Asterix calculus - classical computation in detail
|
|
- Virginia Ray
- 5 years ago
- Views:
Transcription
1 Asterix calculus - classical computation in detail (denoted X ) Dragiša Žunić 1 Pierre Lescanne 2 1 CMU Qatar 2 ENS Lyon Logic and Applications - LAP th conference, September 19-23, Dubrovnik Croatia
2 Outline Sequent calculus, proofs and programs The X calculus : explicit erasure and duplication Strong normalisation property
3 Outline Sequent calculus, proofs and programs Implicit structural rules the X calculus Explicit structural rules the X calculus The X calculus : explicit erasure and duplication Logical setting From sequent proofs to terms The syntax and reduction rules Diagrammatic view Strong normalisation property
4 The framework Classical logic Sequent calculus of G.Gentzen (two important formalisms : sequent calculus and natural deduction calculus) The Curry-Howard correspondence (the relation between proof theory and the programming language theory) The goal is to study classical computation, by assigning computational interpretation(s) to classical logic represented in the sequent calculus.
5 The Curry-Howard correspondence The intuitionistic logic and simply typed λ-calculus Proofs Terms Propositions Types Normalization Reduction Classical logic : T. Griffin (1990) M. Parigot (1992) : λµ-calculus (natural deduction) P. L. Curien, H. Herbelin ( ) : λµ µ-calculus (sequent calculus)
6 Related work The two predecessors Classical logic : the X calculus C. Urban (2000) S. van Bakel, S.Lengrand, P. Lescanne (2005) Intuitionistic logic : the λlxr-calculus D. Kesner, S. Lengrand (2005) X λlxr X
7 G3 sequent system for classical logic X -calculus (axiom) Γ, A A, Γ A, Γ, B ( left) Γ, A B Γ A, Γ, A (cut) Γ, A B, ( right) Γ A B, Γ Contexts Γ, are sets Context-sharing style Structural rules implicit
8 G1 sequent system for classical logic X calculus (axiom) A A Γ A, Γ, Γ, A B, Γ, B ( left) Γ, A B, ( right) Γ A B, Γ A, Γ, Γ, Γ, A (cut) Γ (left weakening) Γ, A Γ (right weakening) Γ A, Γ, A, A (left contraction) Γ, A Γ A, A, (right contraction) Γ A,
9 Outline Sequent calculus, proofs and programs Implicit structural rules the X calculus Explicit structural rules the X calculus The X calculus : explicit erasure and duplication Logical setting From sequent proofs to terms The syntax and reduction rules Diagrammatic view Strong normalisation property
10 Computational interpretation of classical proofs Sequent calculus with explicit structural rules (weakening and contraction) 1) Terms in X -calculus are in fact annotations for proofs, 2) Computation in X -calculus corresponds to cut-elimination Weakening as an eraser / Contraction as a duplicator
11 Names The terms are built from names. Two categories of names : x, y, z... in-names α, β, γ... out-names Binders wear hats : x, ŷ, ẑ... α, β, γ... Examples : x.α x x.β β. α [P β γ >α
12 Name Variable In λ-calculus : variables β (λy.xyz)m xmz An arbitrary term M substitutes the variable y In X -calculus : names A name can never be substituted for a term A name can only be renamed
13 G1 sequent system for classical logic (axiom) A A Γ A, Γ, Γ, A B, Γ, B ( left) Γ, A B, ( right) Γ A B, Γ A, Γ, Γ, Γ, A (cut) Γ (left weakening) Γ, A Γ (right weakening) Γ A, Γ, A, A (left contraction) Γ, A Γ A, A, (right contraction) Γ A,
14 The terms correspond to proofs : (caps) x.α :. x : A α : A P :. Γ α : A, Q :. Γ, x : B P :. Γ, x : A α : B, (imp) (exp) P α [y] xq :. Γ, Γ, y : A B, x P α. β :. Γ β : A B, P :. Γ α : A, P α xq :. Γ, Γ, Q :. Γ, x : A (cut) P :. Γ (left eraser) x P :. Γ, x : A P :. Γ (right eraser) P α :. Γ α : A, P :. Γ, y : A, z : A (left dupl.) x < ŷ ẑ P] :. Γ, x : A P :. Γ β : A, γ : A, (right dupl.) [P β γ >α :. Γ α : A,
15 The terms correspond to proofs : (caps) x.α :. x : A α : A P :. Γ α : A, Q :. Γ, x : B P :. Γ, x : A α : B, (imp) (exp) P α [y] xq :. Γ, Γ, y : A B, x P α. β :. Γ β : A B, P :. Γ α : A, P α xq :. Γ, Γ, Q :. Γ, x : A (cut) P :. Γ (left eraser) x P :. Γ, x : A P :. Γ (right eraser) P α :. Γ α : A, P :. Γ, y : A, z : A (left dupl.) x < ŷ ẑ P] :. Γ, x : A P :. Γ β : A, γ : A, (right dupl.) [P β γ >α :. Γ α : A,
16 The terms correspond to proofs : (caps) x.α :. x : A α : A P :. Γ α : A, Q :. Γ, x : B P :. Γ, x : A α : B, (imp) (exp) P α [y] xq :. Γ, Γ, y : A B, x P α. β :. Γ β : A B, P :. Γ α : A, P α xq :. Γ, Γ, Q :. Γ, x : A (cut) P :. Γ (left eraser) x P :. Γ, x : A P :. Γ (right eraser) P α :. Γ α : A, P :. Γ, y : A, z : A (left dupl.) x < ŷ ẑ P] :. Γ, x : A P :. Γ β : A, γ : A, (right dupl.) [P β γ >α :. Γ α : A,
17 The terms correspond to proofs : (caps) x.α :. x : A α : A P :. Γ α : A, Q :. Γ, x : B P :. Γ, x : A α : B, (imp) (exp) P α [y] xq :. Γ, Γ, y : A B, x P α. β :. Γ β : A B, P :. Γ α : A, P α xq :. Γ, Γ, Q :. Γ, x : A (cut) P :. Γ (left eraser) x P :. Γ, x : A P :. Γ (right eraser) P α :. Γ α : A, P :. Γ, y : A, z : A (left dupl.) x < ŷ ẑ P] :. Γ, x : A P :. Γ β : A, γ : A, (right dupl.) [P β γ >α :. Γ α : A,
18 The terms correspond to proofs : (caps) x.α :. x : A α : A P :. Γ α : A, Q :. Γ, x : B P :. Γ, x : A α : B, (imp) (exp) P α [y] xq :. Γ, Γ, y : A B, x P α. β :. Γ β : A B, P :. Γ α : A, P α xq :. Γ, Γ, Q :. Γ, x : A (cut) P :. Γ (left eraser) x P :. Γ, x : A P :. Γ (right eraser) P α :. Γ α : A, P :. Γ, y : A, z : A (left dupl.) x < ŷ ẑ P] :. Γ, x : A P :. Γ β : A, γ : A, (right dupl.) [P β γ >α :. Γ α : A,
19 The X calculus The syntax P, Q ::= x.α capsule x P β. α exporter P α [y] xq importer P α xq cut x P left-eraser P α right-eraser x < ŷ ẑ P] [P β γ >α left-duplicator right-duplicator
20 Linearity In X -calculus only linear terms are considered : every free name occurs only once every binder does bind an occurrence of a free name (and therefore only one) Examples of non-linear terms : x y.β β. α and x.α α Every non-linear term has a linear representation : x (x y.β ) β. α and [ x.α 1 α 2 α 1 α 2 >α
21 Linearity In X -calculus only linear terms are considered : every free name occurs only once every binder does bind an occurrence of a free name (and therefore only one) Examples of non-linear terms : x y.β β. α and x.α α Every non-linear term has a linear representation : x (x y.β ) β. α and [ x.α 1 α 2 α 1 α 2 >α
22 Linearity In X -calculus only linear terms are considered : every free name occurs only once every binder does bind an occurrence of a free name (and therefore only one) Examples of non-linear terms : x y.β β. α and x.α α Every non-linear term has a linear representation : x (x y.β ) β. α and [ x.α 1 α 2 α 1 α 2 >α
23 Linearity In X -calculus only linear terms are considered : every free name occurs only once every binder does bind an occurrence of a free name (and therefore only one) Examples of non-linear terms : x y.β β. α and x.α α Every non-linear term has a linear representation : x (x y.β ) β. α and [ x.α 1 α 2 α 1 α 2 >α
24 Linearity In X -calculus only linear terms are considered : every free name occurs only once every binder does bind an occurrence of a free name (and therefore only one) Examples of non-linear terms : x y.β β. α and x.α α Every non-linear term has a linear representation : x (x y.β ) β. α and [ x.α 1 α 2 α 1 α 2 >α
25 The X calculus Grouping the rules : Logical rules (L-principal names involved) Structural rules (S-principal names involved) (Activation rules) (Deactivation rules) Propagation rules
26 The X calculus The syntax P, Q ::= x.α capsule x P β. α exporter P α [y] xq importer P α xq cut x P left-eraser P α right-eraser x < ŷ ẑ P] [P β γ >α left-duplicator right-duplicator The notion of principal name of a term. 1. L-principal name 2. S principal name
27 The X calculus The syntax P, Q ::= x.α capsule x P β. α exporter P α [y] xq importer P α xq cut x P left-eraser P α right-eraser x < ŷ ẑ P] [P β γ >α left-duplicator right-duplicator The notion of principal name of a term. 1. L principal name 2. S-principal name
28 X : logical rules Renaming : (ren-l) : y.α α xq Q{y/x} (ren-r) : P α x x.β P{β/α} Diagrammatically : (ren-l) : y α x Q y Q α x β (ren-r) : P P β
29 X : logical rules Inserting (ei-insert) : (ŷ P β. α) α x(q γ [x] ẑr) either { (Q γ ŷp) β ẑr Q γ ŷ(p β ẑr) Diagrammatically : y P β Q γ I z R E α x y β Q γ z P R Notice : logical rules define reducing when a cut binds L-principal names
30 X : structural rules Left erasure : ( -eras) : (P α) α xq I Q P O Q, where I Q = I (Q) \x, O Q = O(Q) Diagrammatically : I Q { α x P Q Q } O I Q { O P } Q
31 X : structural rules Left duplication ( -dupl) : ([P α1 α 2 >α) α xq I Q < I Q 1 I Q 2 (P α 1 x 1 Q 1 ) α 2 Ô Q 1 x 2 Q 2 > O Q, O Q 2 Diagrammatically : I Q { P β γ α x Q Q } O I Q { P β γ x Q 2 x Q 1 Q } O
32 Symmetry... The -erasure } O P I P { P α x Q I P { Q } O P An illustration of the symmetry...
33 Symmetry... The -duplication P I { P α x y z Q } O P P I { P 1 α P 2 α y z Q } O P An illustration of the symmetry...
34 Duplication, informally Classical Logic Intuitionistic Logic Component duplicated, interface preserved. Similarly for erasure
35 X : propagation rules (left subgroup) (exp prop) : ( x P γ. α) β ŷr x (P β ŷr) γ. α (imp prop 1 ) : (P α [x] ẑq) β ŷr (P β ŷr) α [x] ẑq, β O(P) (imp prop 2 ) : (P α [x] ẑq) β ŷr P α [x] ẑ(q β ŷr), β O(Q) (cut(c) prop) : (P α x x.β ) β ŷr P α ŷr (cut prop 1 ) : (P α xq) β ŷr (P β ŷr) α xq, β O(P), Q x.β (cut prop 2 ) : (P α xq) β ŷr P α x(q β ŷr), β O(Q), Q x.β (L eras prop) : (x M) β ŷr x (M β ŷr) (R eras prop) : (M α) β ŷr (M β ŷr) α, α β (L dupl prop) : (x< x 1 x2 M]) β ŷr x< x 1 x2 M β ŷr] (R dupl prop) : ([M α 1 >α) β α 2 ŷr [M β ŷr α 1 >α, α 2 α β Propagation rules do not have a diagrammatic representation
36 The X calculus Propagation rules α ^ α ^ x Q
37 The X calculus Propagation rules α ^ α ^ x Q
38 The X calculus Propagation rules α ^ α ^ x Q
39 The X calculus Propagation rules α ^ α ^ x Q
40 The X calculus Propagation rules ^ α α ^ x Q
41 The X calculus Propagation rules ^ α α ^ x Q We may think of α xq as an explicit substitution
42 X : the basic properties 1. Linearity preservation If P is linear and P Q then Q is linear 2. Free names ( interface ) preservation If P Q then N(P) = N(Q) 3. Type preservation computation can be (has to be) seen as proof-transformation If P :. Γ and P P, then P :. Γ 4. Strong normalisation (termination of reduction)
43 Peirce s law in X x.α 1 :. x : A α 1 : A (capsule) x.α 1 β :. x : A α 1 : A, β : B x ( x.α 1 β) β. γ :. α 1 : A, γ : A B (R eraser) (exporter) y.α 2 :. y : A α 2 : A ( x ( x.α 1 β) β. γ) γ [z] ŷ y.α 2 :. z : (A B) A α 1 : A, α 2 : A [( x ( x.α 1 β) β. γ) γ [z] ŷ y.α 2 α 1 α 2 >α :. z : (A B) A α : A ẑ ([( x ( x.α 1 β) β. γ) γ [z] ŷ y.α 2 α 1 α 2 >α) α. δ :. δ : ((A B) A) A (capsule) (importer) (R duplicator) (exporter) (axiom) A A (R weakening) A A, B ( R) A, A B (axiom) A A ( L) (A B) A A, A (R contraction) (A B) A A ( R) ((A B) A) A
44 *X : (axiom) A A (R weakening) A A, B ( R) A, A B (axiom) A A ( L) (A B) A A, A (R contraction) (A B) A A ( R) ((A B) A) A X : (axiom) A A, B ( R) A, A B (axiom) A A ( L) (A B) A A ( R) ((A B) A) A
45 The terms of X are convenient for two-dimensional representation, due to the presence of erasers and duplicators but also the linearity constraints
46 Diagrammatic calculus Syntax P, Q ::= x α x P β α y P Q I E x α P α x Q P α β γ z x y P P α x P
47 Outline Sequent calculus, proofs and programs Implicit structural rules the X calculus Explicit structural rules the X calculus The X calculus : explicit erasure and duplication Logical setting From sequent proofs to terms The syntax and reduction rules Diagrammatic view Strong normalisation property
48 Encoding X in X calculus
49 Encoding preserves types
50 Simulating X reduction
51 Strong normalisation of X -calculus
52 Future work define equivalent terms, the c X calculus concurrency - which process calculus corresponds to this classical cut-elimination? a 3D computational model...
53 Some references S. Ghilezan, P. Lescanne, D. Žunić Comp. interpretation of classical logic with expl. struct. rules, draft. S. Ghilezan, J. Ivetic, P. Lescanne, D. Žunić Intuitionistic sequent-style calculus with explicit struct. rules, P. L. Curien, H. Herbelin The duality of computation, D. Kesner and S. Lengrand Explicit operators for λ-calculus, C. Urban and G. M. Bierman Strong normalisation of cut-elimination in classical logic, P. Wadler Propositions as sessions, E. Robinson Proof nets for classical logic, 2005.
54 Thanks..! Homepage: Dragisa Zunic
Constructive approach to relevant and affine term calculi
Constructive approach to relevant and affine term calculi Jelena Ivetić, University of Novi Sad, Serbia Silvia Ghilezan,University of Novi Sad, Serbia Pierre Lescanne, University of Lyon, France Silvia
More informationApproaches to Polymorphism in Classical Sequent Calculus
Approaches to Polymorphism in Classical Sequent Calculus (ESOP 06, LNCS 3924, pages 84-99, 2006) Alex Summers and Steffen van Bakel Department of Computing, Imperial College London 180 Queen s Gate, London
More informationIntersection and Union Types for X
ITRS 2004 Preliminary Version Intersection and Union Types for X Steffen van Bakel Department of Computing, Imperial College London, 180 Queen s Gate, London SW7 2BZ, UK, svb@doc.ic.ac.uk Abstract This
More informationTERMS FOR NATURAL DEDUCTION, SEQUENT CALCULUS AND CUT ELIMINATION IN CLASSICAL LOGIC
TERMS FOR NATURAL DEDUCTION, SEQUENT CALCULUS AND CUT ELIMINATION IN CLASSICAL LOGIC SILVIA GHILEZAN Faculty of Engineering, University of Novi Sad, Novi Sad, Serbia e-mail address: gsilvia@uns.ns.ac.yu
More informationComputation with Classical Sequents
Computation with Classical Sequents (Mathematical Structures in Computer Science, 18:555-609, 2008) Steffen van Bakel and Pierre Lescanne Department of Computing, Imperial College London, 180 Queen s Gate
More informationA General Technique for Analyzing Termination in Symmetric Proof Calculi
A General Technique for Analyzing Termination in Symmetric Proof Calculi Daniel J. Dougherty 1, Silvia Ghilezan 2 and Pierre Lescanne 3 1 Worcester Polytechnic Institute, USA, dd@cs.wpi.edu 2 Faculty of
More informationIntersection and Union Types for X
ITRS 2004 Preliminary Version Intersection and Union Types for X Steffen van Bakel epartment of Computing, Imperial College London, 180 Queen s Gate, London SW7 2BZ, UK, svb@doc.ic.ac.uk Abstract This
More informationRačunske interpretacije intuicionističke i klasične logike
Računske interpretacije intuicionističke i klasične logike Silvia Ghilezan Univerzitet u Novom Sadu Srbija Sustavi dokazivanja 2012 Dubrovnik, Hrvatska, 28.06.2012 S.Ghilezan () Računske interpretacije
More informationA congruence relation for restructuring classical terms
A congruence relation for restructuring classical terms Dragiša Žunić1 and ierre Lescanne 2 1 Carnegie Mellon Universit in atar 2 Laboratoire de l nformatique du arallélisme, L Ecole Normale Supérieure
More informationA completeness result for λµ
A completeness result for µ Phillipe Audebaud Steffen van Bakel Inria Sophia Antipolis, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis, France E-mail: Philippe.Audebaud@sophia.inria.fr,svb@doc.ic.ac.uk
More informationCLASSICAL CUT-ELIMINATION IN THE π-calculus STEFFEN VAN BAKEL, LUCA CARDELLI, AND MARIA GRAZIA VIGLIOTTI
CLASSICAL CUT-ELIMINATION IN THE π-calculus STEFFEN VAN BAKEL, LUCA CARDELLI, AND MARIA GRAZIA VIGLIOTTI Department of Computing, Imperial College London, 180 Queen s Gate, London SW7 2BZ, UK e-mail address:
More informationThe duality of computation
The duality of computation (notes for the 3rd International Workshop on Higher-Order Rewriting) Hugo Herbelin LIX - INRIA-Futurs - PCRI Abstract. The work presented here is an extension of a previous work
More informationCommand = Value Context. Hans-Dieter Hiep. 15th of June, /17
1/17 Command = Value Context Hans-Dieter Hiep 15th of June, 2018 2/17 Example Consider a total functional programming language. swap :: A B B A swap (a, b) = (b, a) paws :: A B B A paws a = a paws b =
More informationCompleteness and Partial Soundness Results for Intersection & Union Typing for λµ µ
Completeness and Partial Soundness Results for Intersection & Union Typing for λµ µ Steffen van Bakel Department of Computing, Imperial College London, 180 Queen s Gate, London SW7 2BZ, UK Abstract This
More informationCurry-Howard Correspondence for Classical Logic
Curry-Howard Correspondence for Classical Logic Stéphane Graham-Lengrand CNRS, Laboratoire d Informatique de l X Stephane.Lengrand@Polytechnique.edu 2 Practicalities E-mail address: Stephane.Lengrand@Polytechnique.edu
More informationImplementing X. (TermGraph 04, 2004; ENTCS, volume 127(5)) Steffen van Bakel and Jayshan Raghunandan
Implementing X (TermGraph 04, 2004; ENTCS, volume 127(5)) Steffen van Bakel and Jayshan Raghunandan Department of Computing, Imperial College London, 180 Queen s Gate, London SW7 2BZ, UK svb@doc.ic.ac.uk
More informationSilvia Ghilezan and Jelena Ivetić
PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome 82(96) (2007), 85 91 DOI 102298/PIM0796085G INTERSECTION TYPES FOR λ Gtz -CALCULUS Silvia Ghilezan and Jelena Ivetić Abstract. We introduce
More informationSemantics for Propositional Logic
Semantics for Propositional Logic An interpretation (also truth-assignment, valuation) of a set of propositional formulas S is a function that assigns elements of {f,t} to the propositional variables in
More informationStructuring Logic with Sequent Calculus
Structuring Logic with Sequent Calculus Alexis Saurin ENS Paris & École Polytechnique & CMI Seminar at IIT Delhi 17th September 2004 Outline of the talk Proofs via Natural Deduction LK Sequent Calculus
More informationFrom X to π. Representing the Classical Sequent Calculus in the π-calculus. Extended Abstract
From X to π Representing the Classical Sequent Calculus in the π-calculus Extended Abstract (International Workshop on Classical Logic and Computation (CL&C 08), 2008) Steffen van Bakel 1, Luca Cardelli
More informationAdjoint Logic and Its Concurrent Semantics
Adjoint Logic and Its Concurrent Semantics Frank Pfenning ABCD Meeting, Edinburgh, December 18-19, 2017 Joint work with Klaas Pruiksma and William Chargin Outline Proofs as programs Linear sequent proofs
More informationResource control and strong normalisation
esource control and strong normalisation Silvia Ghilezan, Jelena Ivetic, Pierre Lescanne, Silvia Likavec To cite this version: Silvia Ghilezan, Jelena Ivetic, Pierre Lescanne, Silvia Likavec. esource control
More informationThe language X : circuits, computations and Classical Logic
Laboratoire de l Informatique du Parallélisme École Normale Supérieure de Lyon Unité Mixte de Recherche CNRS-INRIA-ENS LYON-UCBL n o 5668 The language X : circuits, computations and Classical Logic Steffen
More informationProofs in classical logic as programs: a generalization of lambda calculus. A. Salibra. Università Ca Foscari Venezia
Proofs in classical logic as programs: a generalization of lambda calculus A. Salibra Università Ca Foscari Venezia Curry Howard correspondence, in general Direct relationship between systems of logic
More informationStrong normalization of the dual classical sequent calculus
Strong normalization of the dual classical sequent calculus Daniel Dougherty, 1 Silvia Ghilezan, 2 Pierre Lescanne 3 and Silvia Likavec 2,4 1 Worcester Polytechnic Institute, USA, dd@wpiedu 2 Faculty of
More informationTerm Graphs, α-conversion and Principal Types for X
Term Graphs, α-conversion and Principal Types for X Steffen van Bakel Jayshan Raghunandan Alexander Summers Department of Computing, Imperial College, 180 Queen s Gate, London SW7 2BZ, UK, {svb,jr200,ajs300m}@doc.ic.ac.uk
More informationExtended Abstract: Reconsidering Intuitionistic Duality
Extended Abstract: Reconsidering Intuitionistic Duality Aaron Stump, Harley Eades III, Ryan McCleeary Computer Science The University of Iowa 1 Introduction This paper proposes a new syntax and proof system
More informationConsequence Relations and Natural Deduction
Consequence Relations and Natural Deduction Joshua D. Guttman Worcester Polytechnic Institute September 9, 2010 Contents 1 Consequence Relations 1 2 A Derivation System for Natural Deduction 3 3 Derivations
More informationEvaluation Driven Proof-Search in Natural Deduction Calculi for Intuitionistic Propositional Logic
Evaluation Driven Proof-Search in Natural Deduction Calculi for Intuitionistic Propositional Logic Mauro Ferrari 1, Camillo Fiorentini 2 1 DiSTA, Univ. degli Studi dell Insubria, Varese, Italy 2 DI, Univ.
More informationAN ALTERNATIVE NATURAL DEDUCTION FOR THE INTUITIONISTIC PROPOSITIONAL LOGIC
Bulletin of the Section of Logic Volume 45/1 (2016), pp 33 51 http://dxdoiorg/1018778/0138-068045103 Mirjana Ilić 1 AN ALTERNATIVE NATURAL DEDUCTION FOR THE INTUITIONISTIC PROPOSITIONAL LOGIC Abstract
More informationLogical Preliminaries
Logical Preliminaries Johannes C. Flieger Scheme UK March 2003 Abstract Survey of intuitionistic and classical propositional logic; introduction to the computational interpretation of intuitionistic logic
More informationSimply Typed λ-calculus
Simply Typed λ-calculus Lecture 1 Jeremy Dawson The Australian National University Semester 2, 2017 Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, 2017 1 / 23 A Brief History of Type Theory First developed
More informationCompleteness and Partial Soundness Results for Intersection & Union Typing for λµ µ
Completeness and Partial Soundness Results for Intersection & Union Typing for λµ µ (Annals of Pure and Applied Logic 161, pp 1400-1430, 2010) Steffen van Bakel Department of Computing, Imperial College
More informationIntroduction to Intuitionistic Logic
Introduction to Intuitionistic Logic August 31, 2016 We deal exclusively with propositional intuitionistic logic. The language is defined as follows. φ := p φ ψ φ ψ φ ψ φ := φ and φ ψ := (φ ψ) (ψ φ). A
More informationReasoning with Higher-Order Abstract Syntax and Contexts: A Comparison
1 Reasoning with Higher-Order Abstract Syntax and Contexts: A Comparison Amy Felty University of Ottawa July 13, 2010 Joint work with Brigitte Pientka, McGill University 2 Comparing Systems We focus on
More informationCategories, Proofs and Programs
Categories, Proofs and Programs Samson Abramsky and Nikos Tzevelekos Lecture 4: Curry-Howard Correspondence and Cartesian Closed Categories In A Nutshell Logic Computation 555555555555555555 5 Categories
More informationIf-then-else and other constructive and classical connectives (Or: How to derive natural deduction rules from truth tables)
If-then-else and other constructive and classical connectives (Or: How to derive natural deduction rules from truth tables) Herman Geuvers Nijmegen, NL (Joint work with Tonny Hurkens) Types for Proofs
More informationClassical Combinatory Logic
Computational Logic and Applications, CLA 05 DMTCS proc. AF, 2006, 87 96 Classical Combinatory Logic Karim Nour 1 1 LAMA - Equipe de logique, Université de Savoie, F-73376 Le Bourget du Lac, France Combinatory
More informationLecture Notes on Classical Linear Logic
Lecture Notes on Classical Linear Logic 15-816: Linear Logic Frank Pfenning Lecture 25 April 23, 2012 Originally, linear logic was conceived by Girard [Gir87] as a classical system, with one-sided sequents,
More informationOn a computational interpretation of sequent calculus for modal logic S4
On a computational interpretation of sequent calculus for modal logic S4 Yosuke Fukuda Graduate School of Informatics, Kyoto University Second Workshop on Mathematical Logic and Its Applications March
More informationUniform Schemata for Proof Rules
Uniform Schemata for Proof Rules Ulrich Berger and Tie Hou Department of omputer Science, Swansea University, UK {u.berger,cshou}@swansea.ac.uk Abstract. Motivated by the desire to facilitate the implementation
More informationHarvard School of Engineering and Applied Sciences CS 152: Programming Languages
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Lecture 17 Tuesday, April 2, 2013 1 There is a strong connection between types in programming languages and propositions
More informationFormalising the Completeness Theorem of Classical Propositional Logic in Agda (Proof Pearl)
Formalising the Completeness Theorem of Classical Propositional Logic in Agda (Proof Pearl) Leran Cai, Ambrus Kaposi, and Thorsten Altenkirch University of Nottingham {psylc5, psxak8, psztxa}@nottingham.ac.uk
More informationThe Curry-Howard Isomorphism
The Curry-Howard Isomorphism Software Formal Verification Maria João Frade Departmento de Informática Universidade do Minho 2008/2009 Maria João Frade (DI-UM) The Curry-Howard Isomorphism MFES 2008/09
More informationThe Calculus of Inductive Constructions
The Calculus of Inductive Constructions Hugo Herbelin 10th Oregon Programming Languages Summer School Eugene, Oregon, June 16-July 1, 2011 1 Outline - A bit of history, leading to the Calculus of Inductive
More informationProof-Carrying Code in a Session-Typed Process Calculus
Proof-Carrying Code in a Session-Typed Process Calculus Frank Pfenning with Luís Caires and Bernardo Toninho Department of Computer Science Carnegie Mellon University 1st International Conference on Certified
More informationIntersection Synchronous Logic
UnB 2007 p. 1/2 Intersection Synchronous Logic Elaine Gouvêa Pimentel Simona Ronchi della Rocca Luca Roversi UFMG/UNITO, 2007 UnB 2007 p. 2/2 Outline Motivation UnB 2007 p. 2/2 Outline Motivation Intuitionistic
More informationConsequence Relations and Natural Deduction
Consequence Relations and Natural Deduction Joshua D Guttman Worcester Polytechnic Institute September 16, 2010 Contents 1 Consequence Relations 1 2 A Derivation System for Natural Deduction 3 3 Derivations
More informationAtomic Cut Elimination for Classical Logic
Atomic Cut Elimination for Classical Logic Kai Brünnler kaibruennler@inftu-dresdende echnische Universität Dresden, Fakultät Informatik, D - 01062 Dresden, Germany Abstract System SKS is a set of rules
More informationSequent Calculi and Abstract Machines
Sequent Calculi and Abstract Machines ZENA M. ARIOLA University of Oregon AARON BOHANNON University of Pennsylvania and AMR SABRY Indiana University We propose a sequent calculus derived from the λµ µ-calculus
More information3.2 Reduction 29. Truth. The constructor just forms the unit element,. Since there is no destructor, there is no reduction rule.
32 Reduction 29 32 Reduction In the preceding section, we have introduced the assignment of proof terms to natural deductions If proofs are programs then we need to explain how proofs are to be executed,
More informationNatural deduction for propositional logic via truth tables
Natural deduction for propositional logic via truth tables Herman Geuvers Nijmegen, NL (Joint work with Tonny Hurkens) Bengt Nordström honorary workshop Marstrand, Sweden April 2016 H. Geuvers - April
More informationProving Completeness for Nested Sequent Calculi 1
Proving Completeness for Nested Sequent Calculi 1 Melvin Fitting abstract. Proving the completeness of classical propositional logic by using maximal consistent sets is perhaps the most common method there
More informationPreuves de logique linéaire sur machine, ENS-Lyon, Dec. 18, 2018
Université de Lorraine, LORIA, CNRS, Nancy, France Preuves de logique linéaire sur machine, ENS-Lyon, Dec. 18, 2018 Introduction Linear logic introduced by Girard both classical and intuitionistic separate
More informationLABELLED DEDUCTION. SEÁN MATTHEWS Max-Planck-Insitut für Informatik. LUCA VIGANÒ University of Freiburg
LABELLED DEDUCTN LABELLED DEDUCTN Edited by DAVD BASN University of Freiburg MARCELL D AGSTN University of Ferrara DV M GABBAY King s College SEÁN MATTHEWS Max-Planck-nsitut für nformatik LUCA VGANÒ University
More informationA Formulae-as-Types Interpretation of Subtractive Logic
A Formulae-as-Types Interpretation of Subtractive Logic TRISTAN CROLARD Laboratory of Algorithmics, Complexity and Logic 1 University of Paris XII, France. E-mail: crolard@univ-paris12.fr Abstract We present
More informationhal , version 1-21 Oct 2009
ON SKOLEMISING ZERMELO S SET THEORY ALEXANDRE MIQUEL Abstract. We give a Skolemised presentation of Zermelo s set theory (with notations for comprehension, powerset, etc.) and show that this presentation
More informationl École Normale Supérieure de Lyon Computing with Sequents and Diagrams in Classical Logic - Calculi X, d X and c X
N d ordre: 447 N attribue par la bibliothèque: 07ENSL0 447 THÈSE en vue d obtention le grade de Docteur de l École Normale Supérieure de Lyon spécialité : Informatique Laboratoire de l Informatique du
More informationStratifications and complexity in linear logic. Daniel Murfet
Stratifications and complexity in linear logic Daniel Murfet Curry-Howard correspondence logic programming categories formula type objects sequent input/output spec proof program morphisms cut-elimination
More informationNon-Idempotent Typing Operators, beyond the λ-calculus
Non-Idempotent Typing Operators, beyond the λ-calculus Soutenance de thèse Pierre VIAL IRIF (Univ. Paris Diderot and CNRS) December 7, 2017 Non-idempotent typing operators P. Vial 0 1 /46 Certification
More informationNotation for Logical Operators:
Notation for Logical Operators: always true always false... and...... or... if... then...... if-and-only-if... x:x p(x) x:x p(x) for all x of type X, p(x) there exists an x of type X, s.t. p(x) = is equal
More informationSequent calculus for predicate logic
CHAPTER 13 Sequent calculus for predicate logic 1. Classical sequent calculus The axioms and rules of the classical sequent calculus are: Axioms { Γ, ϕ, ϕ for atomic ϕ Γ, Left Γ,α 1,α 2 Γ,α 1 α 2 Γ,β 1
More informationImplicit Computational Complexity
Implicit Computational Complexity Simone Martini Dipartimento di Scienze dell Informazione Università di Bologna Italy Bertinoro International Spring School for Graduate Studies in Computer Science, 6
More informationEncoding Graph Transformation in Linear Logic
Encoding Graph Transformation in Linear Logic Paolo Torrini joint work with Reiko Heckel pt95@mcs.le.ac.uk University of Leicester GTLL p. 1 Graph Transformation Graph Transformation Systems (GTS) high-level
More informationDeriving natural deduction rules from truth tables (Extended version)
Deriving natural deduction rules from truth tables (Extended version) Herman Geuvers 1 and Tonny Hurkens 2 1 Radboud University & Technical University Eindhoven, The Netherlands herman@cs.ru.nl Abstract
More informationRelative Hilbert-Post completeness for exceptions
Relative Hilbert-Post completeness for exceptions Dominique Duval with J.-G. Dumas, B. Ekici, D. Pous, J.-C. Reynaud LJK University of Grenoble-Alpes and ENS Lyon November 12., 2015 MACIS 2015, Berlin
More informationHenk Barendregt and Freek Wiedijk assisted by Andrew Polonsky. Radboud University Nijmegen. March 5, 2012
1 λ Henk Barendregt and Freek Wiedijk assisted by Andrew Polonsky Radboud University Nijmegen March 5, 2012 2 reading Femke van Raamsdonk Logical Verification Course Notes Herman Geuvers Introduction to
More informationThe unessential in classical logic and computation
The unessenial in classical logic and compuaion Dragiša Žunić Facul of Economics and Engineering Managemen - Fimek Cvećarska 2, 21000 Novi Sad, Serbia ierre Lescanne Ecole Normale Supérieure de Lon 46
More informationTwo odd things about computation. Daniel Murfet University of Melbourne
Two odd things about computation Daniel Murfet University of Melbourne Two odd things I. Maxwell s demon (1871) Energy cost of computation II. Russell s paradox (1901) Time and space cost of computation
More informationProof search for programming in Intuitionistic Linear Logic. (extended abstract) Campus Scientique - B.P France
Proof search for programming in Intuitionistic inear ogic (extended abstract) D. Galmiche & E. Boudinet CIN-CNS & INIA orraine Campus Scientique - B.P. 239 54506 Vanduvre-les-Nancy Cedex France e-mail:
More informationLecture Notes on Focusing
Lecture Notes on Focusing Oregon Summer School 2010 Proof Theory Foundations Frank Pfenning Lecture 4 June 17, 2010 1 Introduction When we recast verifications as sequent proofs, we picked up a lot of
More informationCharacterizing strong normalization in the Curien-Herbelin symmetric lambda calculus: extending the Coppo-Dezani heritage
Characterizing strong normalization in the Curien-Herbelin symmetric lambda calculus: extending the Coppo-Dezani heritage Daniel J. Dougherty a, Silvia Ghilezan b, Pierre Lescanne c a Worcester Polytechnic
More informationExample. Lemma. Proof Sketch. 1 let A be a formula that expresses that node t is reachable from s
Summary Summary Last Lecture Computational Logic Π 1 Γ, x : σ M : τ Γ λxm : σ τ Γ (λxm)n : τ Π 2 Γ N : τ = Π 1 [x\π 2 ] Γ M[x := N] Georg Moser Institute of Computer Science @ UIBK Winter 2012 the proof
More informationPropositional and Predicate Logic. jean/gbooks/logic.html
CMSC 630 February 10, 2009 1 Propositional and Predicate Logic Sources J. Gallier. Logic for Computer Science, John Wiley and Sons, Hoboken NJ, 1986. 2003 revised edition available on line at http://www.cis.upenn.edu/
More informationLecture Notes on Cut Elimination
Lecture Notes on Cut Elimination 15-317: Constructive Logic Frank Pfenning Lecture 10 October 5, 2017 1 Introduction The entity rule of the sequent calculus exhibits one connection between the judgments
More informationOutline. Overview. Syntax Semantics. Introduction Hilbert Calculus Natural Deduction. 1 Introduction. 2 Language: Syntax and Semantics
Introduction Arnd Poetzsch-Heffter Software Technology Group Fachbereich Informatik Technische Universität Kaiserslautern Sommersemester 2010 Arnd Poetzsch-Heffter ( Software Technology Group Fachbereich
More informationAn Algorithmic Interpretation of a Deep Inference System
n lgorithmic Interpretation of a eep Inference System Kai rünnler and ichard McKinley Institut für angewandte Mathematik und Informatik Neubrückstr 10, H 3012 ern, Switzerland bstract We set out to find
More informationNatural Deduction for Propositional Logic
Natural Deduction for Propositional Logic Bow-Yaw Wang Institute of Information Science Academia Sinica, Taiwan September 10, 2018 Bow-Yaw Wang (Academia Sinica) Natural Deduction for Propositional Logic
More informationBidirectional Decision Procedures for the Intuitionistic Propositional Modal Logic IS4
Bidirectional ecision Procedures for the Intuitionistic Propositional Modal Logic IS4 Samuli Heilala and Brigitte Pientka School of Computer Science, McGill University, Montreal, Canada {sheila1,bpientka}@cs.mcgill.ca
More informationCommutative Locative Quantifiers for Multiplicative Linear Logic
Commutative Locative Quantifiers for Multiplicative Linear Logic Stefano Guerrini 1 and Patrizia Marzuoli 2 1 Dipartimento di Informatica Università degli Studi Roma La Sapienza Via Salaria, 113 00198
More informationModel Theory in the Univalent Foundations
Model Theory in the Univalent Foundations Dimitris Tsementzis January 11, 2017 1 Introduction 2 Homotopy Types and -Groupoids 3 FOL = 4 Prospects Section 1 Introduction Old and new Foundations (A) (B)
More informationLecture 11: Measuring the Complexity of Proofs
IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Advanced Course on Computational Complexity Lecture 11: Measuring the Complexity of Proofs David Mix Barrington and Alexis Maciel July
More informationExtending higher-order logic with predicate subtyping: application to PVS
Extending higher-order logic with predicate subtyping: application to PVS Frédéric Gilbert To cite this version: Frédéric Gilbert. Extending higher-order logic with predicate subtyping: application to
More informationA Proof Theory for Generic Judgments
A Proof Theory for Generic Judgments Dale Miller INRIA/Futurs/Saclay and École Polytechnique Alwen Tiu École Polytechnique and Penn State University LICS 2003, Ottawa, Canada, 23 June 2003 Outline 1. Motivations
More informationAttack of the Exponentials
Attack of the Exponentials Damiano Mazza LIPN, CNRS Université Paris 13, France LL2016, Lyon school: 7 and 8 November 2016 D. Mazza (LIPN) Attack of the Exponentials LL2016 1 / 26 Something is missing...
More informationNESTED DEDUCTION IN LOGICAL FOUNDATIONS FOR COMPUTATION
ÉCOLE POLYTECHNIQUE Thèse de Doctorat Spécialité Informatique NESTED DEDUCTION IN LOGICAL FOUNDATIONS FOR COMPUTATION Présentée et soutenue publiquement par NICOLAS GUENOT le 10 Avril 2013 devant le jury
More informationPropositional and Predicate Logic - V
Propositional and Predicate Logic - V Petr Gregor KTIML MFF UK WS 2016/2017 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - V WS 2016/2017 1 / 21 Formal proof systems Hilbert s calculus
More informationCategorical Proof Theory of Classical Propositional Calculus
Categorical Proof Theory of Classical Propositional Calculus Gianluigi Bellin Martin Hyland Edmund Robinson Christian Urban Queen Mary, University of London, UK University of Cambridge, UK Technical University
More informationAbout categorical semantics
About categorical semantics Dominique Duval LJK, University of Grenoble October 15., 2010 Capp Café, LIG, University of Grenoble Outline Introduction Logics Effects Conclusion The issue Semantics of programming
More informationAnalyzing Extraction. N.B.: Each type has its own set of variables (Church typing).
Analyzing Extraction Carl Pollard Linguistics 602.02 Mar. 8, 2007 (1) TLC Revisited To motivate our approach to analyzing nonlocal phenomena, we first look at a reformulation TLC of that makes explicit
More informationLecture Notes on Combinatory Modal Logic
Lecture Notes on Combinatory Modal Logic 15-816: Modal Logic Frank Pfenning Lecture 9 February 16, 2010 1 Introduction The connection between proofs and program so far has been through a proof term assignment
More informationInvestigations on the Dual Calculus
Investigations on the Dual Calculus Nikos Tzevelekos nikt@comlab.ox.ac.uk Abstract The Dual Calculus, proposed recently by Wadler, is the outcome of two distinct lines of research in theoretical computer
More informationThe duality of computation under focus
The duality of computation under focus Pierre-Louis Curien (CNRS, Paris 7, and INRIA) and Guillaume Munch-Maccagnoni (Paris 7 and INRIA) Abstract. We review the close relationship between abstract machines
More informationA BRIEF INTRODUCTION TO TYPED PREDICATE LOGIC
A BRIEF INTRODUCTION TO TYPED PREDICATE LOGIC Raymond Turner December 6, 2010 Abstract Typed Predicate Logic 1 was developed to o er a unifying framework for many of the current logical systems, and especially
More informationSemantical study of intuitionistic modal logics
Semantical study of intuitionistic modal logics Department of Intelligence Science and Technology Graduate School of Informatics Kyoto University Kensuke KOJIMA January 16, 2012 Abstract We investigate
More informationAn Introduction to Proof Theory
An Introduction to Proof Theory Class 1: Foundations Agata Ciabattoni and Shawn Standefer anu lss december 2016 anu Our Aim To introduce proof theory, with a focus on its applications in philosophy, linguistics
More informationConservation of Information
Conservation of Information Amr Sabry (in collaboration with Roshan P. James) School of Informatics and Computing Indiana University May 8, 2012 Amr Sabry (in collaboration with Roshan P. James) (IU SOIC)
More informationTowards Concurrent Type Theory
Towards Concurrent Type Theory Luís Caires 1, Frank Pfenning 2, Bernardo Toninho 1,2 1 Universidade Nova de Lisboa 2 Carnegie Mellon University Workshop on Types in Language Design and Implementation (TLDI)
More informationThe Suspension Notation as a Calculus of Explicit Substitutions
The Suspension Notation as a Calculus of Explicit Substitutions Gopalan Nadathur Digital Technology Center and Department of Computer Science University of Minnesota [Joint work with D.S. Wilson and A.
More informationA few bridges between operational and denotational semantics of programming languages
A few bridges between operational and denotational semantics of programming languages Soutenance d habilitation à diriger les recherches Tom Hirschowitz November 17, 2017 Hirschowitz Bridges between operational
More information