Abstract
Term-generic logic (TGL) is a first-order logic parameterized with terms defined axiomatically (rather than constructively), by requiring them to only provide generic notions of free variable and substitution satisfying reasonable properties. TGL has a complete Gentzen system generalizing that of first-order logic. A certain fragment of TGL, called Horn 2, possesses a much simpler Gentzen system, similar to traditional typing derivation systems of λ-calculi. Horn 2 appears to be sufficient for defining a whole plethora of λ-calculi as theories inside the logic. Within intuitionistic TGL, a Horn 2 specification of a calculus is likely to be adequate by default. A bit of extra effort shows adequacy w.r.t. classic TGL as well, endowing the calculus with a complete loose semantics.
Supported in part by NSF grants CCF-0448501, CNS-0509321 and CNS-0720512, by NASA contract NNL08AA23C, by the Microsoft/Intel funded Universal Parallel Computing Research Center at UIUC, and by several Microsoft gifts.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abadi, M., et al.: Explicit substitutions. J. Funct. Program. 1(4), 375–416 (1991)
Aczel, P.: Frege structures and the notions of proposition, truth and set. In: The Kleene Symposium, pp. 31–59. North Holland, Amsterdam (1980)
Barendregt, H.: Introduction to generalized type systems. J. Funct. Program. 1(2), 125–154 (1991)
Barendregt, H.P.: The Lambda Calculus. North-Holland, Amsterdam (1984)
Birkhoff, G.: On the structure of abstract algebras. Proceedings of the Cambridge Philosophical Society 31, 433–454 (1935)
Bruijn, N.: λ-calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem. Indag. Math. 34(5), 381–392 (1972)
Diaconescu, R.: Institution-independent Model Theory. Birkhauser, Basel (2008)
Dowek, G., Hardin, T., Kirchner, C.: Binding logic: Proofs and models. In: Baaz, M., Voronkov, A. (eds.) LPAR 2002. LNCS, vol. 2514, pp. 130–144. Springer, Heidelberg (2002)
Fiore, M., Plotkin, G., Turi, D.: Abstract syntax and variable binding. In: Proc. 14th LICS Conf., pp. 193–202. IEEE, Los Alamitos (1999)
Gallier, J.H.: Logic for computer science. Foundations of automatic theorem proving. Harper & Row (1986)
Girard, J.-Y.: Une extension de l’interpretation de Gödel a l’analyse, et son application a l’elimination des coupure dans l’analyse et la theorie des types. In: Fenstad, J. (ed.) 2nd Scandinavian Logic Symposium, pp. 63–92. North Holland, Amsterdam (1971)
Goguen, J., Burstall, R.: Institutions: Abstract model theory for specification and programming. Journal of the ACM 39(1), 95–146 (1992)
Goguen, J., Meseguer, J.: Order-sorted algebra I. Theoretical Computer Science 105(2), 217–273 (1992)
Gunter, C.A.: Semantics of Programming Languages. MIT Press, Cambridge (1992)
Harper, R., Honsell, F., Plotkin, G.: A framework for defining logics. In: Proc. 2nd LICS Conf., pp. 194–204. IEEE, Los Alamitos (1987)
Hofmann, M.: Semantical analysis of higher-order abstract syntax. In: Proc. 14th LICS Conf., pp. 204–213. IEEE, Los Alamitos (1999)
McDowell, R.C., Miller, D.A.: Reasoning with higher-order abstract syntax in a logical framework. ACM Trans. Comput. Logic 3(1), 80–136 (2002)
Meseguer, J.: General logics. In: Ebbinghaus, H.-D., et al. (eds.) Proceedings, Logic Colloquium 1987, pp. 275–329. North-Holland, Amsterdam (1989)
Miller, D., Nadathur, G., Pfenning, F., Scedrov, A.: Uniform proofs as a foundation for logic programming. Ann. Pure Appl. Logic 51(1-2), 125–157 (1991)
Mitchell, J.C.: Foundations for Programming Languages. MIT Press, Cambridge (1996)
Pfenning, F., Elliot, C.: Higher-order abstract syntax. In: PLDI 1988, pp. 199–208. ACM Press, New York (1988)
Pitts, A.M.: Nominal logic: A first order theory of names and binding. In: Kobayashi, N., Pierce, B.C. (eds.) TACS 2001. LNCS, vol. 2215, pp. 219–242. Springer, Heidelberg (2001)
Popescu, A., Roşu, G.: Term-generic logic. Tech. Rep. Univ. of Illinois at Urbana-Champaign UIUCDCS-R-2009-3027 (2009)
Reynolds, J.C.: Towards a theory of type structure. In: Robinet, B. (ed.) Programming Symposium. LNCS, vol. 19, pp. 408–423. Springer, Heidelberg (1974)
Roşu, G.: Extensional theories and rewriting. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1066–1079. Springer, Heidelberg (2004)
Sannella, D., Tarlecki, A.: Foundations of Algebraic Specifications and Formal Program Development. To appear in Cambridge University Press (Ask authors for current version at tarlecki@mimuw.edu.pl)
Sun, Y.: An algebraic generalization of Frege structures - binding algebras. Theoretical Computer Science 211(1-2), 189–232 (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Popescu, A., Roşu, G. (2009). Term-Generic Logic. In: Corradini, A., Montanari, U. (eds) Recent Trends in Algebraic Development Techniques. WADT 2008. Lecture Notes in Computer Science, vol 5486. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03429-9_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-03429-9_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03428-2
Online ISBN: 978-3-642-03429-9
eBook Packages: Computer ScienceComputer Science (R0)