Skip to main content
SpringerLink
Log in

From IF to BI

A tale of dependence and separation

  • Published:
Synthese Aims and scope Submit manuscript

Abstract

We take a fresh look at the logics of informational dependence and independence of Hintikka and Sandu and Väänänen, and their compositional semantics due to Hodges. We show how Hodges’ semantics can be seen as a special case of a general construction, which provides a context for a useful completeness theorem with respect to a wider class of models. We shed some new light on each aspect of the logic. We show that the natural propositional logic carried by the semantics is the logic of Bunched Implications due to Pym and O’Hearn, which combines intuitionistic and multiplicative connectives. This introduces several new connectives not previously considered in logics of informational dependence, but which we show play a very natural rôle, most notably intuitionistic implication. As regards the quantifiers, we show that their interpretation in the Hodges semantics is forced, in that they are the image under the general construction of the usual Tarski semantics; this implies that they are adjoints to substitution, and hence uniquely determined. As for the dependence predicate, we show that this is definable from a simpler predicate, of constancy or dependence on nothing. This makes essential use of the intuitionistic implication. The Armstrong axioms for functional dependence are then recovered as a standard set of axioms for intuitionistic implication. We also prove a full abstraction result in the style of Hodges, in which the intuitionistic implication plays a very natural rôle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Armstrong, W. W. (1974). Dependency structures of data base relationships. In Information processing 74. Proc. IFIP Congress (pp. 580–583). Netherlands: North Holland.

  • Biering, B., Birkedal, L., Trop-Smith, N. (2007). BI-hyperdoctrines, higher-order separation logic, and abstraction. ACM Transactions on Programming Languages and Systems, 29(5), (to appear).

  • Cameron P., Hodges W. (2001) Some combinatorics of imperfect information. Journal of Symbolic Logic 66(2): 673–684

    Article  Google Scholar 

  • Curry H.B., Feys R. (1958) Combinatory logic. Volume 1, studies in logic and the foundations of mathematics. Netherlands, North Holland

    Google Scholar 

  • Davey B.A., Priestley H.A. (2002) Introduction to lattices and order (2nd ed). Cambridge University Press, Cambridge

    Google Scholar 

  • Fagin R. (1977) Functional dependencies in a relational data base and propositional logic. IBM Journal of Research and Development 21(6): 543–544

    Article  Google Scholar 

  • Girard J.-Y. (1987) Linear logic. Theoretical Computer Science 50: 1–102

    Article  Google Scholar 

  • Henkin L. (1950) Completeness in the theory of types. Journal of Symbolic Logic 15: 81–91

    Article  Google Scholar 

  • Henkin, L. (1961). Some remarks on infinitely long formulas. In Infinitistic methods. Proc. symposium on foundations of mathematics (pp. 167–183). Pergamon.

  • Hintikka J. (1998) The principles of mathematics revisited. Cambridge University Press, Cambridge

    Google Scholar 

  • Hintikka J. (2002) Hyperclassical logic (a.k.a. IF logic) and its implications for logical theory. Bulletin of Symbolic Logic 8(3): 404–423

    Article  Google Scholar 

  • Hintikka J., Sandu G. et al (1989) Informational independence as a semantical phenomenon. In: Fenstad J.E. (eds) Logic, methodology and philosophy of science VIII. Elsevier, London, pp 571–589

    Chapter  Google Scholar 

  • Hintikka J., Sandu G. (1996) Game-theoretical semantics. In: van Benthem J., ter Meulen A. (eds) Handbook of logic and language. Elsevier, London

    Google Scholar 

  • Hodges W. (1997a) Compositional semantics for a language of imperfect information. Logic Journal of the IGPL 5(4): 539–563

    Article  Google Scholar 

  • Hodges W. (1997b) Some strange quantifiers. In: Mycielski G.R.J., Salomaa A. (eds) Structures in logic and computer science. Lecture notes in computer science (Vol. 1261). Springer, New York, pp 51–65

    Google Scholar 

  • Lang, S. (1964). Algebraic numbers. Addison-Wesley.

  • Lawvere F.W. (1969) Adjointness in foundations. Dialectica 23: 281–296

    Article  Google Scholar 

  • Lindström P. (1969) On extensions of elementary logic. Theoria 35: 1–11

    Article  Google Scholar 

  • Milner R. (1977) Fully abstract models of typed lambda-calculi. Theoretical Computer Science 4: 1–22

    Article  Google Scholar 

  • Mitchell J.C. (1996) Foundations for programming languages. MIT Press, USA

    Google Scholar 

  • Mitchell W.P.R., Simmons H. (2001) Monoid based semantics for linear formulas. Journal of Symbolic Logic 66(4): 1597–1619

    Article  Google Scholar 

  • O’Hearn P.W., Pym D.J. (1999) The logic of bunched implications. Bulletin of Symbolic Logic 5(2): 215–244

    Article  Google Scholar 

  • Pitts A. (2000) Categorical logic. In: Abramsky S., Gabbay D., Maibaum T. (eds) Handbook of logic in computer science (Vol. 5). Oxford University Press, Oxford, pp 39–128

    Google Scholar 

  • Plotkin G.D. (1977) LCF considered as a programming language. Theoretical Computer Science 5: 223–255

    Article  Google Scholar 

  • Pym D.J. (2002) The semantics and proof theory of the logic of bunched implications. Applied logic series (Vol. 26). Kluwer, Netherlands

    Google Scholar 

  • Pym D.J., O’Hearn P.W., Yang H. (2004) Possible worlds and resources: The semantics of BI. Theoretical Computer Science 315: 257–305

    Article  Google Scholar 

  • Reynolds, J. (2002). Separation logic: A logic for shared mutable data structures. In Proc. LiCS 2002. IEEE.

  • Rosenthal, K. I. (1990). Quantales and their applications. Pitman research notes in mathematics (No. 234). Longman Scientific and Technical.

  • Scott, D. S. (1969). Outline of a mathematical theory of computation. Technical Monograph PRG-2, Oxford University Computing Laboratory.

  • Tarski A. (1936) Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica 1: 261–405

    Google Scholar 

  • Tarski, A., Vaught, R. (1956) Arithmetical extensions of relational systems. Compositio Mathematica, 81–102

  • Urquhart A. (1972) Semantics for relevant logics. Journal of Symbolic Logic 37(1): 159–169

    Article  Google Scholar 

  • Väänänen J. (2001) Second-order logic and foundations of mathematics. Bulletin of Symbolic Logic 7(4): 504–520

    Article  Google Scholar 

  • Väänänen J. (2007) Dependence logic. London Mathematical Society student texts (Vol. 70). Cambridge University Press, Cambridge

    Google Scholar 

  • Winskel G. (1993) The formal semantics of programming languages. MIT Press, USA

    Google Scholar 

  • Yetter D.N. (1990) Quantales and (noncommutative) linear logic. Journal of Symbolic Logic 55(1): 41–64

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford, OX1 3QD, UK

    Samson Abramsky

  2. ILLC, Amsterdam, The Netherlands

    Jouko Väänänen

Authors
  1. Samson Abramsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jouko Väänänen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Samson Abramsky.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Abramsky, S., Väänänen, J. From IF to BI. Synthese 167, 207–230 (2009). https://doi.org/10.1007/s11229-008-9415-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11229-008-9415-6

Keywords

Search

Navigation