Quantitative semantics of the lambda calculus: Some generalisations of the relational model

We present an overview of some recent work on the quantitative semantics of the λ-calculus. Our starting point is the fundamental degenerate model of linear logic, the relational model MRel. We show that three quantitative semantics of the simply-typed λ-calculus are equivalent: the relational semantics, HO/N game semantics, and the Taylor expansion semantics. We then consider two recent generalisations of the relational model: first, R-weighted relational models where R is a complete commutative semiring, as studied by Laird et al.; secondly, generalised species of structures, as introduced by Fiore et al. In each case, we briefly discuss some applications to quantitative analysis of higher-order programs.

[1]  Antonio Bucciarelli,et al.  A Relational Model of a Parallel and Non-deterministic lambda-Calculus , 2009, LFCS.

[2]  C.-H. Luke Ong,et al.  On Full Abstraction for PCF: I, II, and III , 2000, Inf. Comput..

[3]  A. Joyal.  Une théorie combinatoire des séries formelles , 1981 .

[4]  Ryu Hasegawa,et al.  Two applications of analytic functors , 2002, Theor. Comput. Sci..

[5]  Vincent Danos,et al.  The structure of multiplicatives , 1989, Arch. Math. Log..

[6]  Paul-André Melliès.  Asynchronous games 2: The true concurrency of innocence , 2006, Theor. Comput. Sci..

[7]  David Sands,et al.  Operational Theories of Improvement in Functional Languages (Extended Abstract) , 1991, Functional Programming.

[8]  J. Robin B. Cockett,et al.  Differential categories , 2006, Mathematical Structures in Computer Science.

[9]  Thomas Ehrhard,et al.  Uniformity and the Taylor expansion of ordinary lambda-terms , 2008, Theor. Comput. Sci..

[10]  Antonio Bucciarelli,et al.  Categorical Models for Simply Typed Resource Calculi , 2010, MFPS.

[11]  Thomas Ehrhard,et al.  Finiteness spaces , 2005, Mathematical Structures in Computer Science.

[12]  Patrick Lincoln,et al.  Linear logic , 1992, SIGA.

[13]  Antonio Bucciarelli,et al.  Full Abstraction for the Resource Lambda Calculus with Tests, through Taylor Expansion , 2012, Log. Methods Comput. Sci..

[14]  Hanno Nickau.  Hereditarily Sequential Functionals , 1994, LFCS.

[15]  Hugo Férée,et al.  Game semantics approach to higher-order complexity , 2017, J. Comput. Syst. Sci..

[16]  David Sands,et al.  Total correctness by local improvement in the transformation of functional programs , 1996, TOPL.

[17]  Thomas Ehrhard.  A Finiteness Structure on Resource Terms , 2010, 2010 25th Annual IEEE Symposium on Logic in Computer Science.

[18]  Laurent Regnier,et al.  The differential lambda-calculus , 2003, Theor. Comput. Sci..

[19]  Thomas Ehrhard,et al.  Probabilistic coherence spaces as a model of higher-order probabilistic computation , 2011, Inf. Comput..

[20]  François Lamarche,et al.  Quantitative Domains and Infinitary Algebras , 1992, Theor. Comput. Sci..

[21]  Hugo Fre,et al.  Game semantics approach to higher-order complexity , 2017 .

[22]  Glynn Winskel,et al.  A Presheaf Semantics of Value-Passing Processes , 1996, CONCUR.

[23]  Robin Houston.  Finite products are biproducts in a compact closed category , 2008 .

[24]  Cosimo Laneve,et al.  Solos in Concert , 1999, ICALP.

[25]  Dan R. Ghica,et al.  Slot games: a quantitative model of computation , 2005, POPL '05.

[26]  Martin Hyland,et al.  Some reasons for generalising domain theory , 2010, Mathematical Structures in Computer Science.

[27]  C.-H. Luke Ong,et al.  Plays as Resource Terms via Non-idempotent Intersection Types , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[28]  Michele Pagani,et al.  Parallel Reduction in Resource Lambda-Calculus , 2009, APLAS.

[29]  B Jean,et al.  Distributors at Work , 2000 .

[30]  C.-H. Luke Ong,et al.  Generalised species of rigid resource terms , 2017, 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[31]  Paul-André Melliès.  CATEGORICAL SEMANTICS OF LINEAR LOGIC , 2009 .

[32]  Michele Pagani,et al.  Strong Normalizability as a Finiteness Structure via the Taylor Expansion of \lambda λ -terms , 2016, FoSSaCS.

[33]  Giulio Manzonetto,et al.  Constructing differential categories and deconstructing categories of games , 2013, Inf. Comput..

[34]  Antonio Bucciarelli,et al.  On phase semantics and denotational semantics: the exponentials , 2001, Ann. Pure Appl. Log..

[35]  Jean-Yves Girard,et al.  Linear Logic , 1987, Theor. Comput. Sci..

[36]  Marina Lenisa,et al.  Innocent Game Semantics via Intersection Type Assignment Systems , 2013, CSL.

[37]  C.-H. Luke Ong.  Normalisation by Traversals , 2015, ArXiv.

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

[39]  Thomas Ehrhard,et al.  Differential Interaction Nets , 2005, WoLLIC.

[40]  Glynn Winskel,et al.  A Theory of Recursive Domains with Applications to Concurrency , 1997 .

[41]  Giulio Manzonetto,et al.  Weighted Relational Models of Typed Lambda-Calculi , 2013, 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science.

[42]  Assaf J. Kfoury.  A linearization of the Lambda-calculus and consequences , 2000, J. Log. Comput..

[43]  Thomas Ehrhard,et al.  On Köthe sequence spaces and linear logic , 2002, Mathematical Structures in Computer Science.

[44]  Samson Abramsky,et al.  Linearity, Sharing and State: a fully abstract game semantics for Idealized Algol with active expressions , 1996, Electron. Notes Theor. Comput. Sci..

[45]  Marcelo P. Fiore,et al.  Mathematical Models of Computational and Combinatorial Structures , 2005, FoSSaCS.

[46]  R. Rosebrugh,et al.  Normed Spaces and the Change of Base for Enriched Categories , 2008 .

[47]  Robin Milner,et al.  Fully Abstract Models of Typed lambda-Calculi , 1977, Theor. Comput. Sci..

[48]  Jean-Yves Girard,et al.  Normal functors, power series and λ-calculus , 1988, Ann. Pure Appl. Log..

[49]  P. Freyd.  Algebraically complete categories , 1991 .

[50]  James Laird.  From Qualitative to Quantitative Semantics - By Change of Base , 2017, FoSSaCS.

[51]  J. Laird.  Fixed Points In Quantitative Semantics , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[52]  Thomas Ehrhard,et al.  Böhm Trees, Krivine's Machine and the Taylor Expansion of Lambda-Terms , 2006, CiE.

[53]  Jean-Yves Girard,et al.  The System F of Variable Types, Fifteen Years Later , 1986, Theor. Comput. Sci..

[54]  Pierre-Louis Curien,et al.  A semantics for lambda calculi with resources , 1999, Mathematical Structures in Computer Science.

[55]  C.-H. Luke Ong,et al.  Two-Level Game Semantics, Intersection Types, and Recursion Schemes , 2012, ICALP.

[56]  Glynn Winskel,et al.  Profunctors, open maps and bisimulation , 2004, Mathematical Structures in Computer Science.

[57]  Irène Guessarian,et al.  Algebraic semantics , 1981, Lecture Notes in Computer Science.

[58]  Nicolas Tabareau,et al.  An explicit formula for the free exponential modality of linear logic , 2009, Mathematical Structures in Computer Science.

[59]  Radha Jagadeesan,et al.  Full Abstraction for PCF , 1994, Inf. Comput..

[60]  Thomas Ehrhard,et al.  Probabilistic coherence spaces are fully abstract for probabilistic PCF , 2014, POPL.

[61]  Kim G. Larsen,et al.  Continuous Markovian Logics - Axiomatization and Quantified Metatheory , 2012, Log. Methods Comput. Sci..

[62]  C.-H. Luke Ong,et al.  Nondeterminism in Game Semantics via Sheaves , 2015, 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science.

[63]  Michael Barr,et al.  Algebraically compact functors , 1992 .

[64]  Glynn Winskel,et al.  The cartesian closed bicategory of generalised species of structures , 2008 .

[65]  Daniel de Carvalho,et al.  Execution time of λ-terms via denotational semantics and intersection types , 2009, Mathematical Structures in Computer Science.

[66]  C.-H. Luke Ong,et al.  On Model-Checking Trees Generated by Higher-Order Recursion Schemes , 2006, 21st Annual IEEE Symposium on Logic in Computer Science (LICS'06).

[67]  Pierre Boudes,et al.  Thick Subtrees, Games and Experiments , 2009, TLCA.

[68]  Christopher Strachey,et al.  Toward a mathematical semantics for computer languages , 1971 .

[69]  Thomas Ehrhard,et al.  A convenient differential category , 2010, ArXiv.

[70]  G. M. Kelly,et al.  Coherence for compact closed categories , 1980 .

[71]  James Laird.  Weighted Relational Models for Mobility , 2016, FSCD.

[72]  Thomas Ehrhard,et al.  An introduction to differential linear logic: proof-nets, models and antiderivatives , 2016, Mathematical Structures in Computer Science.

[73]  M. Droste,et al.  Semirings and Formal Power Series , 2009 .

[74]  A. Joyal.  Foncteurs analytiques et espèces de structures , 1986 .

[75]  Gordon D. Plotkin,et al.  Complete axioms for categorical fixed-point operators , 2000, Proceedings Fifteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.99CB36332).

[76]  Paul-André Melliès.  Comparing hierarchies of types in models of linear logic , 2004, Inf. Comput..

[77]  Michele Pagani,et al.  Applying quantitative semantics to higher-order quantum computing , 2013, POPL.