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Slides from Berkeley Colloquium, March 6th, 2015.

- Intuitionistic Modal Logic:

15 Years Later...

Valeria de Paiva

Nuance Communications

Berkeley March 2015

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

1 / 47 - Intuitionistic Modal Logics

...there is no one fundamental logical notion of necessity, nor

consequently of possibility. If this conclusion is valid, the subject

of modality ought to be banished from logic, since propositions

are simply true or false...

[Russell, 1905]

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

2 / 47 - Intuitionistic Modal Logics

One often hears that modal (or some other) logic is pointless

because it can be translated into some simpler language in a

first-order way. Take no notice of such arguments. There is no

weight to the claim that the original system must therefore be

replaced by the new one. What is essential is to single out

important concepts and to investigate their properties.

[Scott, 1971]

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

3 / 47 - Anyways

Intuitionistic Modal Logic and Applications (IMLA)

is a loose association of researchers, meetings

and a certain amount of mathematical common ground.

IMLA stems from the hope that

philosophers,

mathematical logicians

and computer scientists

would share information and tools when investigating

intuitionistic modal logics and modal type theories,

if they knew of each other’s work.

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

4 / 47 - Intuitionistic Modal Logics and Applications IMLA

Workshops:

FLoC1999, Trento, Italy, (Pfenning)

FLoC2002, Copenhagen, Denmark, (Scott and Sambin)

LiCS2005, Chicago, USA, (Walker, Venema and Tait)

LiCS2008, Pittsburgh, USA, (Pfenning, Brauner)

14th LMPS in Nancy, France, 2011 (Mendler, Logan, Strassburger,

Pereira)

UNILOG 2013, Rio de Janeiro, Brazil. (Gurevich, Vigano and Bellin)

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

5 / 47 - Intuitionistic Modal Logics and Applications IMLA

Special volumes:

M. Fairtlough, M. Mendler, Eugenio Moggi (eds.) Modalities in Type

Theory, Mathematical Structures in Computer Science, (2001)

V. de Paiva, R. Gor´

e, M. Mendler (eds.), Modalities in constructive

logics and type theories, Journal of Logic and Computation, (2004)

V. de Paiva, B. Pientka (eds.) Intuitionistic Modal Logic and

Applications (IMLA 2008), Inf. Comput. 209(12): 1435-1436 (2011)

V. de Paiva, M. Benevides, V. Nigam and E. Pimentel (eds.),

Proceedings of the 6th Workshop on Intuitionistic Modal Logic and

Applications (IMLA 2013), Electronic Notes in Theoretical Computer

Science, Volume 300, (2014)

N. Alechina, V. de Paiva (eds.) Intuitionistic Modal Logics

(IMLA2011), Journal of Logic and Computation, (to appear)

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

6 / 47 - Intuitionistic Modal Logic

Basic idea: Modalities over an Intuitionistic Basis

which modalities?

which intuitionistic basis?

why? how?

why so many?

how to choose?

can relate to others?

which are the important theorems?

which are the most useful applications?

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

7 / 47 - Modal Logic

Modalities: the most successful logical framework in CS

Temporal logic, knowledge operators, BDI models, denotational

semantics, effects, security modelling and verification, natural

language understanding and inference, databases, etc..

Logic used both to create logical representation of information and to

reason about it

But usually only classical modalities...

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

8 / 47 - Classical Modalities in Computer Science

Reasoning about [concurrent] programs

Pnueli, The Temporal Logic of Programs, 1977.

ACM Turing Award, 1996.

Reasoning about hardware; model-checking

Clarke, Emerson, Synthesis of Synchronization Skeletons for

Branching Time Temporal Logic, 1981.

Bryant, Clarke, Emerson, McMillan; ACM Kanellakis Award, 1999

Knowledge representation

From frames to KL-ONE to Description Logics

MacGregor87, Baader et al03

Thanks Frank Pfenning!

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

9 / 47 - Intuitionistic Modal Logic

Basic idea: Modalities over an Intuitionistic Basis

which modalities?

which intuitionistic basis?

why? how? my take, based on Curry-Howard correspondence...

why so many?

how to choose?

can relate to others?

which are the important theorems?

which are the most useful applications?

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

10 / 47 - Constructive reasoning in CS

What: Reasoning principles that are safer

if I ask you whether “is there an x such that P(x)?”,

I’m happier with an answer “yes, x0”, 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, classical if need be,

but tell me where...

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

11 / 47 - Constructive Logic

a logical basis for programming via Curry-Howard correspondences

short digression...

Modalities useful in CS

Examples from applications abound (Monadic Language, Separation

Logic, DKAL, etc..)

Constructive modalities ought to be twice as useful?

But which constructive modalities?

Usual phenomenon: classical facts can be ‘constructivized’ in many

different ways. Hence constructive notions multiply

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

12 / 47 - Trivial Case of Curry Howard

Add λ terms to Natural Deduction:

Γ, x : A

t : B

(→ I )

Γ

λx : A.t : A → B

Γ

t : A → B Γ

u : A

(→ E )

Γ

tu : B

Works for conjunction, disjunction too.

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

14 / 47 - Constructive Modal Logics

Operators Box, Diamond (like forall/exists), not interdefinable

How do these two modalities interact?

Depends on expected behavior and on tools you want/can accept to

use

Collection of articles on why is the proof theory of modal logic difficult

child poster of difficulty S5

Adding to syntax: hypersequents, labelled deduction systems, adding

semantics to syntax (many ways...)

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

15 / 47 - Schools of Constructive Modal logic

Control of Hybrid Systems, Nerode et al, from 1990

Logic of Proofs, Justification Logics, Artemov, from 1995

Judgemental Modal Logic, Pfenning et al, from 2001

Separation Logic, Reynolds and O’Hearn

Modalities as Monads, Moggi et al, Lax Logic, Mendler et al, from

1990,

Simpson framework, Negri sequent calculus

Avron hyper-sequents, Dosen’s higer-order sequents, Belnap display

calculus, Bruennler/Strassburger, Poggiolesi and others “Nested

sequents”

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

16 / 47 - What’s the state of play?

IMLA’s goal: functional programmers talking to philosophical

logicians and vice-versa

Not attained, so far

Communities still largely talking past each other

Incremental work on intuitionistic modal logics continues, as well as

some of the research programmes above

Does it make sense to try to change this?

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

17 / 47 - What did I expect fifteen years ago?

Fully worked out Curry-Howard for a collection of intuitionistic modal

logics

Fully worked out design space for intuitionistic modal logic, for

classical 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

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

18 / 47 - Why did I think it would be easy?

Some early successes. Systems: CS4, Lax, CK

CS4: On an Intuitionistic Modal Logic (with Bierman, Studia Logica

2000, conference 1992)

DIML: Explicit Substitutions for Constructive Necessity (with Neil

Ghani and Eike Ritter), ICALP 1998

Lax Logic: Computational Types from a Logical Perspective (with

Benton, Bierman, JFP 1998)

CK: Basic Constructive Modal Logic. (with Bellin and Ritter, M4M

2001), Kripke semantics for CK (with Mendler 2005),

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

19 / 47 - Constructive S4 (CS4)

This is the better behaved modal system, used by G¨

odel and Girard

CS4 motivation is category theory, because of proofs, not simply

provability

Usual intuitionistic axioms plus MP, Nec rule and

Modal Axioms

(A → B) → ( A →

B)

A → A

A →

A

(A → ♦B) → (♦A → ♦B)

A → ♦A

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

20 / 47 - CS4 Sequent Calculus

S4 modal sequent rules already discussed in 1957 by Ohnishi and

Matsumoto:

Γ, A

B

Γ

A

Γ,

A

B

Γ

A

Γ, A

B

Γ

A

Γ, ♦A

B

Γ

♦A

Cut-elimination works, for classical and intuitionistic basis.

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

21 / 47 - CS4 Natural Deduction Calculus

But ND was more complicated.

The rule

Γ

A

Γ

A

(called by Wadler promotion in Linear Logic, where

=!) led to some

controversy.

As presented in Abramsky’s “Computational Interpretation of Linear

Logic” (1993), it leads to calculus that does not satisfy substitution.

A1

A2

C →

A

Given proofs

B

and

1

C , should be able to

A

B

1

C →

A1

C

substitute A1

A1

A2

But a problem, as the promotion rule

B

is not applicable, anymore.

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

22 / 47 - Natural Deduction for CS4

Benton, Bierman, de Paiva and Hyland solved the problem for Linear Logic

in TLCA 1993.

Bierman and de Paiva (Amsterdam 1992, journal 2000) used the same

solution for modal logic.

The solution builds in the substitutions into the rule as

Γ

A1, . . . , Γ

Ak

A1, . . . , Ak

B ( I)

Γ

B

Prawitz uses a notion of essentially modal subformula to guarantee

substitutivity in his monograph.

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

23 / 47 - Natural Deduction for CS4

Usual Intuitionistic ND rules plus:

Γ

A1, . . . , Γ

Ak

A1, . . . , Ak

B

Γ

A

( I )

( E )

Γ

B

Γ

A

Γ

A1, . . . , Γ

Ak , Γ

♦B A1 . . . Ak , B

♦C

Γ

A

(♦E )

(♦I )

Γ

♦C

Γ

♦A

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

24 / 47 - CS4: Properties/Theorems

Axioms satisfy Deduction Thm, are equivalent to sequents,

Sequents satisfy cut-elimination, sub-formula property

ND is equivalent to sequents

ND satisfies normalization, ND assigns λ-terms CH equivalent

Categorical model: monoidal comonad plus box-strong monad

Issue with Prawitz formulation: idempotency of comonad not

warranted...

Problems with system:

Impurity of rules?

Commuting conversions, eek!

what about other modal logics?

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

25 / 47 - Variations on CS4 I: Dual Intuitionistic and Modal Logic

Following Linear Logic, can define a dual system for

-only modal logic.

DIML, after Barber and Plotkin’s DILL, in ICALP 1998.

Γ, x : A, Γ |∆

xM : A

Γ|∆, x : A, ∆

xI : A

Γ|

t : A

Γ|∆

t

( I )

i :

Ai Γ, xi : Ai |∆

u : B

( E )

Γ|∆

t :

A

Γ|∆

let t1, . . . , tn be

x1, . . . , xn in u : B

Less ‘impurity’ on rules, less commuting conversions, but what about ♦?

what about other modal systems?

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

26 / 47 - Variations on CS4 II: Lax Logic

Computational Types from a Logical Perspective, JFP 1998

Motivation: Moggi’s computational lambda calculus, an intuitionistic

modal metalanguage for denotational semantics for programming language

features: non-termination, differing evaluation strategies, non-determinism,

side-effects are examples.

Curry-Howard ‘backwards’ to get the logic: intuitionistic modal logic with

a degenerate possibility, Curry 1952

Modal Axioms

A → ♦A

♦♦A → ♦A

(A → B) → (♦A → ♦B)

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

27 / 47 - Lax Modality Examples

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

28 / 47 - System Lax

Also called CL-logic (for computational lambda calculus)

Better behaved typed lambda-calculus

Definition: the logic CH-equivalent to a strong monad in a CCC

Semantic distinction: computations and values,

If A models values of a type, then T (A) is the

object that models computations of the type A

T is a curious possibility-like modality, Curry 1952,

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

29 / 47 - Lax logic Properties

Axioms, sequents and ND are equivalent

Deduction theorem holds, as does substitution and subject reduction

The term calculus associated is strongly normalizing

The reduction system given is confluent

Cut elimination holds (Curry 1952)

Lax logic (PLL) categorical models as expected

Lax logic (PLL) Kripke models as expected

Fairtlough and Mendler application: hardware correctness, up to

constraints

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

30 / 47 - Constructive K: a difficult one

Constructive K comes from proof-theoretical intuitions provided by

Natural Deduction formulations of logic

Already CS4 does not satisfy distribution of possibiliity over

disjunction: ♦(A ∨ B) ∼

= ♦A ∨ ♦B and ♦⊥ ∼

= ⊥

Modal Axioms

(A → B) → ( A →

B)

(A → B) → ♦A → ♦B

( A × ♦B) → ♦(A × B)

Sequent rules not as symmetric as in constructive S4, harder to model

Γ

A

Γ, A

B

Γ

A

Γ, ♦A

♦B

Note: only one rule for each connective, also ♦ depends on

.

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

31 / 47 - Constructive K Properties

Dual-context only for Box fragment

For Box-fragment, OK. Have subject reduction, normalization and

confluence for associated lambda-calculus.

Have categorical models, but too unconstrained?

Kripke semantics OK

No syntax in CK style for Diamonds...

No ideas for uniformity of systems...

More work necessary here...

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

32 / 47 - What every one else was doing?

Simpson: The Proof Theory and Semantics of Intuitionistic Modal

Logic (1994) a great summary of previous work and a very robust

system for geometric theories in Natural Deduction for intuitionistic

modal logic

Intuition from ”possible world semantics” interpreted in an

intuitionistic metatheory

Justified by faithfulness of translation into intuitionistic first-order,

recovers many of the systems already in the literature

Strong normalization and confluence proved for all the systems

Normalization used to establish completeness of cut-free sequent

calculi and decidability of some of the systems

Systems that are decidable also satisfy ”finite model property”

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

34 / 47 - What every one else was doing?

Arnon Avron (1996) Hypersequents (based on Pottinger and Mints)

Martini and Masini 2-sequents (1996)

Dosen’s higher-order sequents (1985)

Display calculus (Belnap 1982, Kracht, Gore’, survey by Wansing

2002)

multiple-sequent (more than one kind of sequent arrow) Indrejcazk

(1998)

labelled sequent calculus Negri (2005)

Nested sequents: Bruennler (2009), Hein, Stewart and Stouppa,

Strassburger et al,

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

35 / 47 - What I wanted

constructive modal logics with axioms, sequents and natural

deduction formulations

Satisfying cut-elimination, finite model property, (strong)

normalization, confluence and decidability

with algebraic, Kripke and categorical semantics

With translations between formulations and proved

equivalences/embeddings

Translating proofs more than simply theorems

A broad view of constructive and/or modality

If possible limitative results

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

36 / 47 - Simpson’s desiderata

IML is a conservative extension of IPL.

IML contains all substitutions instances of theorems of IPL and is

closed under modus ponens.

If A ∨ B is a theorem of IML either A is a theorem or B is a theorem

too. (Disjunction Property)

Box

and Diamond ♦ are independent in IML

Adding excluded middle to IML yields a standard classical modal logic

(Intuitionistic) Meaning of the modalities, wrt IML is sound and

complete

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

37 / 47 - Avron’s desiderata

A generic proof-theoretical framework should:

Be able to handle a great diversity of logics. Expect to get the ones

logicians have used already

Be independent of any particular semantics

Structures should be built from formulae in the logic and not too

complicated, should yield a “real” subformula property

Rules of inference should have a small fixed number of premisses, and

a local nature of application

Rules for conjunction, disjunction, implication and negation should be

as standard as possible

Proof systems constructed should give us a better understanding of

the corresponding logics and the differences between them

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

38 / 47 - Some divergence: Distribution of Diamond over

Disjunction

distribution of possibility over disjunction binary and nullary: CS4 vs.

IS4 (Simpson)

Example (Distribution)

♦(A ∨ B) → ♦A ∨ ♦B

♦⊥ → ⊥

This is canonical for classical modal logics

Many constructive systems don’t satisfy it

Should it be required for constructive ones or not?

Consequence: adding excluded middle gives you back classical modal

logic or not?

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

39 / 47 - Some divergence: labelled vs. unlabelled systems

proof system should have semantics as part of the syntax?

Example (Introduction of Box)

Γ

A

Γ [xRy ]

y : A

vs.

Γ

A

Γ

x :

A

The introduction rule for

must express that if A holds at every

world y visible from x then

A holds at x .

if, on the assumption that y is an arbitrary world visible from x , we

can show that A holds at y then we can conclude that

A holds at x .

Simpson’s systems have two kinds of hypotheses, x : A which means

that the modal formula A is true in the world x and xRy , which says

that world y is accessible from world x

How reasonable is it to have your proposed semantics as part of your

syntax?

Proof-theoretic properties there, but no categorical semantics?

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

40 / 47 - More divergence: modularity of framework?

The framework of ordinary sequents is not capable of handling all

interesting logics. There are logics with nice, simple semantics

and obvious interest for which no decent, cut-free formulation

seems to exist... Larger, but still satisfactory frameworks should,

therefore, be sought. Avron (1996)

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

41 / 47

Would love if we could modify minimally the system and obtain the other

modal logics. At least the ones in the ”modal cube”. - IK and CK cubes

Hypersequents, 2-sequents, labelled sequents, nested sequents, display

calculi are modular

Cut-elimination for cubes below, syntax works, but very

complicated?...Curry-Howard for CK cube, OK!

Kripke semantics for CK Mendler and Scheele

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

42 / 47 - Conclusions

Panorama of Curry-Howard for constructive modal logics, as I see it.

Plenty of recent work on pure syntax from Bruennler, Strassburger

and many others

Many applications of the ideas of constructive modal logic

Many interesting papers on FRP, see Jagadhesan et al, Jeffrey, Sergei

Winitzki, etc

Still lacking an over-arching framework, is it possible?

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

43 / 47 - Summary

Constructive modal logics are interesting for programmers, logicians

and philosophers. Shame they don’t talk to each other.

At least two families CK and IK, different properties. Hard to produce

good proof theory for them: many augmentations of sequent systems.

S5 (classical or intuitionistic) main example

So far IK better for model theory, CK better for lambda-calculus, but

want both, plus categorical semantics too

Further work

New preprint on fibrational view of CS4.

Can extend it to CK? I am sure we can do it for Linear Logic, but

gains?

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

44 / 47 - Some References I

G. Bierman, V de Paiva

On an Intuitionistic Modal Logic

Studia Logica (65):383-416, 2000.

N. Benton, G. Bierman, V de Paiva

Computational Types from a Logical Perspective I.

Journal of Functional Programming, 8(2):177-193. 1998 .

N. Ghani, V de Paiva, E. Ritter

Explicit Substitutions for Constructive Necessity

ICALP’98 Proceedings, LNCS 1443, 1998

G. Bellin, V de Paiva, E. Ritter

Basic Constructive Modal Logic

Methods for the Modalities, 2001

Valeria de Paiva (Nuance)

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Berkeley March 2015

45 / 47 - Some References II

A. Simpson

The Proof Theory and Semantics of Intuitionistic Modal Logic

PhD thesis, Edinburgh, 1994.

A. Avron

The method of hypersequents in the proof theory of propositional

non-classical logics

http://www.math.tau.ac.il, 1996

H. Wansing

Sequent Systems for Modal Logic

Handbook of Philosophical Logic Volume 8, 2002

S. Negri

Proof analysis in modal logic.

Journal of Philosophical Logic, 34:507544, 2005.

Valeria de Paiva (Nuance)

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Berkeley March 2015

46 / 47 - Some References III

S Marin and L. Strassburger

Label-free Modular Systems for Classical and Intuitionistic Modal

Logics

AiML, 2014.

M. Mendler and S Scheele

On the Computational Interpretation of CKn for Contextual

Information Processing

Informaticae 130, 2014.

Valeria de Paiva (Nuance)

Intuitionistic Modal Logic: 15 Years Later...

Berkeley March 2015

47 / 47