[PPT] - Modal Logics for Updating, Sharing or Composing St ephane Demri PowerPoint Presentation

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Modal Logics for Updating, Sharing or Composing

St´ ephane Demri CNRS, LSV, ENS Paris-Saclay Highlights’20, september 2020

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The role of updates in non-classical logics

Behavioural properties of transition systems expressed in

temporal logics.

q2

C1,N2 turn=1

q1

T1,N2 turn=1

q3

T1,T2 turn=1

q4

C1,T2 turn=1

q0

N1,N2 turn=0

q5

N1,T2 turn=2

q6

N1,C2 turn=2

q7

T1,C2 turn=2

Separation logics: extensions of Hoare-Floyd logic for

(concurrent) programs with mutable data structures.

x y

Logics of public announcements can update the knowledge

states in view of announcements made in the logical language.

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Modal logics updating models are popular!

Second-order modal logics (∀p)

[Fine, Theoria 1970]

Logics of public announcements ([φ])

[Plaza, ISMIS’89]

Sabotage modal logics ()

[van Benthem, 2002]

Relation-changing modal logics (sw)

[Fervari, PhD 2014]

Logic with separating modalities LSM (∗)

[Courtault & Galmiche & Pym, TCS 2016] 3

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This talk

Recent developments on modal logics

with built-in update mechanisms based

n composition.
Relationships with other logical formalisms such as

second-order modal logics, separation logics, team logics,. . . See e.g. [Gr¨

adel et al., 2020] relating separation and team logics.

Results about decidability, computational complexity,

expressive power from joint works with

B. Bednarczyk M. Deters
R. Fervari
A. Mansutti

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Plan of the talk

1 Modal logics for Updating or Composing 2 Foundations: the Logic of Bunched Implications BI 3 Second-Order Modal Logics (with Tree Semantics) 4 Modal Separation Logics 5 New Proposal: Description Logics and Updates

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Modal logics in a nutshell

Formulae: φ ::= p | ¬φ | φ ∧ φ | ♦φ | φ.
Kripke-style structures M = (W , R, V ):
W : non-empty set of worlds.
R ⊆ W × W : accessibility relation.
V : PROP → P(W ): valuation.

w q p, q p q | = ♦♦p ∧ ♦♦¬p ∧ ¬p

Satisfaction relation:
M, w |

= p

def

⇔ w ∈ V (p).

M, w |

= ♦φ

def

⇔ there is w ′ s.t. (w, w ′) ∈ R and M, w ′ | = φ.

M, w |

= φ

def

⇔ for all w ′ s.t. (w, w ′) ∈ R, M, w ′ | = φ.

Modal logics for Updating or Composing

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How to update pointed Kripke-style structures ?

Bottom line: changing the pointed model with ♦.

w q p, q p q | = ♦¬q w′ q p, q p q | = ¬q

Each element from (W , R, V ) could potentially be changed.

(approach advocated in [Aucher et al. ENTCS 2009])

Changing

– W requires the power of some 2nd logic. – R requires the power of some dyadic 2nd logic. – V requires the power of some monadic 2nd logic.

Modal logics for Updating or Composing

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Examples: sabotage and announcement

Saboting the model with (deleting exactly one edge).

w q p, q p q | = ♦ ⊥ w q p, q p q | = ♦ ⊥

See e.g., [van Benthem, 2005; L¨

ding & Rohde, FST&TCS’03]
Removing states with the public announcement [φ].

w q p, q p q | = [♦♦p ∨ ♦q]3p w q p, q p q | = 3p

See e.g.,[Plaza, ISMIS’89]

Modal logics for Updating or Composing

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Other logical formalisms

Propositional quantification ∀p in modal/temporal logics.
Second-order modal logics.

[Bull, JSL 1969; Fine, Theoria 1970]

Quantified CTL with tree semantics.

See [Laroussinie & Markey, LMCS 2014]

Tree-like models and compositions.
Static ambient logics with composition operator .

[Cardelli & Gordon, POPL’00]

Modal separation logic for resource trees.

[Biri & Galmiche, JLC 2006]

Modalities and abstract models based on resources.
Modal relevant logics of processes.

[Dams, PhD thesis 90]

Exploitation of a modality for BI in [Pym, Book 2002], see also

modal BI in [Pym & Tofts, FAC 2006].

Modal extensions of BI.

[Courtault & Galmiche, LFCS’13]

Modal logics for Updating or Composing

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Foundations: Logic of Bunched Implications BI

Foundations: the Logic of Bunched Implications BI

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An abstract view based on resources

Logic of bunched implications BI introduced in

[O’Hearn & Pym, BSL 99]

Boolean BI has classical additive connectives.
BI, Boolean BI and bunched logics defined proof-theoretically

but completeness with different types of resource models.

[Pym, Book 2002; Galmiche et al., MSCS 2005; Docherty, PhD 2019] [Jipsen & Litak, arXiv 2018]

Ingredients for a simple model of resources

[Pym & Tofts, FAC 2006]

– a set R of resource elements, – partial composition ◦ : R × R ⇀ R, – comparing resource elements with ⊑, – zero resource element e.

Foundations: the Logic of Bunched Implications BI

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Boolean BI – the semantics side

Abstract models with composition: BBI-frame (M, ◦, e)
M is a non-empty set,
binary function ◦ : M × M → P(M) such that ◦ is

commutative and associative,

e ∈ M and e ◦ m = {m} for all m ∈ M.
Formulae

φ, ψ ::= I | p | φ ∧ ψ | ¬φ | φ ∗ ψ | φ − ∗ ψ

Satisfaction relation (m ∈ M, V : PROP → P(M)).

m | =V I iff m = e m | =V p iff m ∈ V (p) m | =V φ1 ∗ φ2 iff for some m1, m2 ∈ M, we have m ∈ m1 ◦ m2, m1 | =V φ1 and m2 | =V φ2 m | =V φ1 − ∗ φ2 iff for all m′, m′′ ∈ M such that m′′ ∈ m ◦ m′, if m′ | =V φ1 then m′′ | =V φ2.

Foundations: the Logic of Bunched Implications BI

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Abstract view leading to undecidability but . . .

A formula φ is valid iff for all BBI-models (M, ◦, e, V ) and for

all m ∈ M, we have m | =V φ.

Validity problem for Boolean BI is undecidable.

[Kurucz & N´ emeti & Sain & Simon, JoLLI 1995] [Brotherson & Kanovich; Larchey & Galmiche, LiCS’10]

Decidable concretisations such as separation logics, modal

logics of trees, ambient logics (including modal extensions).

See related structures from substructural logics.
Pieces of information in [Urquhart, JSL 1972].
Information frames (P, ◦, 1, ⊑) for substructural logics.

[D’Agostino & Gabbay, JAR 1994]

Routley-Meyer frames for relevance logics (W , R) with ternary
R. See e.g. [Meyer APAL 2004; Restall, Handbook 2006]

Foundations: the Logic of Bunched Implications BI

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Classical sharing interpretations [Pym, Book 2002]

Separation logics for the verification of program with pointers.

[Reynolds, LiCS’02]

– Separation logics are concretisations of (Boolean) BI. – Memory state (s, h) with s : PVAR → Val, h : Loc ⇀fin Val. – Disjoint heaps when dom(h1) ∩ dom(h2) = ∅ and disjoint union h1 ⊎ h2.

= ⊎

– (s, h) | = φ1 ∗ φ2 iff ∃h1, h2 s.t. h = h1 ⊎ h2, (s, h1) | = φ1, (s, h2) | = φ2.

Petri net semantics for linear logic adjusted to BI’s resource

interpretation with (Nn, +, ⊑, 0) (with n ⊑ m iff n ⇒∗ m).

[Engberg & Winskel, APAL 1997; Pym et al., TCS 2004]

Static ambient logics have models that are finite edge-labelled

trees with composition [Calcagno et al., TLDI’03].

=

Foundations: the Logic of Bunched Implications BI

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Second-order modal logics (with tree semantics)

Second-Order Modal Logics (with Tree Semantics)

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Propositional quantification in modal logics

Changing valuations with ∃ p.

w q p, q p q | = ∃ p p w p, q q q | = p p

See e.g.,[Fine, Theoria 1970]

QK formulae: φ ::= p | ¬φ | φ ∧ φ | ♦φ | φ | ∃ p φ.
M, w |

= ∃ p φ iff there is a p-variant M′ s.t. M′, w | = φ.

Second-order quantification is handy!
To design algorithms for ATL with strategy contexts.

[Laroussinie & Markey, IC 2015]

Relationships with epistemic reasoning.

[Belardinelli & van der Hoek, AAAI’16]

Enriching the modal µ-calculus for control synthesis.

[Riedweg & Pinchinat, MFCS’03]

Second-Order Modal Logics (with Tree Semantics)

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Undecidable logics QL

Variants second-order modal logics QS4, QS5, etc.

See e.g. [Kripke, JSL 1959; Fine, PhD 1969; Kaplan, JSL 1970]

For any modal logic L between K and S4, the satisfiability

problem for QL is undecidable.

[Fine, PhD thesis ’69, Theoria 1970]

The satisfiability problem for QS5 is decidable and QS5 as

expressive as graded modal logic GS5.

Second-Order Modal Logics (with Tree Semantics)

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Moving to tree-like models for QL

Modal logic K characterised by finite tree models and QK is undecidable.

What about complexity of QK on finite tree models (QKt)?

Modal logic S4 characterised by models (W , R∗, V ) s.t. (W , R) is a finite-branching tree with all branches infinite.

. . . . . . . . . . . . . . . . . . . . . . . .

+R∗

What about complexity of QS4 on such tree models (QS4t)?

(QS4 on tree models already considered in [Zach, JPL 2004])

Second-Order Modal Logics (with Tree Semantics)

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Tree semantics in QCTLt [Laroussinie & Markey, LMCS 2014]

Satisfiability problem for QCTLt (tree semantics):

input: a QCTL formula φ.

utput: 1 iff there is a finite total Kripke structure whose

tree unfolding satisfies φ.

| = φ 01 p 011 0111 . . . . . . 0112 p . . . 012 0121 p . . . . . . 02 p 021 . . . . . . 1 2

Equivalently, finite-branching trees with all branches infinite.
SAT(QCTLt) is Tower-complete.

[Laroussinie & Markey, LMCS 2014]

Second-Order Modal Logics (with Tree Semantics)

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Tower-completeness

MSO over the infinite tree SωS is decidable.

[Rabin, TAMS 1969]

Tower upper bound for QKt and QS4t is a consequence of

Rabin’s Theorem.

Tower-hardness for SAT(QKt) and SAT(QS4t).

[Bednarczyk & Demri, LiCS’19]

More Tower-hardness results about fragments of QCTLt

and related logics can be found in the papers

[LMCS 2014; LiCS’19; Mansutti, FoSSaCS’20]

Second-Order Modal Logics (with Tree Semantics)

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Local nominals captured in QKt

Nominals from hybrid (modal) logics: propositional variables

that hold true in a unique world.

[Areces & Blackburn & Marx, JSL 2001]

Qφ: there is a unique world satisfying φ [Fine, PhD 1969].

= ♦kx∧¬∃ p (♦k(x∧p)∧♦k(x∧¬p)).

A toolkit for introducing local nominals x.

– @k

xφ

def

= ♦k(x ∧ φ): this unique descendant satisfies φ. – diff-nom(x1, . . . , xα, k): α distinct descendants at depth k. diff-nom(x1, . . . , xα, k)

def

=

i∈[1,α]

nom(xi, k) ∧

i<j∈[1,α]

¬@k

xixj.

Second-Order Modal Logics (with Tree Semantics)

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Enforcing an exponential number of children

Simulation of first-order quantification on a given set of nodes
f bounded depth.
At most 2n children (♦≤2n⊤ in graded modal logics):

∃ p0, . . . , pn−1  ∀x, y diff-nom(x, y, 1) → ¬(

i∈[0,n−1]

@1

xpi ↔ @1 ypi)

 

[David & Laroussinie & Markey, CONCUR’16]

Second-Order Modal Logics (with Tree Semantics)

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How to prove Tower-hardness for SAT(QKt)

Uniform elementary reduction from k-nexptime-complete

tiling problems Tilingk.

t(0, n)=n and t(k + 1, n)=2t(k,n).
Tilingk:

input:

(T , H, V) (tile types, horizontal and vertical

matching relations),

c = t0, t1, . . . , tn−1 ∈ T n: initial condition.
utput: 1 iff the grid [0, t(k, n) − 1] × [0, t(k, n) − 1] can

be tiled (with usual hori/verti. constraints)?

Second-Order Modal Logics (with Tree Semantics)

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Enforcing large numbers of children

[Bednarczyk & Demri, LiCS’19]

Type k Type (k-1) Type (k-2) Type 0

Second-Order Modal Logics (with Tree Semantics)

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Modal Separation Logics

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Modal separation logic MSL(∗, ♦, =)

Modal separation logics: Kripke-style semantics with modal

and separating connectives.

MSL(∗, ♦, =) inspired from [Demri & Deters, ToCL 2015].
Formulae: φ ::= p | emp | ¬φ | φ ∧ φ | ♦φ | =φ | φ ∗ φ.
Models M = N, R, V are tree-like (heap graphs):
R ⊆ N × N is finite and weakly functional (deterministic),
V : PROP → P(N).

Modal Separation Logics

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Semantics

Disjoint unions M1 ⊎ M2.

M, l | = p

def

⇔ l ∈ V (p) M, l | = ♦φ

def

⇔ M, l′ | = φ, for some l′ ∈ N such that (l, l′) ∈ R M, l | = =φ

def

⇔ M, l′ | = φ, for some l′ ∈ N such that l′ = l M, l | = emp

def

⇔ R = ∅ M, l | = φ1 ∗ φ2

def

⇔ N, R1, V , l | = φ1 and N, R2, V , l | = φ2, for some partition {R1, R2} of R Separating ∗ as prop. quantification: mark the source nodes

| = φ1 ∗ φ2 p p p | = φp

1 ∗ φ¬p 2

Modal Separation Logics

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Expressing simple properties

size ≥ k

def

= ¬emp ∗ · · · ∗ ¬emp

k times

.

The model is a loop of length 2 visiting the current location:

size ≥ 2 ∧ ¬size ≥ 3 ∧ ♦♦♦⊤∧ ¬(¬emp ∗ ♦♦♦⊤) ∧ ¬♦(¬emp ∗ ♦♦♦⊤) (not expressible in modal logic Alt1)

l

Modal Separation Logics

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The tradition of adding modalities to resource logics

Modal separation logic first named in [Zeilberger, draft 2005].
Quantification over heaps.

[Nishimura, AMAST’06]

s, h | = ♦φ iff for some h′, dom(h′) = dom(h) and s, h′ | = φ.

Modal extensions of BI [Courtault & Galmiche, LFCS’13].
Modal/temporal/epistemic extensions of bunched logics.

[Pym & Tofts, FAC 2006; Kamide, TCS 2013; Kimmel, PhD 2018]

Modal separation logic for resource trees.

[Biri & Galmiche, JLC 2006], see also [Conforti, PhD 2005]

Modal Kripke resource models with neighbourhood functions.

[Porello & Troquard, JANCL 2015]

Modal Separation Logics

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Tower-completeness of SAT(MSL(∗, ♦, =))

Tower upper bound for SAT(MSL(∗, ♦, =)) by reduction

into satisfiability for the weak MSO theory of (D, f, =).

Linear model:

l0 l1 . . . ln

There is a formula φ∃ls s.t. M |

= φ∃ls iff M is linear.

Star-free expressions e ::= a | ε | e ∪ e | ee | ∼ e.

– Nonemptiness problem is Tower-complete.

[Meyer & Stockmeyer, STOC’73; Schmitz, ToCT 2016]

– Encoding words by linear models. a1 a2 a1 ⊲

l0 l1 p1 l2 p2 l3 p1 , l0

MSL(∗, ♦, =) satisfiability problem is Tower-hard.

[Demri & Fervari, AiML’18]

Modal Separation Logics

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Results for variant logics

The satisfiability problems for MSL(∗, ♦) and MSL(∗, =) are

np-complete.

Undecidable variant logics.

– MSL(∗, ♦, =) + magic wand − ∗.

[Demri & Fervari, JLC 2019]

– MSL(∗, ♦) over all frames.

[Areces et al., JLC 2018]

More Tower-hardness.

– Modal logic for heaps MLH(∗).

[Demri & Deters, TOCL 2015]

– MSL(∗, ♦, =) restricted to φ ::= ⊤ | ¬φ | φ ∧ φ | Uφ | φ ∗ φ.

[Mansutti, FoSSaCS’20]

– MSL(∗, ♦−1).

[Bednarczyk et al., LiCS’20]

Modal Separation Logics

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Modal logic ML(∗) on finite forests (sister logic MSL(∗, ♦−1))

Formulae: φ ::= p | φ ∧ φ | ¬φ | ♦φ | φ ∗ φ.
Models are finite Kripke-style forests equipped with the

composition from separation logic.

= +

M, w | = φ1 ∗ φ2 iff there are M1, M2 such that M = M1 + M2, M1, w | = φ1 and M2, w | = φ2.

Recent results

[Bednarczyk et al., LiCS’20]

– Satisfiability problem for ML(∗) is Tower-complete. – ML(∗) strictly less expressive than graded modal logic (modalities are ♦≥k). – See also results about ML( ) with from static ambient logic.

Modal Separation Logics

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A Proposal: Dynamic Axioms in DLs

New Proposal: Description Logics and Updates

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Description logics and updates

Description logics are well-known logical formalisms for

knowledge representation.

[Baader et al., Book 2017]

Known updates in DLs, mainly at the level of ABoxes.

See e.g. [Liu et al., KR’06]

How to specify the evolution of the satisfaction of GCIs

(C ⊑ C ′) or assertions (C(a)) when the current interpretation is updated?

New framework based on separating connectives from

separation logics introduced in [Bednarczyk et al., IJCAI’20].

New Proposal: Description Logics and Updates

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ALC and EL in a nutshell

ALC concepts: C, C ′ := ⊤ | A | ¬C | C ⊓ C ′ | ∃r.C
Assertions: C(a), r(a, b).
General concept inclusion (GCI): C1 ⊑ C2.
Knowledge base K = (T , A): T finite set of GCIs and A

finite set of assertions.

Interpretation I = (∆I, ·I) ∈ I

I | = C1 ⊑ C2

def

⇔ C I

1 ⊆ C I 2

I | = C(a)

def

⇔ aI ∈ C I I | = r(a, b)

def

⇔ (aI, bI) ∈ rI

EL defined as the restriction of ALC to ⊓, ∃, ⊤ and to the

concept names.

New Proposal: Description Logics and Updates

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A framework with dynamic axioms – BI stricks back!

KB with dynamic axioms Kda = (T , A, D)

D is a dynamic box (DBox).

Partial composition operator ⊕ : I × I → I with AC ⊕.

In this talk: ⊕ deals with the disjointness of (arbitrary) role interpretations only.

Positive dynamic axioms (PDAs)

U, V := ⊤ | C(a) | r(a, b) | C⊑D | U⊔V | U⊓V | U∗V | U−

⊛V

I | = U1 ∗ U2 iff there are I1, I2 s.t. I = I1 ⊕ I2, I1 | = U1 and I2 | = U2 I | = U1 −

⊛ U2

iff there is I′ s.t. I ⊕ I′ is defined, I′ | = U1 and I ⊕ I′ | = U2.

Dynamic axioms (closure under Boolean operators)

U, V ::= U | ¬U | U ⊔ V | U ⊓ V

New Proposal: Description Logics and Updates

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Consistency problem with dynamic axioms

[Bednarczyk et al, IJCAI’20]

A KB for EL is always consistent.
A KB with PDAs for EL is not always consistent.

r(a, b) ⊓ (r(a, b) −

⊛ ≈⊤

(⊤ ⊑ ⊤))
Consistency of a KB with PDAs characterised by the

non-derivability of ⊥ in a simple calculus.

Logic \ Dynamic axioms Positive DAs DAs EL in PTime undecidable ALC ExpTime-complete undecidable

The consistency problem for ALC with role inclusion axioms

r1 ◦ · · · ◦ rn ⊑ s is undecidable [Baldoni et al., TABLEAUX’98].

CRIAs encoded by dynamic axioms for ALC and for EL.

New Proposal: Description Logics and Updates

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Concluding remarks

Rich framework of modal logics with updates based on

composition operators.

Concrete logical formalisms on the theme “Modalities and BI”

developped by D. Galmiche, D. Pym and colleagues.

Recent developments improve our understanding.

Must-read forthcoming [Mansutti, PhD 2020]

Potential research directions.

– Tractable fragments of modal separation logics. – Relationships with team logics. – Decidable separation logics with partial use of separating implication. – Many more directions, for instance related to DLs and updates.

New Proposal: Description Logics and Updates