements are doubletons and which has at most one element, has a choice function.} }

@article{fourman82b, author = {Michael P. Fourman}, title = {A model for the theory of choice sequences (CS)}, note = {(abstract) 82-T-03-80}, pages = {183}, journal = {Abstracts Amer. Math. Soc.}, year = {1982} }

@inproceedings{fourman82c, author = {Michael P. Fourman}, title = {Notions of Choice Sequence}, booktitle = {L.E.J. Brouwer Centenary Symposium}, editor = {D. van Dalen and A. Troelstra}, publisher = {North-Holland}, year = {1982}, pages = {91--105}, url = {http://homepages.inf.ed.ac.uk/mfourman/research/publications/pdf/fourman82-notions-of-choice-sequence.pdf}, keyword = {topoi, intuitionism, sheaf model, choice sequence, Grothendieck topology, Beth model; lawless sequence}, abstract = {We use sheaf models to undertake a constructive analysis of the effects of admitting non-constructive choice sequences to mathematics.}, note = {Zbl 0545.03036 (Review by M. Beeson) The mathematical contribution of this paper is the construction of sheaf models for various theories of choice sequences. These models are defined when a category is given, together with a notion of which morphisms "cover" a given object (for the cognoscenti, a Grothendieck topology). The semantics is like Beth models; in fact they form a special case in which the objects of the category are the nodes of the Beth model and there is just one arrow between every pair of order-related nodes. Several specific models are considered; the two simplest are equivalent to the Scott-Moschovakis model, and the Kreisel-Troelstra elimination of lawless sequences, respectively. Another model, found independently by Grayson and by Moerdijk and van der Hoeven, corresponds to the Kreisel-Troelstra elimination of choice sequences. Several choice and continuity principles are verified in these models. The author mentions that Dragalin had, unknown to him at the time, also constructed similar models in 1974. On the philosophical side, the author hopes these models provide a conceptual basis for theories of choice sequences, i.e. a coherent and precise notion of choice sequence.} }

@inproceedings{fourman82d, author = {Michael P. Fourman and R.J. Grayson}, title = {Formal Spaces}, booktitle = {L.E.J. Brouwer Centenary Symposium. Proceedings of the Conference held in Noordwijkerhout, 8--13 June, 1981. }, editor = {D. van Dalen and A. Troelstra}, publisher = {North-Holland}, series = {Studies in Logic and the Foundations of Mathematics}, volume = {110}, abstract = {This paper is an introduction to the constructive theory of locales (or 'pointless spaces') from the point of view of intuitionistic propositional theories. It is shown how such a theory gives rise to a 'Lindenbaum algebra' which is the dual of a locale (called the formal space of the theory), and how properties of the theory are reflected in properties of its formal space. In particular, the question of whether the formal space really is a space (i.e. whether it has enough points) becomes the question of whether the theory is complete (i.e. has enough models), and some well-known examples of this phenomenon are discussed. The paper is marred by an unfortunate wrong definition: for a map of locales to be open, it is not sufficient (unless Y is the one-point space) merely for to have a left adjoint.}, keywords = {constructive theory of locales, pointless spaces, intuitionistic propositional theories, Lindenbaum algebra}, year = {1982}, pages = {107--122} }

@article{fourman83, author = {Michael P. Fourman}, title = {T1 spaces over topological sites}, pages = {223-224}, journal = {J. Pure and Applied Algebra}, volume = {27}, number = {3}, year = {1983}, month = {March}, url = {http://dx.doi.org/10.1016/0022-4049(83)90017-8} }

@inproceedings{fourman84a, author = {Michael P. Fourman}, title = {Continuous truth {I}: non-constructive objects}, booktitle = {Proc. Logic Colloquium, Florence 1982}, editor = {G. Lolli and G. Longo and A. Marcja}, series = {Stud. Logic Found. Math. 112}, publisher = {Elsevier Science Publishers B.V. (North-Holland)}, year = {1984}, pages = {161-180}, url = {http://homepages.inf.ed.ac.uk/mfourman/research/publications/pdf/fourman84-continuous-truth-I.pdf}, note = {Invited Paper}, abstract = {We give a general theory of the logic of potentially infinite objects, derived from a theory of meaning for statements concerning these objects. The paper has two main parts which may be read independently but are intended to complement each other. The first part is philosophical. In it, we discuss the theory of meaning. We believe that even the stunchest realist must view potential infinities operationally. The second part is formal. In it we consider the interpretation of logic in the gros topos of sheaves over the category of separable locales equipped with the open cover topology. We show that general priciples of continuity, local choice and local compactness hold for these models. We conclude with a brief discussion of the philosophical significance of our formal results. They allow us to reconcile our explanation of meaning with the "equivalence thesis", that 'snow is white' is true iff snaow is white.} }

@article{fourman84b, author = {Michael P. Fourman and P.Freyd and A.Scedrov}, title = {Toposes and modal logic}, pages = {1443}, journal = {J. Symbolic Logic}, volume = {49}, number = {4}, month = {December}, year = {1984}, note = {Abstracts of the Annual Meeting of the Association for Symbolic Logic: Boston 1983}, url = {http://links.jstor.org/sici?sici=0022-4812\%28198412\%2949\%3A4\%3C1441\%3AAMOTAF\%3E2.0.CO\%3B2-4} }

@article{fourman85a, author = {Michael P. Fourman}, title = {Evolving Layout}, journal = {I.E.E. Digest}, pages = {3/1--3/4}, volume = {41}, year = {1985}, note = {Colloquium on VLSI Design Methodologies} }

@inproceedings{fourman85b, author = {Michael P. Fourman}, title = {Compaction of Symbolic Layout using Genetic Algorithms}, booktitle = {ICGA}, crossref = {DBLP:conf/icga/1985}, url = {http://homepages.inf.ed.ac.uk/mfourman/research/publications/pdf/fourman85-compaction-using-genetic-algorithms.pdf}, year = {1985}, pages = {141--153} }

@proceedings{DBLP:conf/icga/1985, editor = {John J. Grefenstette}, title = {Proceedings of the 1st International Conference on Genetic Algorithms, Pittsburgh, PA, USA, July 1985}, booktitle = {ICGA}, publisher = {Lawrence Erlbaum Associates}, year = {1985}, isbn = {0-8058-0426-9}, bibsource = {DBLP, http://dblp.uni-trier.de} }

@inproceedings{fourman85c, author = {Michael P. Fourman and C.M. Fourman and G. Musgrave and A.P. Ambler}, title = {An approach to hardware description for the VIDA}, booktitle = {Proc. Silicon Design Conference}, editor = {A.P. Ambler}, address = {Wembley}, year = {1985}, pages = {67--85} }

@inproceedings{fourman86a, author = {Michael P. Fourman}, title = {Strategies for redundancy in {WSI}}, booktitle = {Proc. International Workshop on Wafer-Scale Integration}, editor = {C. Jessop and W. Moore}, publisher = {Adam Hilger}, address = {Boston}, year = {1986}, pages = {72-81} }

@inproceedings{fourman86b, author = {Michael P. Fourman and S. Vickers}, title = {Theories as Categories}, booktitle = {Category Theory and Computer Programming}, editor = {Pitt and Abramsky and Poign\'{e} and Rydeheard}, publisher = {Springer-Verlag}, address = {Berlin}, year = {1986}, pages = {434--448}, note = {Lecture Notes in Computer Science 240} }

@inproceedings{fourman86c, author = {Michael P. Fourman}, title = {Verification using Higher-order Specifications and Transformations}, booktitle = {Proc. Silicon Design Conference}, editor = {A.P. Ambler}, address = {Wembley}, month = {July}, year = {1986} }

@inproceedings{fourman87a, author = {Michael P. Fourman and R.M. Zimmer}, title = {Modular Design as Algebraic Composition}, booktitle = {Intelligent CAD Systems I: Theoretical and Methodological Aspects}, editor = {P.J.W. Ten Hagen and T. Tomiyama}, publisher = {Springer-Verlag}, address = {Berlin}, year = {1987}, pages = {176--185}, note = {Proceedings Eurographics Workshop on Intelligent CAD} }

@inproceedings{fourman87b, author = {Michael P. Fourman and W.J. Palmer and R.M. Zimmer}, title = {Using Higher-Order Functions to Describe Hardware}, booktitle = {Electronic Design Automation}, editor = {A.P. Ambler}, address = {Wembley}, year = {1987}, pages = {87--94} }

@inproceedings{fourman87c, author = {Michael P. Fourman and R. Holte and W.J. Palmer and R.M. Zimmer}, title = {Top-down Design as Bottom-up Proof}, booktitle = {Electronic Design Automation}, editor = {A.P. Ambler}, address = {Wembley}, year = {1987}, pages = {617--628} }

@inproceedings{fourman87d, author = {Michael P. Fourman and R. Harris and G. Musgrave}, title = {Core Tools for the Next Generation of Electronics {CAD}}, booktitle = {Electronic Design Automation}, editor = {A.P. Ambler}, address = {Wembley}, year = {1987}, pages = {335--340} }

@inproceedings{fourman88a, author = {Michael P. Fourman and W.J. Palmer and R.M. Zimmer}, title = {Core Tools for the Next Generation of Electronics {CAD}}, booktitle = {Alvey Technical Conference}, address = {Swansea}, year = {1988}, pages = {428--430} }

@inproceedings{fourman88b, author = {Michael P. Fourman and W.J. Palmer and R.M. Zimmer}, title = {Proof and Synthesis}, booktitle = {International Conference on Circuit Design}, address = {Rye, New York}, year = {1988}, pages = {600--603}, url = {http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=25771} }

@inproceedings{fourman88c, author = {Michael P. Fourman and R. Holte and W.J. Palmer and R.M. Zimmer}, title = {Interactive Behavioural Synthesis}, booktitle = {Electronic Design Automation}, editor = {A.P. Ambler}, address = {Wembley}, year = {1988}, pages = {317--329} }

@inproceedings{fourman89a, author = {Michael P. Fourman and Eleanor M. Mayger}, title = {Formally based systems design---{I}nteractive hardware scheduling}, year = {1989}, pages = {101--112}, address = {Munich, Germany}, booktitle = {Proceedings of {VLSI} '89} }

@inproceedings{fourman89b, author = {S. Finn and M. Fourman and M. Francis and R. Harris}, title = {Formal system design---interactive synthesis based on computer-assisted formal reasoning}, booktitle = {IMEC-IFIP International Workshop on Applied Formal Methods for Correct VLSI Design, Volume 1}, year = {1989}, month = {November}, editor = {Luc Claesen}, pages = {97--110}, publisher = {Elsevier Science Publishers, B.V. North-Holland, Amsterdam}, address = {Houthalen, Belgium} }

@inproceedings{fourman90, author = {Michael P. Fourman}, title = {Formal System Design}, booktitle = {Formal Methods for {VLSI} Design}, year = {1990}, editor = {J{\o}rgen Staunstrup}, publisher = {North-Holland}, pages = {191--236} }

@inproceedings{fourman92b, author = {Michael P. Fourman}, title = {Formal Design}, booktitle = {Formal Design Methods for {VLSI}}, editor = {J{\o}rgen Staunstrup}, publisher = {North-Holland}, year = {1992} }

@inproceedings{fourman91, author = {Michael P. Fourman and R. Hexsel}, title = {Formal Synthesis}, booktitle = {IV Higher Order Workshop, Banff 1990}, year = {1991}, editor = {Graham Birtwistle}, publisher = {Springer-Verlag}, pages = {245--264} }

@inproceedings{mayger91, author = {E.M. Mayger and Michael P. Fourman}, title = {Integration of Formal Methods with System Design}, booktitle = {{IFIP TC 10/WG} 10.5 International Conference on Very Large Scale Integration}, year = {1991}, editor = {A. Halaas and P.B. Denyer}, address = {Edinburgh, Scotland}, publisher = {Elsevier}, month = {August} }

@inproceedings{chapman95, author = {N. Chapman and S. Finn and Michael P. Fourman}, title = {Datatypes in {L2}}, booktitle = {Higher Order Logic Theorem Proving and its Applications. 7th International Workshop Proceedings}, year = {1994}, editor = {T.F. Melham and J. Camillieri}, publisher = {Springer-Verlag; Berlin, Germany}, address = {Valetta, Malta}, month = {September}, pages = {128--43}, note = {Invited Paper} }

@misc{finn90logic, author = {S. Finn and Michael P. Fourman}, title = {Logic Manual for the {LAMBDA} System}, text = {Simon Finn and Michael P. Fourman. Logic Manual for the LAMBDA System 3.2. Abstract Hardware Ltd., November 1990.}, year = {1990} }

@misc{finn93abstract, author = {S. Finn and M. Fourman}, title = {The {LAMBDA} Logic. Abstract Hardware Limited, September 1993. In LAMBDA 4.3 Reference Manuals.}, year = {1993} }

@inproceedings{phoa92, author = {Wesley Phoa and Michael Fourman}, title = {A proposed categorical semantics for Pure {ML}}, booktitle = {{ICALP} '92 International Colloquium on Automata, Languages, and Programming, Wien Austria}, year = {1992}, publisher = {Springer-Verlag}, editor = {W. Kuich}, series = {Springer {LNCS}}, pages = {533--544}, doi = {10.1007/3-540-55719-9_102}, url = {http://www.lfcs.inf.ed.ac.uk/reports/92/ECS-LFCS-92-213/index.html}, note = {LFCS report ECS-LFCS-92-213} }

@inproceedings{fourman92c, author = {Michael P. Fourman}, title = {The {CAD} tool of the future}, editor = {J. van Leeuwen}, booktitle = {Information Processing 92}, year = {1992}, publisher = {Elsevier}, note = {discussion paper} }

@inproceedings{fourman95, author = {Michael P. Fourman and Hayo Thieleke}, title = {A Proposed Categorical Semantics for {ML} Modules}, booktitle = {Category Theory and Computer Science}, editor = {David Pitt and David E. Rydeheard and Peter Johnstone}, year = {1995}, publisher = {Springer Berlin / Heidelberg}, series = {Lecture Notes in Computer Science}, volume = {953/1995}, pages = {240--252}, isbn = {3-540-60164-3}, doi = {10.1007/3-540-60164-3_30}, issn = {0302-9743 (Print) 1611-3349 (Online)} }

@incollection{fourman96a, author = {Michael P. Fourman}, title = {Proof and design}, booktitle = {Deductive Program Design}, pages = {397-439}, publisher = {Springer}, year = {1996}, editor = {Manfred Broy}, volume = {152}, series = {NATO Advanced Science Institute, Series F: Computer and System Sciences}, address = {Marktoberdorf Germany}, month = {July}, url = {http://www.lfcs.inf.ed.ac.uk/reports/95/ECS-LFCS-95-319/index.html}, note = {Proceedings of the NATO Advanced Study Institute on Deductive Program Design, also available as LFCS report ECS-LFCS-95-319} }

@article{fourman96b, author = {Simon Finn and Michael P. Fourman and John Longley}, title = {Partial Functions in a Total Setting}, journal = {J. Automated Reasoning}, year = {1997}, volume = {18}, note = {Zbl. 0870.68136}, number = {1}, url = {http://www.lfcs.inf.ed.ac.uk/reports/96/ECS-LFCS-96-341/index.html}, pages = {85-104} }

@article{fourman96c, author = {Michael P. Fourman}, title = {Book Review}, journal = {J. Functional Programming}, volume = {6}, number = {1}, year = {1996}, pages = {189--193}, note = {Three books on {ML}} }

@incollection{fourman96d, author = {Nick Chapman and Simon Finn and Michael Fourman}, editor = {Carlos Delgado-Kloos and Werner Damm}, title = {Reasoning about {VHDL}}, booktitle = {Practical Formal Methods for Hardware Design}, publisher = {Springer}, year = {1996}, volume = {1}, series = {Research Reports Esprit}, note = {Project 6128, FORMAT}, isbn = {3-540-62007-9 } }

@inproceedings{fourman2000-propositional-planning, author = {Michael P. Fourman}, title = {Propositional Planning}, booktitle = {Workshop on Model-Theoretic Aproaches to Planning, AIPS 2000}, year = {2000}, month = {April}, page = {10--17}, note = {EDI-INF-RR-0034 \texttt{http://www.inf.ed.ac.uk/publications/report/0034.html}}, url = {http://www.inf.ed.ac.uk/publications/report/0034.html} }

@techreport{mayger89dialog, author = {E. Mayger and M. Francis and R. Harris and G. Musgrave and M. Fourman}, title = {Dialog -- linking formal proof to the design environment}, institution = {Abstract Hardware Ltd.}, address = {Uxbridge, U. K.}, year = {1989} }

@incollection{fourman02, author = {Michael P. Fourman}, title = {informatics}, booktitle = {International Encyclopedia of Information and Library Science (second edition)}, publisher = {Routledge}, year = {2002}, editor = {John Feather and Paul Sturges}, isbn = {01315259010}, url = {http://www.inf.ed.ac.uk/publications/report/0139.html}, note = {Available as EDI-INF-RR-0139} }

@misc{camurati89, author = {P. Camurati and S. Chin and M. Fourman and C. Pixley and P. Prinetto and A. Takahara}, title = {Formal verification: is it practical for realworld design}, text = {Paolo Camurati, Shiu-Kai Chin, Michael Fourman, Carl Pixley, Paolo Prinetto, and Atsushi Takahara. Formal verification: is it practical for realworld design? IEEE Design and Test of Computers, 6(6):50:58, December 1989. Roundtable discussion held at ICCD. BIBLIOGRAPHY 261}, year = {1989} }

@inproceedings{fourman01, author = {Michael P. Fourman}, title = {Propositional Reasoning}, editor = {Tiziana Margaria and Wang Yi}, booktitle = {Tools and Algorithms for the Construction and Analysis of Systems, 7th International Conference, TACAS 2001 Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2001 Genova, Italy, April 2-6, 2001, Proceedings}, publisher = {Springer}, series = {Lecture Notes in Computer Science}, volume = {2031}, year = {2001}, month = {June}, isbn = {3-540-41865-2}, bibsource = {DBLP, http://dblp.uni-trier.de}, note = {Abstract of Invited Lecture}, pages = {23} }

@inproceedings{fourman07a, author = {Michael P. Fourman}, title = {Local Perspectives on Actions}, booktitle = {A Meeting of the Minds: Proceedings of the Workshop on Logic, Rationality and Interaction}, editor = {van Benthem, Johan and Shier Ju and Frank Veltman }, address = {Beijing}, month = {August}, year = {2007}, url = {http://homepages.inf.ed.ac.uk/mfourman/research/publications/pdf/fourman2007-local-perspectives-on-actions.pdf}, note = {ISBN 978-1-904987-48-2 } }

@inproceedings{fourman07b, author = {Michael P. Fourman}, title = {Logics for Action}, crossref = {DBLP:conf/iicai/2007}, pages = {1223-1237}, url = {http://homepages.inf.ed.ac.uk/mfourman/research/publications/pdf/fourman2007-logics-for-action.pdf} }

@proceedings{DBLP:conf/iicai/2007, editor = {Bhanu Prasad}, title = {Proceedings of the 3rd Indian International Conference on Artificial Intelligence, Pune, India, December 17-19, 2007}, booktitle = {IICAI}, publisher = {IICAI}, year = {2007}, month = {December}, isbn = {978-0-9727412-2-4} }

@unpublished{fourman07c, author = {Michael P. Fourman}, title = {Categorical Perspectives}, year = {2007}, month = {October}, url = {http://homepages.inf.ed.ac.uk/mfourman/research/publications/pdf/fourman2007-categorical-perspectives.pdf} }

@inproceedings{fourman2013continuous, author = {Michael P. Fourman}, title = {Continuous Truth {II}}, year = {2013}, editor = {Leonid Libkin}, address = {Darmstadt}, booktitle = {Proc. {W}o{LLIC} 2013}, publisher = {Springer}, series = {{LNCS}}, month = {August}, note = {to appear}, abstract = {In the late 1960s, Dana Scott first showed how the Stone\,-Tarski topological interpretation of Heyting's calculus could be extended to model intuitionistic analysis; in particular Brouwer's continuity principle. In the early '80s we and others outlined a general treatment of non-constructive objects, using \emph{sheaf models}---constructions from topos theory---to model not only Brouwer's non-classical conclusions, but also his creation of ``new mathematical entities''. These categorical models are intimately related to, but more general than Scott's topological model. The primary goal of this paper is to consider the question of iterated extensions. Can we derive new insights by repeating the second act? In \emph{Continuous Truth I}, presented at Logic Colloquium '82 in Florence, we showed that general principles of continuity, local choice and local compactness hold in the gros topos of sheaves over the category of separable locales equipped with the open cover topology. We touched on the question of iteration. Here we develop a more general analysis of iterated categorical extensions, that leads to a reflection schema for statements of predicative analysis. We also take the opportunity to revisit some aspects of both Continuous Truth I and Formal Spaces (Fourman \& Grayson 1982), and correct two long-standing errors therein. }, keywords = {sheaf model, logic, intuitionism, predicative, analysis, topos.}, url = {http://homepages.inf.ed.ac.uk/mfourman/research/publications/pdf/fourman2013-continuous-truth-II.pdf} }

@inproceedings{fourman2013hitting, author = {Michael P. Fourman}, title = {Hitting {B}uneman Circles}, year = {2013}, address = {Edinburgh}, publisher = {Springer}, series = {{LNCS}}, booktitle = {Peter {B}uneman Festschrift 2013}, month = {October}, note = {to appear}, abstract = {We discuss Peter Buneman's suggestion that a fibre connection to the internet --- a hub --- should be available within every circle enclosing a population of at least 2,000 people (a b-circle). This poses the problem of finding a small set, $H$, of hubs, such that \emph{every} b-circle contains a hub. We show that a greedy algorithm does not lead to an optimal set of hubs. Instead it models market forces, which are naturally greedy. An unfettered market will exploit the most profitable communities and, just like a greedy algorithm, leave gaps that it is uneconomic to fill. We describe a geometric heuristic for the discovery of efficient hub placements satisfying a purely combinatorial analogue of Buneman's criterion, and apply it to illustrate the inherent inefficiency of a market-led broadband policy.}, keywords = {hitting set, approximation algorithm, facilities location, broadband, market forces}, url = {http://homepages.inf.ed.ac.uk/mfourman/research/publications/pdf/fourman2013-hitting-buneman-circles.pdf} }

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