A unifying framework for continuity and complexity in higher types

We set up a parametrised monadic translation for a class of call-by-value functional languages, and prove a corresponding soundness theorem. We then present a series of concrete instantiations of our translation, demonstrating that a number of fundamental notions concerning higher-order computation, including termination, continuity and complexity, can all be subsumed into our framework. Our main goal is to provide a unifying scheme which brings together several concepts which are often treated separately in the literature. However, as a by-product, we also obtain (i) a method for extracting moduli of continuity for closed functionals of type $(\mathbb{N}\to\mathbb{N})\to\mathbb{N}$ definable in (extensions of) System T, and (ii) a characterisation of the time complexity of bar recursion.

[1]  Helmut Schwichtenberg,et al.  On bar recursion of types 0 and 1 , 1979, Journal of Symbolic Logic.

[2]  Chuangjie Xu,et al.  A syntactic approach to continuity of T-definable functionals , 2019, Log. Methods Comput. Sci..

[3]  Mark Bickford,et al.  Validating Brouwer's continuity principle for numbers using named exceptions , 2017, Mathematical Structures in Computer Science.

[4]  Philip Wadler,et al.  The essence of functional programming , 1992, POPL '92.

[5]  Thierry Coquand,et al.  On the computational content of the axiom of choice , 1994, The Journal of Symbolic Logic.

[6]  Marc Bezem,et al.  Strong normalization of barrecursive terms without using infinite terms , 1985, Arch. Math. Log..

[7]  Eugenio Moggi,et al.  Notions of Computation and Monads , 1991, Inf. Comput..

[8]  A. Troelstra.  Metamathematical investigation of intuitionistic arithmetic and analysis , 1973 .

[9]  Paulo Oliva,et al.  A Direct Proof of Schwichtenberg's Bar Recursion Closure Theorem , 2018, J. Symb. Log..

[10]  Y. Ershov.  Model of Partial Continuous Functionals , 1977 .

[11]  Ulrich Berger,et al.  Continuous Semantics for Strong Normalization , 2005, CiE.

[12]  H. Schwichtenberg.  Proof Theory: Some Applications of Cut-Elimination , 1977 .

[13]  Mircea-Dan Hernest.  Synthesis of Moduli of Uniform Continuity by the Monotone Dialectica Interpretation in the Proof-system MinLog , 2007, Electron. Notes Theor. Comput. Sci..

[14]  Paulo Oliva.  Understanding and Using Spector's Bar Recursive Interpretation of Classical Analysis , 2006, CiE.

[15]  G.D. Plotkin,et al.  LCF Considered as a Programming Language , 1977, Theor. Comput. Sci..

[16]  U. Kohlenbach.  Analysing proofs in analysis , 1996 .

[17]  Marc Bezem,et al.  Strongly majorizable functionals of finite type: A model for barrecursion containing discontinuous functionals , 1985, Journal of Symbolic Logic.

[18]  Ulrich Berger,et al.  Continuous semantics for strong normalisation , 2006, Mathematical Structures in Computer Science.

[19]  Helmut Von Vogel.  Ein starker Normalisationssatz für die bar-rekursiven Funktionale , 1977, Arch. Math. Log..

[20]  Paulo Oliva,et al.  On Spector's bar recursion , 2012, Math. Log. Q..

[21]  Ulrich Berger,et al.  Strong normalisation for applied lambda calculi , 2005, ArXiv.

[22]  Jonathan C. Shultis.  On the Complexity of Higher-Order Programs ; CU-CS-288-85 , 1985 .

[23]  Ulrich Berger,et al.  A computational interpretation of open induction , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[24]  Tatsuji Kawai,et al.  Representing definable functions of HAω by neighbourhood functions , 2019, Ann. Pure Appl. Log..

[25]  Theodore Seok Kim.  Cost Semantics for Plotkin's PCF , 2016 .

[26]  Martín Hötzel Escardó,et al.  System T and the Product of Selection Functions , 2011, CSL.

[27]  Daniel R. Licata,et al.  Denotational cost semantics for functional languages with inductive types , 2015, ICFP.

[28]  W. A. Howard,et al.  Functional interpretation of bar induction by bar recursion , 1968 .

[29]  Ralph Benzinger,et al.  Automated higher-order complexity analysis , 2004, Theor. Comput. Sci..

[30]  C. Spector.  Provably recursive functionals of analysis: a consistency proof of analysis by an extension of princ , 1962 .

[31]  Ulrich Kohlenbach,et al.  Applied Proof Theory - Proof Interpretations and their Use in Mathematics , 2008, Springer Monographs in Mathematics.

[32]  Stephen Brookes,et al.  A denotational approach to measuring complexity in functional programs , 2003 .

[33]  Thomas Powell.  A functional interpretation with state , 2018, LICS.

[34]  Jennifer Paykin,et al.  A static cost analysis for a higher-order language , 2012, PLPV.

[35]  D. Normann.  The countable functionals , 1980 .

[36]  David Sands,et al.  Calculi for time analysis of functional programs , 1990 .

[37]  Thierry Coquand,et al.  A Computational Interpretation of Forcing in Type Theory , 2012, Epistemology versus Ontology.