justification logic for constructive modal logic
play

Justification logic for constructive modal logic Sonia Marin With - PowerPoint PPT Presentation

Nov 09, 2022 •434 likes •928 views

Justification logic for constructive modal logic Sonia Marin With Roman Kuznets and Lutz Straburger Inria, LIX, Ecole Polytechnique IMLA17 July 17, 2017 The big picture The big picture Justification logic: G odel: What is the

  1. Justification logic for constructive modal logic Sonia Marin With Roman Kuznets and Lutz Straßburger Inria, LIX, ´ Ecole Polytechnique IMLA’17 July 17, 2017

  2. The big picture

  3. The big picture Justification logic: G¨ odel: What is the classical provability semantics of intuitionistic logic? Artemov: Logic of Proofs gives an operational view of this S4 type of provability. ✷ A ❀ t : A ❀ t is a proof of A

  4. The big picture Justification logic: G¨ odel: What is the classical provability semantics of intuitionistic logic? Artemov: Logic of Proofs gives an operational view of this S4 type of provability. ✷ A ❀ t : A ❀ t is a proof of A Semantics: Peano arithmetics or epistemic possible worlds models Extensions: realisation of logics below and above S4

  5. The big picture Justification logic: G¨ odel: What is the classical provability semantics of intuitionistic logic? Artemov: Logic of Proofs gives an operational view of this S4 type of provability. ✷ A ❀ t : A ❀ t is a proof of A Semantics: Peano arithmetics or epistemic possible worlds models Extensions: realisation of logics below and above S4 Intuitionistic variants: Some investigations toward ◮ realisation theorems (Artemov/Steren and Bonelli), ◮ epistemic semantics (Marti and Studer), ◮ and arithmetical completeness (Artemov and Iemhoff), but where the modal language is restricted to the ✷ modality.

  6. The big picture Justification logic: G¨ odel: What is the classical provability semantics of intuitionistic logic? Artemov: Logic of Proofs gives an operational view of this S4 type of provability. ✷ A ❀ t : A ❀ t is a proof of A Semantics: Peano arithmetics or epistemic possible worlds models Extensions: realisation of logics below and above S4 Intuitionistic variants: Some investigations toward ◮ realisation theorems (Artemov/Steren and Bonelli), ◮ epistemic semantics (Marti and Studer), ◮ and arithmetical completeness (Artemov and Iemhoff), but where the modal language is restricted to the ✷ modality. However, intuitionistically ✸ cannot simply be viewed as the dual of ✷ .

  7. What are we doing here? Justifying ✸ : We start with Artemov’s treatment of the ✷ -fragment of intuitonistic modal logic.

  8. What are we doing here? Justifying ✸ : We start with Artemov’s treatment of the ✷ -fragment of intuitonistic modal logic. ✷ being read as provability, we propose to read ✸ as consistency. µ : A ✸ A ❀ ❀ µ is an witness of A

  9. What are we doing here? Justifying ✸ : We start with Artemov’s treatment of the ✷ -fragment of intuitonistic modal logic. ✷ being read as provability, we propose to read ✸ as consistency. µ : A ✸ A ❀ ❀ µ is an witness of A Intuitionistic modal logic?

  10. What are we doing here? Justifying ✸ : We start with Artemov’s treatment of the ✷ -fragment of intuitonistic modal logic. ✷ being read as provability, we propose to read ✸ as consistency. µ : A ✸ A ❀ ❀ µ is an witness of A Intuitionistic modal logic? The program: represent the operational side of the intuitionistic ✸ .

  11. What are we doing here? Justifying ✸ : We start with Artemov’s treatment of the ✷ -fragment of intuitonistic modal logic. ✷ being read as provability, we propose to read ✸ as consistency. µ : A ✸ A ❀ ❀ µ is an witness of A Intuitionistic modal logic? The program: represent the operational side of the intuitionistic ✸ . The focus: on constructive versions of modal logic.

  12. Constructive modal logic Formulas: A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A Logic CK : Intuitionistic Propositional Logic

  13. Constructive modal logic Formulas: A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A | ✷ A | ✸ A Logic CK : Intuitionistic Propositional Logic A k 1 : ✷ ( A ⊃ B ) ⊃ ( ✷ A ⊃ ✷ B ) + + necessitation: − − − k 2 : ✷ ( A ⊃ B ) ⊃ ( ✸ A ⊃ ✸ B ) ✷ A (Wijesekera/Bierman and de Paiva/Mendler and Scheele)

  14. Justification logic Justification logic adds proof terms directly inside its language. ✷ A t : A t is a proof of A ❀ ❀

  15. Justification logic Justification logic adds proof terms directly inside its language. ✷ A t : A t is a proof of A ❀ ❀ In the constructive version, we also add witness terms into the language. ✸ A ❀ µ : A ❀ µ is a witness of A

  16. Justification logic Modal formulas: A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A | ✷ A A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A | t : A Justification formulas: Grammar of terms: t ::= | | ( t · t ) | ( t + t ) | ! t c x c : proof constants x : proof variables · : application + : sum ! : proof checker

  17. Justification logic for constructive modal logic Modal formulas: A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A | ✷ A | ✸ A A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A | t : A | µ : A Justification formulas: Grammar of terms: t ::= | | ( t · t ) | ( t + t ) | ! t c x c : proof constants x : proof variables · : application + : sum ! : proof checker

  18. Justification logic for constructive modal logic Modal formulas: A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A | ✷ A | ✸ A A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A | t : A | µ : A Justification formulas: Grammar of terms: t ::= | | ( t · t ) | ( t + t ) | ! t c x µ ::= α | t ⋆ µ | ( µ ⊔ µ ) c : proof constants x : proof variables · : application + : sum ! : proof checker

  19. Justification logic for constructive modal logic Modal formulas: A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A | ✷ A | ✸ A A ::= ⊥ | a | A ∧ A | A ∨ A | A ⊃ A | t : A | µ : A Justification formulas: Grammar of terms: t ::= | | ( t · t ) | ( t + t ) | ! t c x µ ::= α | t ⋆ µ | ( µ ⊔ µ ) c : proof constants x : proof variables α : witness variables · : application ⋆ : execution + : sum ⊔ : disjoint witness union ! : proof checker

  20. Justification logic for constructive modal logic Axiomatisation JCK : taut: Complete finite set of axioms for intuitionistic propositional logic jk ✷ : t : ( A ⊃ B ) ⊃ ( s : A ⊃ t · s : B ) sum: s : A ⊃ ( s + t ) : A and t : A ⊃ ( s + t ) : A A ⊃ B A A is an axiom instance mp − ian − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − B c 1 : . . . c n : A

  21. Justification logic for constructive modal logic Axiomatisation JCK : taut: Complete finite set of axioms for intuitionistic propositional logic jk ✷ : t : ( A ⊃ B ) ⊃ ( s : A ⊃ t · s : B ) jk ✸ : t : ( A ⊃ B ) ⊃ ( µ : A ⊃ t ⋆ µ : B ) sum: s : A ⊃ ( s + t ) : A and t : A ⊃ ( s + t ) : A union: µ : A ⊃ ( µ ⊔ ν ) : A and ν : A ⊃ ( µ ⊔ ν ) : A A ⊃ B A A is an axiom instance mp − ian − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − B c 1 : . . . c n : A

  22. Justification logic for constructive modal logic Axiomatisation JCK : taut: Complete finite set of axioms for intuitionistic propositional logic jk ✷ : t : ( A ⊃ B ) ⊃ ( s : A ⊃ t · s : B ) jk ✸ : t : ( A ⊃ B ) ⊃ ( µ : A ⊃ t ⋆ µ : B ) sum: s : A ⊃ ( s + t ) : A and t : A ⊃ ( s + t ) : A union: µ : A ⊃ ( µ ⊔ ν ) : A and ν : A ⊃ ( µ ⊔ ν ) : A A ⊃ B A A is an axiom instance mp − ian − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − B c 1 : . . . c n : A

  23. The machinery Application: jk ✷ : t : ( A ⊃ B ) ⊃ ( s : A ⊃ t · s : B ) If t is a proof of A ⊃ B and s is a proof of A , then t · s is a proof of B .

  24. The machinery Application: jk ✷ : t : ( A ⊃ B ) ⊃ ( s : A ⊃ t · s : B ) If t is a proof of A ⊃ B and s is a proof of A , then t · s is a proof of B . jk ✸ : t : ( A ⊃ B ) ⊃ ( µ : A ⊃ t ⋆ µ : B ) Witness execution: If t is a proof of A ⊃ B and µ is a witness for A , then the same model denoted t ⋆ µ is also a witness for B .

  25. The machinery Application: jk ✷ : t : ( A ⊃ B ) ⊃ ( s : A ⊃ t · s : B ) If t is a proof of A ⊃ B and s is a proof of A , then t · s is a proof of B . jk ✸ : t : ( A ⊃ B ) ⊃ ( µ : A ⊃ t ⋆ µ : B ) Witness execution: If t is a proof of A ⊃ B and µ is a witness for A , then the same model denoted t ⋆ µ is also a witness for B . s : A ⊃ ( s + t ) : A , µ : A ⊃ ( µ ⊔ ν ) : B , . . . Sum and union: We adopt Artemov’s + to incorporate monotonicity of reasoning, and also transpose it on the witness side with ⊔ .

  26. The machinery Application: jk ✷ : t : ( A ⊃ B ) ⊃ ( s : A ⊃ t · s : B ) If t is a proof of A ⊃ B and s is a proof of A , then t · s is a proof of B . jk ✸ : t : ( A ⊃ B ) ⊃ ( µ : A ⊃ t ⋆ µ : B ) Witness execution: If t is a proof of A ⊃ B and µ is a witness for A , then the same model denoted t ⋆ µ is also a witness for B . s : A ⊃ ( s + t ) : A , µ : A ⊃ ( µ ⊔ ν ) : B , . . . Sum and union: We adopt Artemov’s + to incorporate monotonicity of reasoning, and also transpose it on the witness side with ⊔ . Iterated axiom necessitation and modus ponens:

  27. The machinery Justification logic can internalise its own reasoning.

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Formalizing Classical Modal Logic in Constructive Logic Christian Doczkal Gert Smolka

Formalizing Classical Modal Logic in Constructive Logic Christian Doczkal Gert Smolka

Formalizing Classical Modal Logic in Constructive Logic Christian Doczkal Gert Smolka Programming Systems Lab, Saarland University Coq-3 Workshop, Nijmegen, August 26, 2011 Christian Doczkal (Saarland University) Classical Modal Logic in

464 views • 32 slides
Justification Logic Who? Natalia Kotsani - based on the work of S. Artemov (The Logic of

Justification Logic Who? Natalia Kotsani - based on the work of S. Artemov (The Logic of

Justification Logic Who? Natalia Kotsani - based on the work of S. Artemov (The Logic of Justification, 2008) When? 24 October 2012 Table of contents 1. Introduction An example What is Justification Logic Why Justification Logic

392 views • 20 slides
Modal logic Benzm uller/Rojas, 2014 Artificial Intelligence 2 What is Modal Logic?

Modal logic Benzm uller/Rojas, 2014 Artificial Intelligence 2 What is Modal Logic?

Modal logic Benzm uller/Rojas, 2014 Artificial Intelligence 2 What is Modal Logic? Narrowly, traditionally: modal logic studies reasoning that involves the use of the expressions necessarily and possibly . More widely: modal logic

376 views • 36 slides
Justification Logic Who? Natalia Kotsani - based on the work of S. Artemov (The Logic of

Justification Logic Who? Natalia Kotsani - based on the work of S. Artemov (The Logic of

Justification Logic Who? Natalia Kotsani - based on the work of S. Artemov (The Logic of Justification, 2008) When? 28 November 2012 Table of contents 3.Basic Systems Justification Logic J 0 Logical Awareness Constants Specification Basic

293 views • 15 slides
Introduction to Temporal Logic Mehdi Dastani BBL-521 M.M.Dastani@uu.nl Modal Logic Modal logic

Introduction to Temporal Logic Mehdi Dastani BBL-521 M.M.Dastani@uu.nl Modal Logic Modal logic

Introduction to Temporal Logic Mehdi Dastani BBL-521 M.M.Dastani@uu.nl Modal Logic Modal logic is developed to model various concepts and phenomena. Logic of and (necessity / possibility) Various flavours: epistemic /

547 views • 18 slides
Introduction to modal logic Jos Proena HASLab - INESC TEC Universidade do Minho Braga,

Introduction to modal logic Jos Proena HASLab - INESC TEC Universidade do Minho Braga,

Introduction to modal logic Jos Proena HASLab - INESC TEC Universidade do Minho Braga, Portugal Maro, 2016 Whats in a logic? Modal Logic Bisimulation and modal equivalence Temporal logic Hybrid logic A logic A language i.e. a

1.44k views • 64 slides
Introduction to modal logic Lus Soares Barbosa Jos Proena HASLab - INESC TEC Universidade

Introduction to modal logic Lus Soares Barbosa Jos Proena HASLab - INESC TEC Universidade

Introduction to modal logic Lus Soares Barbosa Jos Proena HASLab - INESC TEC Universidade do Minho Braga, Portugal February/March 2018 Whats in a logic? Modal Logic Bisimulation and modal equivalence Richer modal logics A logic

823 views • 59 slides
Temporal and Modal Logic Based on paper: E.A. Emerson. Temporal and Modal Logic J. van Leeuwen,

Temporal and Modal Logic Based on paper: E.A. Emerson. Temporal and Modal Logic J. van Leeuwen,

Temporal and Modal Logic Based on paper: E.A. Emerson. Temporal and Modal Logic J. van Leeuwen, editor, Handbook of Theoretical Computer Science, Volume B: Formal Models and Semantics, pages 9951072, Elsevier, 1990. Temporal and Modal

554 views • 27 slides
Modal dal Logic ic Submitted to Prof . Lubomr Popelnsk, Masaryk University Prepared by

Modal dal Logic ic Submitted to Prof . Lubomr Popelnsk, Masaryk University Prepared by

Outline Introduction Kripke's Formulation of Modal Logic Frames and Forcing Modal Tableaux Soundness and completeness Modal Axioms and special Accessibility Relations Modal dal Logic ic Submitted to Prof . Lubomr Popelnsk, Masaryk

582 views • 34 slides
Model Theory of Modal Logic Lecture 1: A brief introduction to modal logic Valentin Goranko

Model Theory of Modal Logic Lecture 1: A brief introduction to modal logic Valentin Goranko

Introduction Model Theory of Modal Logic Lecture 1: A brief introduction to modal logic Valentin Goranko Technical University of Denmark Third Indian School on Logic and its Applications Hyderabad, 25 January, 2010 Introduction Some course

600 views • 29 slides
The Expressive Power of Backround Modal Dependence Logic Modal logic Team semantics Modal

The Expressive Power of Backround Modal Dependence Logic Modal logic Team semantics Modal

The Expressive Power of Modal Dependence Logic Jonni Virtema The Expressive Power of Backround Modal Dependence Logic Modal logic Team semantics Modal dependence logic Jonni Virtema Modal definability Succinctness Japan Advanced

769 views • 37 slides
On the relation between possibilistic logic and modal logics of belief Mohua Banerjee 1 , Didier

On the relation between possibilistic logic and modal logics of belief Mohua Banerjee 1 , Didier

Possibilistic vs. modal logic Minimal Epistemic Logic Generalized Possibilistic Logic GPL with objective formulas On the relation between possibilistic logic and modal logics of belief Mohua Banerjee 1 , Didier Dubois 2 , Lluis Godo 3 , Henri

611 views • 23 slides
Why is modal logic decidable Petros Potikas NTUA 9/5/2017 Petros Potikas (NTUA) Modal logic

Why is modal logic decidable Petros Potikas NTUA 9/5/2017 Petros Potikas (NTUA) Modal logic

Why is modal logic decidable Petros Potikas NTUA 9/5/2017 Petros Potikas (NTUA) Modal logic decidability 9/5/2017 1 / 26 Outline Introduction 1 Syntax 2 Semantics 3 Modal logic vs. First-Order Logic 4 Petros Potikas (NTUA) Modal

1.28k views • 94 slides
Modal logic for concurrent processes Lu s Soares Barbosa Interaction & Concurrency

Modal logic for concurrent processes Lu s Soares Barbosa Interaction & Concurrency

Modal logic for concurrent processes Lu s Soares Barbosa Interaction & Concurrency Course Unit (Lcc) Universidade do Minho, 23.III.2018 Modal languages Hennessy-Milner logic Modal equivalence and bissimulation Motivation Systems

461 views • 32 slides
T alk outline 1. Higher-Order Modal Logic (HOML) 2. Automating HOML 3. Evaluation 4. Example /

T alk outline 1. Higher-Order Modal Logic (HOML) 2. Automating HOML 3. Evaluation 4. Example /

Theorem Provers for Every Normal Modal Logic 1 T obias Gleiner Alexander Steen Christoph Benzmller Freie Universitt Berlin 21st Int. Conf. on Logic for Programming, Artificial Intelligence and Reasoning 1This work has been supported by

852 views • 53 slides
Intuitionistic and Modal Logic Dick de Jongh Institute for Logic, Language and Computation

Intuitionistic and Modal Logic Dick de Jongh Institute for Logic, Language and Computation

Intuitionistic and Modal Logic Dick de Jongh Institute for Logic, Language and Computation University of Amsterdam dickdj@science.uva.nl Dick de Jongh Days in Logic 08 Overview Introduction to intuitionism Proofs and proof systems

1.71k views • 128 slides
Constructive Action and Discipline Background MOU signed in March 2015 Added to CBA,

Constructive Action and Discipline Background MOU signed in March 2015 Added to CBA,

Constructive Action and Discipline Background MOU signed in March 2015 Added to CBA, Article 13 on 9/1/2016 Purpose: Decrease response time to address Faculty member issues Give supervisors tools to engage with Faculty members

532 views • 12 slides
Overview of Sustainability & Rating Systems in Qatar 20 th November 2018 Presented by: K

Overview of Sustainability & Rating Systems in Qatar 20 th November 2018 Presented by: K

Overview of Sustainability & Rating Systems in Qatar 20 th November 2018 Presented by: K Venkatesan, CSP, CMIOSH, GSAS-CGP, LEED AP Corporate HSE & Sustainability Manager Outline What is Sust stain ainability ability Why Sust

1.56k views • 38 slides
Regional industry forum Dubbo Roads and Maritime Services Alistair Lunn Director Western

Regional industry forum Dubbo Roads and Maritime Services Alistair Lunn Director Western

Regional industry forum Dubbo Roads and Maritime Services Alistair Lunn Director Western Region Roads and Maritime Services Welcome to Country The Hon. Melinda Pavey Minister for Roads, Maritime and Freight Roy Wakelin-King, AM

434 views • 28 slides
PRESS RELEASE FOR IMMEDIATE RELEASE CIC Honours Outstanding Volunteers and Organisations at the

PRESS RELEASE FOR IMMEDIATE RELEASE CIC Honours Outstanding Volunteers and Organisations at the

PRESS RELEASE FOR IMMEDIATE RELEASE CIC Honours Outstanding Volunteers and Organisations at the First Construction Industry Volunteer Award Presentation Ceremony Hong Kong 15 July 2018 The Construction Industry Council (CIC) held the first

395 views • 4 slides
Critical Heritage Studies and the Future of Europe_Towards an integrated, interdisciplinary and

Critical Heritage Studies and the Future of Europe_Towards an integrated, interdisciplinary and

Critical Heritage Studies and the Future of Europe_Towards an integrated, interdisciplinary and transnational training model in cultural heritage research and management [CHEurope] CHEurope focuses on developing a new integrated theoretical and

470 views • 12 slides
Notices under the Your speakers: Occupational Health David Catanese and Safety Act 2004 :

Notices under the Your speakers: Occupational Health David Catanese and Safety Act 2004 :

Notices under the Your speakers: Occupational Health David Catanese and Safety Act 2004 : Principal Solicitor Protecting your agency Andrew Palmer and its reputation Barrister Hayley Petrony Assistant Victorian Government Solicitor Ph

413 views • 18 slides
April 13, 2011 Report on Faculty Diversity and Office of Strategic Planning and Institutional

April 13, 2011 Report on Faculty Diversity and Office of Strategic Planning and Institutional

Faculty Meeting April 13, 2011 Report on Faculty Diversity and Office of Strategic Planning and Institutional Diversity Office Structure and Function Structure/Composition Vice President for Strategic Planning and Institutional Diversity

724 views • 11 slides
PRECISION MEDICINE AUTUMN 2020 ASSAF KLEIN, CEO DISCLAIMER IMPORTANT NOTICE This presentation

PRECISION MEDICINE AUTUMN 2020 ASSAF KLEIN, CEO DISCLAIMER IMPORTANT NOTICE This presentation

A GAME CHANGER IN ENDOSCOPIC ULTRASOUND GUIDED BIOPSY BETTER BIOPSIES TO ADVANCE PRECISION MEDICINE AUTUMN 2020 ASSAF KLEIN, CEO DISCLAIMER IMPORTANT NOTICE This presentation includes certain statements and of an offer to buy, the C

875 views • 13 slides

More recommend