3.
?
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...
4.
?
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?
5.
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…
6.
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…
10.
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
11.
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.
13.
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
14.
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
15.
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
16.
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
18.
More than papers, ideas
Logic
of Proofs
Judgemental Modal Logic
Applications to type theories
Applications to Security
Separation Logic
19.
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
20.
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
21.
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
22.
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
23.
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
24.
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…]
25.
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…]
26.
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
27.
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…
28.
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...)
29.
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
30.
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
31.
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?
32.
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…)
33.
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
34.
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?...
35.
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!
36.
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
37.
How do you think about DLs?
A
sublogic of FOL? Or a sublogic of Modal
logic?
38.
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
39.
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
40.
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
41.
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
42.
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..
43.
Some systems…
With Hermann&
Alexandre
Poss distributes
classical
Poss doesn’t
distribute
With
Alechina
44.
(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
46.
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.
Be the first to comment