Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Moss’ coalgebraic logic for preordered coalgebras Marta B´ ılkov´ a joint work with Jiˇ r´ ı Velebil MB, JV ALCOP, 18 April 2013 1/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Coalgebras in Set A T -coalgebra is a c : X − → TX MB, JV ALCOP, 18 April 2013 2/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Coalgebras in Set A T -coalgebra is a c : X − → TX Examples → Q A × 2 Deterministic and nondeterministic automata Q − → ( P Q ) A × 2 and Q − Kripke frames for modal logics: c : X − → P X ... many others MB, JV ALCOP, 18 April 2013 2/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Moss’ logic in Set Cover modality An expressive language for T -coalgebras — extend language of BA with a single modality ∇ : T L − → L MB, JV ALCOP, 18 April 2013 3/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Moss’ logic in Set Cover modality An expressive language for T -coalgebras — extend language of BA with a single modality ∇ : T L − → L Semantics For T : Set − → Set and a coalgebra c : X − → TX x � ∇ α iff c ( x ) T � α Where T : Rel − → Rel is a lifting of T MB, JV ALCOP, 18 April 2013 3/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Moss’ logic in Set Results for T finitary and preserving weak pulbacks expressivity for bisimulation axiomatization by modal distributive laws completeness nice proof theory MB, JV ALCOP, 18 April 2013 3/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Base for finitary functors base of α in TX is the smallest finite subset Y of X , with α in TY . � base X ( α ) = { Z ⊆ fin X | α ∈ TZ } If T weakly preserves preimages and intersections a , base is a natural transformation base X : TX − → P ω X a P. Gumm, From T-coalgebras to filter structures and transition systems, Calco 05 MB, JV ALCOP, 18 April 2013 4/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Base for finitary functors base of α in TX is the smallest finite subset Y of X , with α in TY . � base X ( α ) = { Z ⊆ fin X | α ∈ TZ } If T weakly preserves preimages and intersections a , base is a natural transformation base X : TX − → P ω X a P. Gumm, From T-coalgebras to filter structures and transition systems, Calco 05 MB, JV ALCOP, 18 April 2013 4/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity The category Pre Pre has preordered sets as objects, and monotone maps as arrows. Preorders and 2-functors form a 2-category Pre The 2-cell f − → g given by the point-wise order. Factorisation system Pre possesses a factorisaton system ( E , M ) (in the sense of category theory enriched in posets) consisting of E = monotone surjections and M = monotone maps reflecting order . MB, JV ALCOP, 18 April 2013 5/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity The category Pre Pre has preordered sets as objects, and monotone maps as arrows. Preorders and 2-functors form a 2-category Pre The 2-cell f − → g given by the point-wise order. Factorisation system Pre possesses a factorisaton system ( E , M ) (in the sense of category theory enriched in posets) consisting of E = monotone surjections and M = monotone maps reflecting order . MB, JV ALCOP, 18 April 2013 5/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity The category Pre Pre has preordered sets as objects, and monotone maps as arrows. Preorders and 2-functors form a 2-category Pre The 2-cell f − → g given by the point-wise order. Factorisation system Pre possesses a factorisaton system ( E , M ) (in the sense of category theory enriched in posets) consisting of E = monotone surjections and M = monotone maps reflecting order . MB, JV ALCOP, 18 April 2013 5/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Functors in Pre T ∂ is the dual of T , defined T ∂ A = ( TA op ) op L X = [ X op , 2], thus L ∂ X = [ X , 2] op = U X Kripke Polynomial Functors T ::= const X | Id | T ∂ | T + T | T × T | L T MB, JV ALCOP, 18 April 2013 6/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Coalgebras in Pre A T -coalgebra is a monotone map c : X − → TX MB, JV ALCOP, 18 April 2013 7/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Coalgebras in Pre A T -coalgebra is a monotone map c : X − → TX Examples Deterministic and nondeterministic ordered automata a Frames for positive modal logics Frames for distributive substructural logics a D. Perrin, J.E. Pin, Infinite words, 2004. MB, JV ALCOP, 18 April 2013 7/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Coalgebras in Pre A T -coalgebra is a monotone map c : X − → TX Deterministic ordered automata are coalgebras for the functor Id A × 2, i.e. → Q A × 2 c : Q − MB, JV ALCOP, 18 April 2013 7/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Coalgebras in Pre A T -coalgebra is a monotone map c : X − → TX Nondeterministic ordered automata are coalgebras → ( PQ ) A × 2 c : Q − MB, JV ALCOP, 18 April 2013 7/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Coalgebras in Pre A T -coalgebra is a monotone map c : X − → TX Frames for positive modal logics with adjoint modalities � X such that R ( x , y ) ∧ x ′ ≤ x ∧ y ≤ y ′ − R : X → R ( x ′ y ′ ) ✕ are coalgebras for the functors U and L , i.e. c : X − → U X and d : X − → L X defined by c ( x ) = { y | R ( x , y ) } and d ( x ) = { y | R ( y , x ) } MB, JV ALCOP, 18 April 2013 7/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Coalgebras in Pre A T -coalgebra is a monotone map c : X − → TX Frames for distributive substructural logics R : X � X × X ✕ (the fusion fragment) are coalgebras for the functor L ( Id × Id ), i.e. c : X − → L ( X × X ) MB, JV ALCOP, 18 April 2013 7/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Moss’ logic in Pre Cover modality An expressive language for T -coalgebras — extend language of DL with ∇ : T ∂ L − → L MB, JV ALCOP, 18 April 2013 8/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Moss’ logic in Pre Cover modality An expressive language for T -coalgebras — extend language of DL with ∇ : T ∂ L − → L Semantics For T : Pre − → Pre and a coalgebra c : X − → TX x � ∇ α iff c ( x ) T ∂ � α Where T ∂ : Rel(Pre) − → Rel(Pre) is a lifting of T ∂ a a Bilkova, Kurz, Petri¸ san, Velebil CALCO 2010 MB, JV ALCOP, 18 April 2013 8/32
Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Moss’ logic in Pre Results for T finitary and preserving exact squares expressivity for simulation axiomatization by distributive laws – this talk completeness – not yet nice proof theory – not yet MB, JV ALCOP, 18 April 2013 8/32
� � � � � Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Subobjects Functor Sub Sub( X ) is the essentially small preorder of all M -subobjects of X preordered by the existence of a factorisation h Z � Z ′ f f ′ X Sub fin ( X ) is the full subpreorder of Sub( X ) spanned by f : Z � � X having Z finite MB, JV ALCOP, 18 April 2013 9/32
� � � � Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Subobjects Sub on maps Z h ∗ ( Z ) � � h ∗ ( f ) f � Y X h Sub( h ) on f : Z � � X is given by the factorisation of hf . MB, JV ALCOP, 18 April 2013 9/32
� � � � � � � � � � Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity Base T preserving monotone maps in M induces a functor T ∗ X : Sub fin ( X ) − → Sub( TX ) α : � T : Pre − → Pre admits a base , if a free object � � X (called Z � the base of α ) w.r.t. T ∗ X exists on every α : Z � � TX For each α with Z finite there is � α such that for each f ′ � Z ′ Z � TZ ′ Z � iff α f ′ α � Tf ′ TX X MB, JV ALCOP, 18 April 2013 10/32
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