Modal Logics for Updating, Sharing or Composing St´ ephane Demri CNRS, LSV, ENS Paris-Saclay Highlights’20, september 2020
The role of updates in non-classical logics • Behavioural properties of transition systems expressed in temporal logics. N 1 , N 2 T 1 , N 2 turn =0 N 1 , T 2 turn =1 turn =2 q 0 q 1 q 5 q 2 q 3 q 6 C 1 , N 2 N 1 , C 2 turn =1 turn =2 T 1 , T 2 turn =1 q 4 q 7 C 1 , T 2 T 1 , C 2 turn =1 turn =2 • Separation logics: extensions of Hoare-Floyd logic for (concurrent) programs with mutable data structures. y x • Logics of public announcements can update the knowledge states in view of announcements made in the logical language. 2
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
This talk Recent developments on modal logics • with built-in update mechanisms based on 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 4
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 5
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. p | = ♦♦ p ∧ ♦♦ ¬ p ∧ � ¬ p w q p , q q • Satisfaction relation: def • M , w | = p ⇔ w ∈ V ( p ). ⇔ there is w ′ s.t. ( w , w ′ ) ∈ R and M , w ′ | def • M , w | = ♦ φ = φ . ⇔ for all w ′ s.t. ( w , w ′ ) ∈ R , M , w ′ | def • M , w | = � φ = φ . 6 Modal logics for Updating or Composing
How to update pointed Kripke-style structures ? • Bottom line: changing the pointed model with ♦ . p p | = ♦ ¬ q | = ¬ q w w ′ q p , q q 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. 7 Modal logics for Updating or Composing
Examples: sabotage and announcement • Saboting the model with � (deleting exactly one edge). p p | = �♦� ⊥ | = ♦� ⊥ w w q p , q q q p , q q See e.g., [van Benthem, 2005; L¨ oding & Rohde, FST&TCS’03] • Removing states with the public announcement [ φ ]. p p = [ ♦♦ p ∨ ♦ q ] � 3 p | = � 3 p | w w q p , q q q p , q q See e.g., [Plaza, ISMIS’89] 8 Modal logics for Updating or Composing
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] 9 Modal logics for Updating or Composing
Foundations: Logic of Bunched Implications BI 10 Foundations: the Logic of Bunched Implications BI
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 . 11 Foundations: the Logic of Bunched Implications BI
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 | m ∈ V ( p ) = V p iff m | = V φ 1 ∗ φ 2 iff for some m 1 , m 2 ∈ M , we have m ∈ m 1 ◦ m 2 , m 1 | = V φ 1 and m 2 | = V φ 2 for all m ′ , m ′′ ∈ M such that m ′′ ∈ m ◦ m ′ , m | = V φ 1 − ∗ φ 2 iff if m ′ | = V φ 1 then m ′′ | = V φ 2 . 12 Foundations: the Logic of Bunched Implications BI
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] 13 Foundations: the Logic of Bunched Implications BI
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( h 1 ) ∩ dom( h 2 ) = ∅ and disjoint union h 1 ⊎ h 2 . ⊎ = – ( s , h ) | = φ 1 ∗ φ 2 iff ∃ h 1 , h 2 s.t. h = h 1 ⊎ h 2 , ( s , h 1 ) | = φ 1 , ( s , h 2 ) | = φ 2 . • Petri net semantics for linear logic adjusted to BI’s resource n ⇒ ∗ � interpretation with ( N n , + , ⊑ , 0 ) (with � n ⊑ � m iff � 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] . = 14 Foundations: the Logic of Bunched Implications BI
Second-order modal logics (with tree semantics) 15 Second-Order Modal Logics (with Tree Semantics)
Propositional quantification in modal logics • Changing valuations with ∃ p . p | = ∃ p � p | = � p w w q p , q q p , q q p q See e.g., [Fine, Theoria 1970] • QK formulae: φ ::= p | ¬ φ | φ ∧ φ | ♦ φ | � φ | ∃ p φ . = ∃ p φ iff there is a p -variant M ′ s.t. M ′ , w | • 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] 16 Second-Order Modal Logics (with Tree Semantics)
Undecidable logics Q L • 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 Q L is undecidable. [Fine, PhD thesis ’69, Theoria 1970] • The satisfiability problem for QS5 is decidable and QS5 as expressive as graded modal logic GS5. 17 Second-Order Modal Logics (with Tree Semantics)
Moving to tree-like models for Q L Modal logic K characterised by finite tree models and QK is undecidable. • What about complexity of QK on finite tree models (QK t )? Modal logic S4 characterised by models + R ∗ ( W , R ∗ , V ) s.t. ( W , R ) is a finite-branching . . . . . . tree with all branches infinite. . . . . . . . . . . . . . . . . . . • What about complexity of QS4 on such tree models (QS4 t )? (QS4 on tree models already considered in [Zach, JPL 2004] ) 18 Second-Order Modal Logics (with Tree Semantics)
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