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Inhabitation in simply typed lambda-calculus through a lambda-calculus for proof search



A new approach to inhabitation problems in simply typed lambda-calculus is shown, dealing with both decision and counting problems. This approach works by exploiting a representation of the search space generated by a given inhabitation problem, which is in terms of a lambda-calculus for proof search that the authors developed recently. The representation may be seen as extending the Curry–Howard representation of proofs by lambda terms. Our methodology reveals inductive descriptions of the decision problems, driven by the syntax of the proof-search expressions, and produces simple, recursive decision procedures and counting functions. These allow to predict the number of inhabitants by testing the given type for syntactic criteria. This new approach is comprehensive and robust: based on the same syntactic representation, we also derive the state-of-the-art coherence theorems ensuring uniqueness of inhabitants.



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Alves, S. and Broda, S. (2015). A short note on type-inhabitation: Formula-trees vs. game semantics. Information Processing Letters 115 (11) 908911.
Aoto, T. and Ono, H. (1994). Uniqueness of normal forms in →, ∧-fragment of NJ. Technical report, Research Report IS-RR-94-0024F.
Barendregt, H., Dekkers, W. and Statman, R. (2013). Lambda Calculus with Types. Perspectives in Logic, Cambridge University Press.
Ben-Yelles, C.-B. (1979). Type Assignment in the Lambda-Calculus: Syntax & Semantics. PhD thesis, University College of Swansea.
Bourreau, P. and Salvati, S. (2011). Game semantics and uniqueness of type inhabitance in the simply-typed λ-calculus. In: Ong, L. (ed.), Proceedings of TLCA 2011, LNCS, vol. 6690, Springer, 61–75.
Broda, S. and Damas, L. (2005). On long normal inhabitants of a type. Journal of Logic and Computation 15 (3) 353390.
Bucciarelli, A., Kesner, D. and Rocca, S. R. D. (2014). The inhabitation problem for non-idempotent intersection types. In: Díaz, J., Lanese, I., and Sangiorgi, D. (eds.), Proceedings of IFIP TCS 2014, LNCS, vol. 8705, Springer, 341–354.
David, R. and Zaionc, M. (2009). Counting proofs in propositional logic. Archive for Mathematical Logic 48 (2) 185199.
Dowek, G. and Jiang, Y. (2009). Enumerating proofs of positive formulae. The Computer Journal 52 (7) 799807.
Dudenhefner, A. and Rehof, J. (2017). Intersection type calculi of bounded dimension. In: Proceedings of POPL 2017, ACM, 653–665.
Espírito Santo, J., Matthes, R., and Pinto, L. (2013). A coinductive approach to proof search. In: Baelde, D. and Carayol, A. (eds.), Proceedings of FICS 2013, vol. 126, EPTCS, 28–43.
Espírito Santo, J., Matthes, R., and Pinto, L. (2016). A coinductive approach to proof search through typed lambda-calculi. Available at
Hindley, J. R. (1997). Basic Simple Type Theory, vol. 42. Cambridge Tracts in Theoretical Computer Science, Cambridge University Press.
Hopcroft, J. E. and Ullman, J. D. (1979). Introduction to Automata Theory, Languages and Computation, Addison-Wesley.
Liang, C. and Miller, D. (2009). Focusing and polarization in linear, intuitionistic, and classical logic. Theoretical Computer Science 410 47474768.
Miller, D. and Nadathur, G. (2012). Programming with Higher-Order Logic, Cambridge University Press.
Mints, G. (1979). A coherence theorem for cartesian closed categories (abstract). The Journal of Symbolic Logic 44 453454.
Mints, G. (1992). A simple proof for the coherence theorem for cartesian closed categories. In: Selected papers in proof theory, vol. 3, 213220 North-Holland Publishing Co., Amsterdam.
Schubert, A., Dekkers, W. and Barendregt, H. P. (2015). Automata theoretic account of proof search. In: Kreutzer, S. (ed.), Proceedings CSL 2015, LIPIcs, vol. 41, Schloss Dagstuhl, 128–143.
Takahashi, M., Akama, Y. and Hirokawa, S. (1996). Normal proofs and their grammar. Information and Computation 125 (2) 144153.
Wells, J. B. and Yakobowski, B. (2004). Graph-based proof counting and enumeration with applications for program fragment synthesis. In: Proceedings of LOPSTR 2004, LNCS, vol. 3573, Springer, 262–277.

Inhabitation in simply typed lambda-calculus through a lambda-calculus for proof search



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