Constructive Modal Logics, once again Valeria de Paiva Nuance Communications Ilha Grande, August 2013

or why it is not working?...

? � Since the early 90s I have been worrying

about constructive modal logics � The reason I started thinking about it was

Linear Logic � Linear logic’s !,? modalities are specially

difficult to model and their (sequent calculus) rules are exactly S4 modal rules

� But S4 is only one of the many modal logics...

? �  S4 is only one of the many modal logics

people use. � Which others would be useful? � Can we model them all? Most of them? � How can we prove things in modal logic? � There are several ways of trying to

improve both the proof theory and the model theory of modal logics.

� What are their pros and cons of each?

Other people were thinking similar thoughts… Over the last 14 years we’ve managed to organize six IMLA workshops to discuss constructive modal logics and their applications…

Constructive Modalities? �  The most successful logical framework ever in

CS �  Temporal logic, knowledge operators, BDI

models, security issues, AI, natural language understanding and inference, databases, etc..

�  Logic used both to create logical representation of information and to reason about it

�  Usually classical modalities…

Google Trends

Google trends 2

Google trends 3

Constructive Modalities? �  Constructive logic: a logical basis for

programming via Curry-Howard correspondences

�  Modalities extremely useful �  Constructive modalities twice as useful? �  examples from applications abound �  Which constructive modalities? �  Usual phenomenon: classical facts can be

construed in many different ways constructively

Timeline for IMLA (Intuitionistic Modal Logic & Applications)

� Six workshops were held as part of: �  FLoC1999, Trento, Italy, �  FLoC2002, Copenhagen, Denmark, �  LiCS2005, Chicago, USA, �  LiCS2008, Pittsburgh, USA, �  14th LMPS in Nancy, France, �  UNILOG 2013, Rio de Janeiro, Brazil.

IMLA 1999 �  Pfenning: Judgemental Modal Logic �  Aczel: Russell-Prawitz modality �  Artemov: Logic of Proofs �  Ferrari, Fiorentini, Miglioli: T-intermediate systems �  Goubault-Larrecq: Simplicial Models �  Hilken, Rydeheard: Sheaf models �  Holm: �  Benaissa, Moggi, Taha, Sheard: Run-time analysis �  Despeyroux, Leleu: HOAS �  Howe: proof search in Lax Logic

IMLA 2002 Dana Scott, Realizability and modality Steve Awodey and Andrej Bauer, Propositions as [types] Claudio Hermida Outlook on relational modalities and simulations Gianluigi Bellin, Towards a formal pragmatics: Olivier Brunet A modal logic for observation-based knowledge representation  Giovanni Sambin, Open truth and closed falsity Davoren; Coulthard; Moor; Goré and A. Nerode Topological semantics for intuitionistic modal logics, and spatial discretisation by A/D maps Maria Emilia Maietti, and Eike Ritter Modal run-time analysis revisited

IMLA 2005 David Walker, Checking Properties of Pointer Programs Aleksandar Nanevski,  A Modal Calculus for Named Control Effects Chun-chieh Shan, A Computational Interpretation of Classical S4 Modal Logic Gianluigi Bellin, A Term Calculus for Dual Intuitionistic Logic  Yde Venema, Intuitionistic Modal Logic: observations from algebra and duality Patrick Girard, Labeling Sequents: motivations and applications Charles Stewart, On the inferential role semantics of modal logic SPECIAL CONTRIBUTION: Gödel's Interpretation of Intuitionism William W. Tait Valeria de Paiva, Constructive Description Logics: work in progress

IMLA 2008 Invited Talk: Frank Pfenning Kensuke Kojima, Atsushi Igarashi: On Constructive Linear-Time Logic Rene Vestergaard, Pierre Lescanne, Hiroakira Ono: Constructive rationality implies backward induction for conscientious players Simon Kramer: Reducing Provability to Knowledge in Multi-Agent Systems Invited Talk: Torben Brauner Neelakantan Krishnaswami: A Modal Sequent Calculus for Propositional Separation Logic Didier Galmiche, Yakoub Salhi: Calculi for an Intuitionistic Hybrid Modal Logic Kurt Ranalter: Two-sequent K and simple fibrations Deepak Garg: Principal-centric Reasoning in Constructive Authorization Logic

IMLA 2011 INVITED SPEAKER: Michael Mendler Robert Simmons/Bernardo Toninho INVITED SPEAKER: Luiz Carlos Pereira INVITED SPEAKER: Brian Logan Gianluigi Bellin INVITED SPEAKER: Lutz Strassburger Newton M. Peron and Marcelo E. Coniglio Giuseppe Primiero

IMLA 2013

�  Gurevich �  Steren, Bonelli �  Freitas, Viana �  Dodo’, Joao Marcos �  Vigano �  Bellin �  Kramer �  Crouch, de Paiva �  Pouliasis, Primiero �  Figallo, Pelaitay

More than papers, ideas � Logic of Proofs � Judgemental Modal Logic � Applications to type theories � Applications to Security � Separation Logic

Intuitionistic Modal Logic �  The programmes of the meetings indicate some of

the lines of research �  E. Moggi: Computational Lambda Calculus �  A. Nerode: control of hybrid systems �  S. Artemov: logic of proofs �  M. Mendler: modal logic for hardware verification �  A. Simpson: world-enriched proof system �  G. Bellin: pragmatics and co-intuitionism �  F. Pfenning: judgemental modal logic, apps �  Benton, Bierman, Sheard, Taha: modalities in FP

What’s the state of play? �  IMLA was created with the goal of making

functional programmers “talk” to philosophical logicians and vice-versa

� Goal not attained � Communities still largely talking past each

other �  Incremental work on intuitionistic modal logics

continues, as do some big research programs

What’s the state of play? Research programs � Artemov: Justification logic and variants � Pfenning: linear and S4 modal logics for

applications � Bellin: co-intuitionistic framework for pragmatics � Avron et al: hyper-sequents � Nerode: topological methods � Vigano/Gurevich: modals for security � De Paiva/Mendler: modals for KR

What’s the state of play? New lines… � Hybrid and descriptive constructive logics:

generalizing the model theory � Coalgebraic modal logics � Deep inference for modal logics � Hyper-sequents and other variants (Lahev,

Salhi, Poggiolesi, etc..) for proof search � Focusing as a generic tool �  Light logics, complexity-oriented � Process algebra-oriented logics for

concurrency, security

What did I expect? � Fully worked out Curry-Howard for a collection

of intuitionistic modal logics � Fully worked out design space for classic logic

and how to move from intuitionistic modal to classic modal

� Full range of applications of modal type systems

� Fully worked out dualities for desirable systems � Collections of implementations for proof

search/proof normalization

What have we got? �  Simpson 1994 a good summary of previous work �  Piecemeal systems from Fitch 1948, with attempts to

framework, (WolterZ, Negri,…) �  An Intuitionistic basis with modalities bolted on

top? too many design decisions �  Possible to classify solutions? �  Was the plan, not done �  will instead catalog my own attempts, as in 2009… �  [As Gilles says, maybe when you’re ready to bury a

project as a dead-end, it returns…]

Possible to classify solutions? �  Well… �  Analogy �  Semantics �  Translations �  Others…

Classification was the plan for this talk, not done �  will instead catalog my own attempts, as in 2009… �  [As Gilles says, maybe when you’re ready to bury a

project as a dead-end, it returns…]

Constructive reasoning � What: Reasoning principles that are safer �  if I ask you whether “There is x such that P(x)”, �  I'm happier with an answer “yes, x_0”, than with

an answer “yes, for all x it is not the case that not P(x)”.

� Why: want reasoning to be as precise and safe as possible

� How: constructive reasoning as much as possible, but classical if need be, pragmatism

A Skewed Timeline �  Beginning of 20th century: Debates over constructive or

classical logics/mathematics �  Modal logics from 1920's - Lewis �  Kripke-like semantics in the 60s. �  Connections constructive/modal logic: – Algebraic McKinsay/Tarski 30s – Kripke semantics, for both 65 – Modal type theories, 90's �  constructive and modal together: �  Fitch 1948 MIPC, Bull 1966, Prawitz 1965, Curry, Fisher-Servi

80's, Bozic-Dosen, 84, Wolter/Zacharyaschev 88 Simpson, Gabbay, Masini/Martini early 90's Mendler, Fairtlough, Bierman/dePaiva, etc

�  Goldblatt History of Modal Logic…

Constructive modal logics �  Basic ideas:

– Box, Diamond are like forall/exists – Intuitionistic logic is like S4-modal logic,

�  – where A-->B=BoxA→B – Combining modalities not that easy...

�  To have ``intuitionistic modal logic” need to have two modalities, how do they interact? It depends on expected behavior

Commuting squares possibilities (Plotkin/Stirling) �  Adding to syntax: hypersequents, labelled

deduction systems, adding semantics to syntax (many ways...)

Personal program � What I want: � constructive modal logics with axioms,

sequents and natural deduction formulations �  with algebraic, Kripke and categorical

semantics � With translations between formulations and

proved equivalences/embeddings � Translating proofs more than simply theorems

broad view of constructive and/or modality �  If possible limitative results

Simpson’s Desiderata �  IML is a conservative extension of IPL. �  IML contains all substitutions instances of theorems

of IPL and is closed under modus ponen. �  Adding excluded middle to IML yields a standard

classical modal logic �  If “A or B” is a theorem of IML either A is a theorem

or B is a theorem too. �  Box and Diamond are independent in IML. �  (Intuitionistic) Meaning of the modalities, wrt it IML is

sound and complete

How the desiderata diverge? � Mostly because he did what he wanted… � Then output behavior diverges. � Main point: distribution of possibility over

disjunction binary and nullary

� This is canonical for classical modal logics �  Is it required for constructive ones or not? � Consequence: adding excluded middle gives

you back classical modal logic or not?

Extensions: Description and Hybrid Logics � Closely associated with modal logics � Both classes tend to be classical logics � We discuss both constructive hybrid logics

(Brauner/dePaiva 03) and constructive description logics (dePaiva05) in turn.

�  (sometimes generalizing helps to decide on the initial system…)

What are hybrid logics? � Extension of modal logic, where we make part

of the syntax of the formulae the worlds at which they're evaluated.

� Add to basic modal logic second kind of propositional symbols (nominals) and satisfaction operators

� A nominal is assumed to be true at exactly one world

� A formula like a:A where a is a nominal and A is a formula is called a satisfaction statement

Constructive Hybrid Logic? � Brauner/dePaiva ('03, '05) � Which kind of constructive? � Depends on kind of constructive modal logic � Many choices for syntax and for models. � Our choice: modal base Simpson-style, Natural

Deduction style. � Results: IHL as a ND system, models, soundness

and completeness, extensions to geometric theories

� Open problem: hybrid system CK style?...

What Are Description Logics? �  A family of logic based Knowledge Representation

formalisms �  – Descendants of semantic networks and KL-ONE

– Describe domain in terms of concepts (classes), roles (properties, relationships) and individuals Distinguished by: �  Formal semantics (typically model theoretic) �   Decidable fragments of FOL (often contained in C2) �   Closely related to Propositional Modal, Hybrid &

Dynamic Logics �   Closely related to Guarded Fragment – Provision of

inference services �   Decision procedures for key problems (satisfiability,

subsumption, etc) �   Implemented systems (highly optimised) Thanks Ian Horrocks!

Description Logic Basics �  Concepts (formulae/unary predicates) �  – E.g., Person, Doctor, HappyParent, etc. �  Roles (modalities/relations) �  – E.g., hasChild, loves

Individuals (nominals/constants) �  – E.g., John, Mary, Italy �  Operators (for forming concepts and roles) restricted so

that: �  –  Satisfiability/subsumption is decidable and, if possible,

of low complexity �  –  No need for explicit use of variables �  –  Features such as counting (graded modalities)

succinctly expressed

How do you think about DLs? � A sublogic of FOL? Or a sublogic of Modal

logic?

DLs via translation �  Into first-order logic t1:ALC → FOL � concept C maps to C(x), role R maps to

relation, quantifiers the point �  Into modal logic t2:ALC → Kn, roles into boxes,

diamonds

Constructivizing DLs… � DL can be defined via t1 translation into FOL To

constructivize it transform FOL into IFOL � Call system IALC � DL can be defined via t2 translation into

multimodal K (Schilds91) � Need to choose a constructive K � Using IK (Simpson) call system iALC, using CK

(Mendler & de Paiva) call system cALC

Two kinds of constructive K

�  If Simpson’s IKàiALC, �  if Mendler/de Paiva CKàcALC � Difference: distribution of possibility over

disjunction and nullary one: � Dia (A or B) → Dia A or Dia B �  Dia (false) → false

Choosing constructive K? �  IK framework, geometric theories � CK only two CK and CS4… � Can show IK is a theory in CK (Mendler/

Scheele) obtained by adding two missing axioms

� Can show IK-like version of CK??? Need to

How far are we? � Starting points are too diverse � Work progressing along individual lines, only � Work in lambda-calculus still fragmented �  maybe logicians modalities really aren’t useful

for computing and vice-versa..

Some systems… With Hermann& Alexandre

With Alechina

Poss distributes

Poss doesn’t distribute

classical

(lack of) Conclusions �  IMLA has not fulfilled its aim � Constructive modal logic is a very productive

field, with new systems coming up every day � Applications abound, theory that explains it not

so much � Are these systems any good? �  I have not clear criteria to offer at the

moment… � Still working on this

Thanks for listening to the ramble…

References �  Natural Deduction and Context as (Constructive) Modality (V. de

Paiva). In Proceedings of the 4th International and Interdisciplinary Conference CONTEXT 2003, Stanford, CA, USA, Springer Lecture Notes in Artificial Intelligence, vol 2680, 2003.

�  Constructive CK for Contexts (M. Mendler, V de Paiva), In Proceedings of the Worskhop on Context Representation and Reasoning, Paris, France, July 2005.

�  Intuitionistic Hybrid Logic (T. Brauner, V. de Paiva), Presented at Methods for Modalities 3, LORIA, Nancy, France, September 22-23, 2003. Full paper in Journal of Applied Logic 2005

�  Modalities in Constructive Logics and Type Theories Preface to the special issue on Intuitionistic Modal Logic and Application of the Journal of Logic and Computation, volume 14, number 4, August 2004. Guest Editors: Valeria de Paiva, Rajeev Gore' and Michael Mendler.

�  Constructive Description Logics: what, why and how. (extended draft) Presented at Context Representation and Reasoning, Riva del Garda, August 2006.

Are there substantial obstacles?