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A categorical model for 2-PDAs with states J urgen Koslowski Department of Theoretical Computer Science Technical University Braunschweig cmat14, Coimbra (2014-01-25) http://www.iti.cs.tu-bs.de/koslowj/RESEARCH J urgen Koslowski

  1. Categorical approaches to LTSs (0) Most categorical approaches have focussed on labled transition systems (LTSs), that form the core of FAs (disregarding initial/final states): � ! , ℓ � (faithful graph morphism) G Σ s ℓ G 0 G 1 Σ (jointly mono) t � s , t � ℓ G 0 × G 0 G 1 Σ (jointly mono, relation G 0 × G 0 Σ) L Σ × G 0 G 0 (non-obvious relation) s where G = ( G 1 G 0 ) is a finite graph, Σ is an alphabet, and ! t Σ = (Σ 1) is a single-node graph with hom-set Σ . ! J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 4 / 17

  2. Categorical approaches to LTSs (0) Most categorical approaches have focussed on labled transition systems (LTSs), that form the core of FAs (disregarding initial/final states): � ! , ℓ � (faithful graph morphism) G Σ s ℓ G 0 G 1 Σ (jointly mono) t � s , t � ℓ G 0 × G 0 G 1 Σ (jointly mono, relation G 0 × G 0 Σ) L (Σ × G 0 ) P G 0 (coalgebra, Aczel and Mendler [1989]) s where G = ( G 1 G 0 ) is a finite graph, Σ is an alphabet, and ! t Σ = (Σ 1) is a single-node graph with hom-set Σ . ! J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 4 / 17

  3. Categorical approaches to LTSs (0) Most categorical approaches have focussed on labled transition systems (LTSs), that form the core of FAs (disregarding initial/final states): � ! , ℓ � (faithful graph morphism) G Σ s ℓ G 0 G 1 Σ (jointly mono) t � s , t � ℓ G 0 × G 0 G 1 Σ (jointly mono, relation G 0 × G 0 Σ) L (Σ × G 0 ) P G 0 (coalgebra, Aczel and Mendler [1989]) L G 0 × Σ (non-obvious relation, textbook LTS?) G 0 s where G = ( G 1 G 0 ) is a finite graph, Σ is an alphabet, and ! t Σ = (Σ 1) is a single-node graph with hom-set Σ . ! J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 4 / 17

  4. Categorical approaches to LTSs (0) Most categorical approaches have focussed on labled transition systems (LTSs), that form the core of FAs (disregarding initial/final states): � ! , ℓ � (faithful graph morphism) G Σ s ℓ G 0 G 1 Σ (jointly mono) t � s , t � ℓ G 0 × G 0 G 1 Σ (jointly mono, relation G 0 × G 0 Σ) L (Σ × G 0 ) P G 0 (coalgebra, Aczel and Mendler [1989]) L G 0 × Σ (textbook LTS!) G 0 P s where G = ( G 1 G 0 ) is a finite graph, Σ is an alphabet, and ! t Σ = (Σ 1) is a single-node graph with hom-set Σ . ! J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 4 / 17

  5. Categorical approaches to LTSs (0) Most categorical approaches have focussed on labled transition systems (LTSs), that form the core of FAs (disregarding initial/final states): � ! , ℓ � (faithful graph morphism) G Σ s ℓ G 0 G 1 Σ (jointly mono) t � s , t � ℓ G 0 × G 0 G 1 Σ (jointly mono, relation G 0 × G 0 Σ) L (Σ × G 0 ) P G 0 (coalgebra, Aczel and Mendler [1989]) L G 0 × Σ (textbook LTS!) G 0 P L Σ G 0 × G 0 (reversed obvious relation) s where G = ( G 1 G 0 ) is a finite graph, Σ is an alphabet, and ! t Σ = (Σ 1) is a single-node graph with hom-set Σ . ! J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 4 / 17

  6. Categorical approaches to LTSs (0) Most categorical approaches have focussed on labled transition systems (LTSs), that form the core of FAs (disregarding initial/final states): � ! , ℓ � (faithful graph morphism) G Σ s ℓ G 0 G 1 Σ (jointly mono) t � s , t � ℓ G 0 × G 0 G 1 Σ (jointly mono, relation G 0 × G 0 Σ) L (Σ × G 0 ) P G 0 (coalgebra, Aczel and Mendler [1989]) L G 0 × Σ (textbook LTS!) G 0 P L Σ ( G 0 × G 0 ) P (this looks promising) s where G = ( G 1 G 0 ) is a finite graph, Σ is an alphabet, and ! t Σ = (Σ 1) is a single-node graph with hom-set Σ . ! J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 4 / 17

  7. Categorical approaches to LTSs (0) Most categorical approaches have focussed on labled transition systems (LTSs), that form the core of FAs (disregarding initial/final states): � ! , ℓ � (faithful graph morphism) G Σ s ℓ G 0 G 1 Σ (jointly mono) t � s , t � ℓ G 0 × G 0 G 1 Σ (jointly mono, relation G 0 × G 0 Σ) L (Σ × G 0 ) P G 0 (coalgebra, Aczel and Mendler [1989]) L G 0 × Σ (textbook LTS!) G 0 P L Σ ( G 0 , G 0 ) rel (hom-component of a graph morphism) s where G = ( G 1 G 0 ) is a finite graph, Σ is an alphabet, and ! t Σ = (Σ 1) is a single-node graph with hom-set Σ . ! J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 4 / 17

  8. Categorical approaches to LTSs (0) Most categorical approaches have focussed on labled transition systems (LTSs), that form the core of FAs (disregarding initial/final states): � ! , ℓ � (faithful graph morphism) G Σ s ℓ G 0 G 1 Σ (jointly mono) t � s , t � ℓ G 0 × G 0 G 1 Σ (jointly mono, relation G 0 × G 0 Σ) L (Σ × G 0 ) P G 0 (coalgebra, Aczel and Mendler [1989]) L G 0 × Σ (textbook LTS!) G 0 P L Σ ( G 0 , G 0 ) rel (hom-component of a graph morphism) L Σ rel (“finitary” graph morphism) s where G = ( G 1 G 0 ) is a finite graph, Σ is an alphabet, and ! t Σ = (Σ 1) is a single-node graph with hom-set Σ . ! J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 4 / 17

  9. Categorical approaches to LTSs (1) Using the free monoid Σ ⋆ and categories K instead one obtains J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 5 / 17

  10. Categorical approaches to LTSs (1) Using the free monoid Σ ⋆ and categories K instead one obtains (fibre-small faithful functor) K Σ ⋆ (lax functor, Rosenthal [1996]) Σ ⋆ rel J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 5 / 17

  11. Categorical approaches to LTSs (1) Using the free monoid Σ ⋆ and categories K instead one obtains (fibre-small faithful functor) K Σ ⋆ ( K 0 × K 0 ) P Σ ⋆ (lax homomorphism) (lax functor, Rosenthal [1996]) Σ ⋆ rel J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 5 / 17

  12. Categorical approaches to LTSs (1) Using the free monoid Σ ⋆ and categories K instead one obtains (fibre-small faithful functor) K Σ ⋆ ( K 0 × K 0 ) P Σ ⋆ P (quantale-enriched category, Betti [1980]) ( K 0 × K 0 ) P Σ ⋆ (lax homomorphism) (lax functor, Rosenthal [1996]) Σ ⋆ rel J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 5 / 17

  13. Categorical approaches to LTSs (1) Using the free monoid Σ ⋆ and categories K instead one obtains (fibre-small faithful functor) K Σ ⋆ ( K 0 × K 0 ) P Σ ⋆ P (quantale-enriched category, Betti [1980]) ( K 0 × K 0 ) P Σ ⋆ (lax homomorphism) (lax functor, Rosenthal [1996]) Σ ⋆ rel ⊲ The bottom lines would seem to place our subject squarely into the realm of categorical relational algebra. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 5 / 17

  14. Categorical approaches to LTSs (1) Using the free monoid Σ ⋆ and categories K instead one obtains (fibre-small faithful functor) K Σ ⋆ ( K 0 × K 0 ) P Σ ⋆ P (quantale-enriched category, Betti [1980]) ( K 0 × K 0 ) P Σ ⋆ (lax homomorphism) (lax functor, Rosenthal [1996]) Σ ⋆ rel ⊲ The bottom lines would seem to place our subject squarely into the realm of categorical relational algebra. L ⊲ Morphisms of coalgebras G 0 (Σ × G 0 ) P turn out to be functional bisimulations, while spans are needed to model general bisimulations. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 5 / 17

  15. Categorical approaches to LTSs (1) Using the free monoid Σ ⋆ and categories K instead one obtains (fibre-small faithful functor) K Σ ⋆ ( K 0 × K 0 ) P Σ ⋆ P (quantale-enriched category, Betti [1980]) ( K 0 × K 0 ) P Σ ⋆ (lax homomorphism) (lax functor, Rosenthal [1996]) Σ ⋆ rel ⊲ The bottom lines would seem to place our subject squarely into the realm of categorical relational algebra. L ⊲ Morphisms of coalgebras G 0 (Σ × G 0 ) P turn out to be functional bisimulations, while spans are needed to model general bisimulations. ⊲ Joyal, Winskel and Nielsen [1994] as well as Cockett and Spooner [1997] approach bisimulations synthetically; in an enriched context this has been done by Schmitt and Worytkiewicz [2006]. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 5 / 17

  16. Remarks J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 6 / 17

  17. Remarks The outer bijection persists, when Σ / Σ ⋆ is replaced by an arbitrary graph/category X , giving rise to a Grothendieck-type construction. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 6 / 17

  18. Remarks The outer bijection persists, when Σ / Σ ⋆ is replaced by an arbitrary graph/category X , giving rise to a Grothendieck-type construction. But not all intermediate stages admit a similar generalization, in particular not coalgebra. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 6 / 17

  19. Remarks The outer bijection persists, when Σ / Σ ⋆ is replaced by an arbitrary graph/category X , giving rise to a Grothendieck-type construction. But not all intermediate stages admit a similar generalization, in particular not coalgebra. Oplax transformations as morphisms between lax functors into rel translate into simulations between LTSs (JK, several talks since 2003, Soboci´ nski [2012]). J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 6 / 17

  20. Remarks The outer bijection persists, when Σ / Σ ⋆ is replaced by an arbitrary graph/category X , giving rise to a Grothendieck-type construction. But not all intermediate stages admit a similar generalization, in particular not coalgebra. Oplax transformations as morphisms between lax functors into rel translate into simulations between LTSs (JK, several talks since 2003, Soboci´ nski [2012]). So far we have ignored initial/final states. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 6 / 17

  21. Remarks The outer bijection persists, when Σ / Σ ⋆ is replaced by an arbitrary graph/category X , giving rise to a Grothendieck-type construction. But not all intermediate stages admit a similar generalization, in particular not coalgebra. Oplax transformations as morphisms between lax functors into rel translate into simulations between LTSs (JK, several talks since 2003, Soboci´ nski [2012]). So far we have ignored initial/final states. We’d prefer a categorical interpretation rather than selecting arbitrary subsets of states. But the attempt to use simulations from, resp., into a special LTS fails. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 6 / 17

  22. Remarks The outer bijection persists, when Σ / Σ ⋆ is replaced by an arbitrary graph/category X , giving rise to a Grothendieck-type construction. But not all intermediate stages admit a similar generalization, in particular not coalgebra. Oplax transformations as morphisms between lax functors into rel translate into simulations between LTSs (JK, several talks since 2003, Soboci´ nski [2012]). So far we have ignored initial/final states. We’d prefer a categorical interpretation rather than selecting arbitrary subsets of states. But the attempt to use simulations from, resp., into a special LTS fails. Instead, one has to use modules rather than oplax natural transfor- D mations, from, resp., into the discrete lax functor Σ ⋆ rel . J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 6 / 17

  23. Remarks The outer bijection persists, when Σ / Σ ⋆ is replaced by an arbitrary graph/category X , giving rise to a Grothendieck-type construction. But not all intermediate stages admit a similar generalization, in particular not coalgebra. Oplax transformations as morphisms between lax functors into rel translate into simulations between LTSs (JK, several talks since 2003, Soboci´ nski [2012]). So far we have ignored initial/final states. We’d prefer a categorical interpretation rather than selecting arbitrary subsets of states. But the attempt to use simulations from, resp., into a special LTS fails. Instead, one has to use modules rather than oplax natural transfor- D mations, from, resp., into the discrete lax functor Σ ⋆ rel . In the context of graphs this means that instead of Σ we need to consider the reflexive graph Σ ǫ with hom-set Σ + { ǫ } . J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 6 / 17

  24. Moving up the Chomsky hierarchy: Walters’ approach J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 7 / 17

  25. Moving up the Chomsky hierarchy: Walters’ approach ⊲ Some approaches to describe at least real-time PDAs by coalgebraic methods are presently under way, but they seem to be very intricate. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 7 / 17

  26. Moving up the Chomsky hierarchy: Walters’ approach ⊲ Some approaches to describe at least real-time PDAs by coalgebraic methods are presently under way, but they seem to be very intricate. ⊲ Instead, we slightly extend Walters’ [1989] categorification of a certain type of context-free grammars (CFGs), which functionally separates terminals (= elements of Σ ) from variables, J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 7 / 17

  27. Moving up the Chomsky hierarchy: Walters’ approach ⊲ Some approaches to describe at least real-time PDAs by coalgebraic methods are presently under way, but they seem to be very intricate. ⊲ Instead, we slightly extend Walters’ [1989] categorification of a certain type of context-free grammars (CFGs), which functionally separates terminals (= elements of Σ ) from variables, rather than (classically) lumping them together and forming a free monoid. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 7 / 17

  28. Moving up the Chomsky hierarchy: Walters’ approach ⊲ Some approaches to describe at least real-time PDAs by coalgebraic methods are presently under way, but they seem to be very intricate. ⊲ Instead, we slightly extend Walters’ [1989] categorification of a certain type of context-free grammars (CFGs), which functionally separates terminals (= elements of Σ ) from variables, rather than (classically) lumping them together and forming a free monoid. Σ ǫ between finite reflexive graphs as γ ⊲ Walters views morphisms G regular grammars rather than as LTSs. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 7 / 17

  29. Moving up the Chomsky hierarchy: Walters’ approach ⊲ Some approaches to describe at least real-time PDAs by coalgebraic methods are presently under way, but they seem to be very intricate. ⊲ Instead, we slightly extend Walters’ [1989] categorification of a certain type of context-free grammars (CFGs), which functionally separates terminals (= elements of Σ ) from variables, rather than (classically) lumping them together and forming a free monoid. Σ ǫ between finite reflexive graphs as γ ⊲ Walters views morphisms G regular grammars rather than as LTSs. Then morphsms between suitable multi-graphs (edges have finitely many inputs and one output; this yields bottom-up parsing) capture a class of CFGs (in Walters Normal Form (WNF)) that generate all context-free languages. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 7 / 17

  30. Moving up the Chomsky hierarchy: Walters’ approach ⊲ Some approaches to describe at least real-time PDAs by coalgebraic methods are presently under way, but they seem to be very intricate. ⊲ Instead, we slightly extend Walters’ [1989] categorification of a certain type of context-free grammars (CFGs), which functionally separates terminals (= elements of Σ ) from variables, rather than (classically) lumping them together and forming a free monoid. Σ ǫ between finite reflexive graphs as γ ⊲ Walters views morphisms G regular grammars rather than as LTSs. Then morphsms between suitable multi-graphs (edges have finitely many inputs and one output; this yields bottom-up parsing) capture a class of CFGs (in Walters Normal Form (WNF)) that generate all context-free languages. ⊲ Walters wanted to illustrate his construction of the free category with products over a multi-graph. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 7 / 17

  31. Moving up the Chomsky hierarchy: Walters’ approach ⊲ Some approaches to describe at least real-time PDAs by coalgebraic methods are presently under way, but they seem to be very intricate. ⊲ Instead, we slightly extend Walters’ [1989] categorification of a certain type of context-free grammars (CFGs), which functionally separates terminals (= elements of Σ ) from variables, rather than (classically) lumping them together and forming a free monoid. Σ ǫ between finite reflexive graphs as γ ⊲ Walters views morphisms G regular grammars rather than as LTSs. Then morphsms between suitable multi-graphs (edges have finitely many inputs and one output; this yields bottom-up parsing) capture a class of CFGs (in Walters Normal Form (WNF)) that generate all context-free languages. ⊲ Walters wanted to illustrate his construction of the free category with products over a multi-graph. However, a more direct way of extracting the generated language becomes available with top-down parsing, hence we revert to co-multi-graphs or cm-graphs, for short. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 7 / 17

  32. Walters’ approach slightly generalized Definition J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 8 / 17

  33. Walters’ approach slightly generalized Definition (0) Any set Σ induces a cm-graph Σ I N with a single node H and Σ + { ǫ } for all hom-sets [ H , H n ] , n ∈ I N . J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 8 / 17

  34. Walters’ approach slightly generalized Definition (0) Any set Σ induces a cm-graph Σ I N with a single node H and Σ + { ǫ } for all hom-sets [ H , H n ] , n ∈ I N . ( ∅ I N is terminal.) J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 8 / 17

  35. Walters’ approach slightly generalized Definition (0) Any set Σ induces a cm-graph Σ I N with a single node H and Σ + { ǫ } for all hom-sets [ H , H n ] , n ∈ I N . ( ∅ I N is terminal.) (1) A CFG ` a la Walters (CFW) γ over Σ is a faithful cm-graph morphism γ G Σ I N with G finite. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 8 / 17

  36. Walters’ approach slightly generalized Definition (0) Any set Σ induces a cm-graph Σ I N with a single node H and Σ + { ǫ } for all hom-sets [ H , H n ] , n ∈ I N . ( ∅ I N is terminal.) (1) A CFG ` a la Walters (CFW) γ over Σ is a faithful cm-graph morphism γ G Σ I N with G finite. ⊲ Terminals (= elements of Σ ) label the edges of Σ I N , while the set B of variables is the set of G -nodes. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 8 / 17

  37. Walters’ approach slightly generalized Definition (0) Any set Σ induces a cm-graph Σ I N with a single node H and Σ + { ǫ } for all hom-sets [ H , H n ] , n ∈ I N . ( ∅ I N is terminal.) (1) A CFG ` a la Walters (CFW) γ over Σ is a faithful cm-graph morphism γ G Σ I N with G finite. ⊲ Terminals (= elements of Σ ) label the edges of Σ I N , while the set B of variables is the set of G -nodes. ⊲ Classical CFG-productions X aY 0 Y 1 . . . Y n − 1 in ǫ -Greibach normal form, that is, a ∈ Σ + { ǫ } , can be expressed by J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 8 / 17

  38. Walters’ approach slightly generalized Definition (0) Any set Σ induces a cm-graph Σ I N with a single node H and Σ + { ǫ } for all hom-sets [ H , H n ] , n ∈ I N . ( ∅ I N is terminal.) (1) A CFG ` a la Walters (CFW) γ over Σ is a faithful cm-graph morphism γ G Σ I N with G finite. ⊲ Terminals (= elements of Σ ) label the edges of Σ I N , while the set B of variables is the set of G -nodes. ⊲ Classical CFG-productions X aY 0 Y 1 . . . Y n − 1 in ǫ -Greibach normal form, that is, a ∈ Σ + { ǫ } , can be expressed by ϕ a H n ) ( X Y 0 . . . Y n − 1 ) γ = ( H J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 8 / 17

  39. Walters’ approach slightly generalized Definition (0) Any set Σ induces a cm-graph Σ I N with a single node H and Σ + { ǫ } for all hom-sets [ H , H n ] , n ∈ I N . ( ∅ I N is terminal.) (1) A CFG ` a la Walters (CFW) γ over Σ is a faithful cm-graph morphism γ G Σ I N with G finite. ⊲ Terminals (= elements of Σ ) label the edges of Σ I N , while the set B of variables is the set of G -nodes. ⊲ Classical CFG-productions X aY 0 Y 1 . . . Y n − 1 in ǫ -Greibach normal form, that is, a ∈ Σ + { ǫ } , can be expressed by ϕ a H n ) ( X Y 0 . . . Y n − 1 ) γ = ( H a or simply as X Y 0 . . . Y n − 1 , since γ is faithful. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 8 / 17

  40. Trees and words J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 9 / 17

  41. Trees and words ⊲ Instead of traditional node-labeled derivation trees, Poincar´ e duality now yields trees with somewhat different building blocks: J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 9 / 17

  42. Trees and words ⊲ Instead of traditional node-labeled derivation trees, Poincar´ e duality now yields trees with somewhat different building blocks: X . . . a Y 0 Y 1 Y n − 1 J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 9 / 17

  43. Trees and words ⊲ Instead of traditional node-labeled derivation trees, Poincar´ e duality now yields trees with somewhat different building blocks: X X vs. a . . . . . . a Y 0 Y 1 Y n − 1 Y 0 Y 1 Y n − 1 J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 9 / 17

  44. Trees and words ⊲ Instead of traditional node-labeled derivation trees, Poincar´ e duality now yields trees with somewhat different building blocks: X X vs. a . . . . . . a Y 0 Y 1 Y n − 1 Y 0 Y 1 Y n − 1 ⊲ for the language recognized by a G - node S , roughly speaking, J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 9 / 17

  45. Trees and words ⊲ Instead of traditional node-labeled derivation trees, Poincar´ e duality now yields trees with somewhat different building blocks: X X vs. a . . . . . . a Y 0 Y 1 Y n − 1 Y 0 Y 1 Y n − 1 ⊲ for the language recognized by a G - node S , roughly speaking, freely extend γ to a cm-functor γ ⋆ between “free cm-categories over cm-graphs” (in analogy to forming free categories over a graphs); J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 9 / 17

  46. Trees and words ⊲ Instead of traditional node-labeled derivation trees, Poincar´ e duality now yields trees with somewhat different building blocks: X X vs. a . . . . . . a Y 0 Y 1 Y n − 1 Y 0 Y 1 Y n − 1 ⊲ for the language recognized by a G - node S , roughly speaking, freely extend γ to a cm-functor γ ⋆ between “free cm-categories over cm-graphs” (in analogy to forming free categories over a graphs); consider the γ -image of the hom-set � S , ǫ � G ⋆ in Σ ⋆ N ; I J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 9 / 17

  47. Trees and words ⊲ Instead of traditional node-labeled derivation trees, Poincar´ e duality now yields trees with somewhat different building blocks: X X vs. a . . . . . . a Y 0 Y 1 Y n − 1 Y 0 Y 1 Y n − 1 ⊲ for the language recognized by a G - node S , roughly speaking, freely extend γ to a cm-functor γ ⋆ between “free cm-categories over cm-graphs” (in analogy to forming free categories over a graphs); consider the γ -image of the hom-set � S , ǫ � G ⋆ in Σ ⋆ N ; I extract words over Σ from the resulting diagrams in Σ ⋆ N ; these I so-called yields constitute the string-language generated by γ and S . J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 9 / 17

  48. Trees and words ⊲ Instead of traditional node-labeled derivation trees, Poincar´ e duality now yields trees with somewhat different building blocks: X X vs. a . . . . . . a Y 0 Y 1 Y n − 1 Y 0 Y 1 Y n − 1 ⊲ for the language recognized by a G - node S , roughly speaking, freely extend γ to a cm-functor γ ⋆ between “free cm-categories over cm-graphs” (in analogy to forming free categories over a graphs); consider the γ -image of the hom-set � S , ǫ � G ⋆ in Σ ⋆ N ; I extract words over Σ from the resulting diagrams in Σ ⋆ N ; these I so-called yields constitute the string-language generated by γ and S . Optionally, one can view Σ I N as a reflexive cm-graph, which results in a somewhat simpler free cm-category Σ ⋆ N . I J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 9 / 17

  49. Trees and words ⊲ Instead of traditional node-labeled derivation trees, Poincar´ e duality now yields trees with somewhat different building blocks: X X vs. a . . . . . . a Y 0 Y 1 Y n − 1 Y 0 Y 1 Y n − 1 ⊲ for the language recognized by a G - node S , roughly speaking, freely extend γ to a cm-functor γ ⋆ between “free cm-categories over cm-graphs” (in analogy to forming free categories over a graphs); consider the γ -image of the hom-set � S , ǫ � G ⋆ in Σ ⋆ N ; I extract words over Σ from the resulting diagrams in Σ ⋆ N ; these I so-called yields constitute the string-language generated by γ and S . Optionally, one can view Σ I N as a reflexive cm-graph, which results in a somewhat simpler free cm-category Σ ⋆ N . I ⊲ As terminals are not limited to leaves, we need to switch from positional ordering of trees to temporal ordering (rotation by π/ 2 indicates this), which requires some notion of current position. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 9 / 17

  50. Strategy: towards 2PDAs J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 10 / 17

  51. Strategy: towards 2PDAs ⊲ CFGs with the constraint of only left-derivations being allowed essentially are single state PDAs that accept by empty stack. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 10 / 17

  52. Strategy: towards 2PDAs ⊲ CFGs with the constraint of only left-derivations being allowed essentially are single state PDAs that accept by empty stack. Hence such pure PDAs may have been disregarded as uninteresting. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 10 / 17

  53. Strategy: towards 2PDAs ⊲ CFGs with the constraint of only left-derivations being allowed essentially are single state PDAs that accept by empty stack. Hence such pure PDAs may have been disregarded as uninteresting. ⊲ Juxtaposing a second stack to the first one, the interface between them determines the current position: J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 10 / 17

  54. Strategy: towards 2PDAs ⊲ CFGs with the constraint of only left-derivations being allowed essentially are single state PDAs that accept by empty stack. Hence such pure PDAs may have been disregarded as uninteresting. ⊲ Juxtaposing a second stack to the first one, the interface between them determines the current position: from here the first element on each side is visible, resp., the information that some stack is empty. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 10 / 17

  55. Strategy: towards 2PDAs ⊲ CFGs with the constraint of only left-derivations being allowed essentially are single state PDAs that accept by empty stack. Hence such pure PDAs may have been disregarded as uninteresting. ⊲ Juxtaposing a second stack to the first one, the interface between them determines the current position: from here the first element on each side is visible, resp., the information that some stack is empty. ⊲ We will employ two stack alphabets (= sets of variables) B and C , which ` a priori need not be disjoint. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 10 / 17

  56. Strategy: towards 2PDAs ⊲ CFGs with the constraint of only left-derivations being allowed essentially are single state PDAs that accept by empty stack. Hence such pure PDAs may have been disregarded as uninteresting. ⊲ Juxtaposing a second stack to the first one, the interface between them determines the current position: from here the first element on each side is visible, resp., the information that some stack is empty. ⊲ We will employ two stack alphabets (= sets of variables) B and C , which ` a priori need not be disjoint. But to indicate the current position, we use color-coded disjoint copies B and C for the lower/upper stack. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 10 / 17

  57. Strategy: towards 2PDAs ⊲ CFGs with the constraint of only left-derivations being allowed essentially are single state PDAs that accept by empty stack. Hence such pure PDAs may have been disregarded as uninteresting. ⊲ Juxtaposing a second stack to the first one, the interface between them determines the current position: from here the first element on each side is visible, resp., the information that some stack is empty. ⊲ We will employ two stack alphabets (= sets of variables) B and C , which ` a priori need not be disjoint. But to indicate the current position, we use color-coded disjoint copies B and C for the lower/upper stack. Their union makes up the set of G - nodes. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 10 / 17

  58. Strategy: towards 2PDAs ⊲ CFGs with the constraint of only left-derivations being allowed essentially are single state PDAs that accept by empty stack. Hence such pure PDAs may have been disregarded as uninteresting. ⊲ Juxtaposing a second stack to the first one, the interface between them determines the current position: from here the first element on each side is visible, resp., the information that some stack is empty. ⊲ We will employ two stack alphabets (= sets of variables) B and C , which ` a priori need not be disjoint. But to indicate the current position, we use color-coded disjoint copies B and C for the lower/upper stack. Their union makes up the set of G - nodes. Moreover, we require the outputs of cm-edges to inherit the input’s color. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 10 / 17

  59. Strategy: towards 2PDAs ⊲ CFGs with the constraint of only left-derivations being allowed essentially are single state PDAs that accept by empty stack. Hence such pure PDAs may have been disregarded as uninteresting. ⊲ Juxtaposing a second stack to the first one, the interface between them determines the current position: from here the first element on each side is visible, resp., the information that some stack is empty. ⊲ We will employ two stack alphabets (= sets of variables) B and C , which ` a priori need not be disjoint. But to indicate the current position, we use color-coded disjoint copies B and C for the lower/upper stack. Their union makes up the set of G - nodes. Moreover, we require the outputs of cm-edges to inherit the input’s color. a ⊲ Transitions take the form AB Γ∆ with A , B not both empty (acceptance by empty stack), a ∈ Σ + { ǫ } , and � Γ , ∆ � ∈ B ⋆ × C ⋆ . J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 10 / 17

  60. Strategy: towards 2PDAs ⊲ CFGs with the constraint of only left-derivations being allowed essentially are single state PDAs that accept by empty stack. Hence such pure PDAs may have been disregarded as uninteresting. ⊲ Juxtaposing a second stack to the first one, the interface between them determines the current position: from here the first element on each side is visible, resp., the information that some stack is empty. ⊲ We will employ two stack alphabets (= sets of variables) B and C , which ` a priori need not be disjoint. But to indicate the current position, we use color-coded disjoint copies B and C for the lower/upper stack. Their union makes up the set of G - nodes. Moreover, we require the outputs of cm-edges to inherit the input’s color. a ⊲ Transitions take the form AB Γ∆ with A , B not both empty (acceptance by empty stack), a ∈ Σ + { ǫ } , and � Γ , ∆ � ∈ B ⋆ × C ⋆ . ǫ ǫ ⊲ Left and right moves AB ǫ AB and AB AB ǫ just change the current position. J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 10 / 17

  61. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  62. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  63. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions a ǫ S ǫ SBC | ǫ BC b ǫ SCA | ǫ CA ǫ S c ǫ S ǫ SAB | ǫ AB J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  64. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions a a ǫ S ǫ SBC | ǫ BC @ A @ ǫ b b ǫ SCA | ǫ CA ǫ S @ B @ ǫ c c ǫ S ǫ SAB | ǫ AB @ C @ ǫ J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  65. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions a a a ǫ S ǫ SBC | ǫ BC @ A @ ǫ A @ ǫ @ b b b ǫ SCA | ǫ CA ǫ S @ B @ ǫ B @ ǫ @ c c c ǫ S ǫ SAB | ǫ AB @ C @ ǫ C @ ǫ @ J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  66. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions a a a ǫ ǫ S ǫ SBC | ǫ BC @ A @ ǫ A @ ǫ @ @ X @ X ǫ b b b ǫ ǫ SCA | ǫ CA ǫ S @ B @ ǫ B @ ǫ @ X @ ǫ X @ c c c ǫ S ǫ SAB | ǫ AB @ C @ ǫ C @ ǫ @ moves! J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  67. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions a a a ǫ ǫ S ǫ SBC | ǫ BC @ A @ ǫ A @ ǫ @ @ X @ X ǫ b b b ǫ ǫ SCA | ǫ CA ǫ S @ B @ ǫ B @ ǫ @ X @ ǫ X @ c c c ǫ S ǫ SAB | ǫ AB @ C @ ǫ C @ ǫ @ moves! with @ ∈ { A , B , C , ǫ } and X ∈ { A , B , C } . J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  68. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions a a a ǫ ǫ S ǫ SBC | ǫ BC @ A @ ǫ A @ ǫ @ @ X @ X ǫ b b b ǫ ǫ SCA | ǫ CA ǫ S @ B @ ǫ B @ ǫ @ X @ ǫ X @ c c c ǫ S ǫ SAB | ǫ AB @ C @ ǫ C @ ǫ @ moves! with @ ∈ { A , B , C , ǫ } and X ∈ { A , B , C } . b a b c c a b c a The derivation of can take the form: J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  69. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions a a a ǫ ǫ S ǫ SBC | ǫ BC @ A @ ǫ A @ ǫ @ @ X @ X ǫ b b b ǫ ǫ SCA | ǫ CA ǫ S @ B @ ǫ B @ ǫ @ X @ ǫ X @ c c c ǫ S ǫ SAB | ǫ AB @ C @ ǫ C @ ǫ @ moves! with @ ∈ { A , B , C , ǫ } and X ∈ { A , B , C } . b a b c c a b c a The derivation of can take the form: A C C B c A B b a ⊘ � b a B B S S S C B ⊘ c b c a ⋆ current position A J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  70. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions a a a ǫ ǫ S ǫ SBC | ǫ BC @ A @ ǫ A @ ǫ @ @ X @ X ǫ b b b ǫ ǫ SCA | ǫ CA ǫ S @ B @ ǫ B @ ǫ @ X @ ǫ X @ c c c ǫ S ǫ SAB | ǫ AB @ C @ ǫ C @ ǫ @ moves! with @ ∈ { A , B , C , ǫ } and X ∈ { A , B , C } . b a b c c a b c a The derivation of can take the form: A C C B c A B b a ⊘ � b a B B S S S C B ⊘ c b c a ⋆ A J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  71. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions a a a ǫ ǫ S ǫ SBC | ǫ BC @ A @ ǫ A @ ǫ @ @ X @ X ǫ b b b ǫ ǫ SCA | ǫ CA ǫ S @ B @ ǫ B @ ǫ @ X @ ǫ X @ c c c ǫ S ǫ SAB | ǫ AB @ C @ ǫ C @ ǫ @ moves! with @ ∈ { A , B , C , ǫ } and X ∈ { A , B , C } . b a b c c a b c a The derivation of can take the form: A C C B c A B b a ⊘ � b a B B S S S C B ⊘ c b c a ⋆ A J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  72. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions a a a ǫ ǫ S ǫ SBC | ǫ BC @ A @ ǫ A @ ǫ @ @ X @ X ǫ b b b ǫ ǫ SCA | ǫ CA ǫ S @ B @ ǫ B @ ǫ @ X @ ǫ X @ c c c ǫ S ǫ SAB | ǫ AB @ C @ ǫ C @ ǫ @ moves! with @ ∈ { A , B , C , ǫ } and X ∈ { A , B , C } . b a b c c a b c a The derivation of can take the form: A C C B c A B b a ⊘ � b a B B S S S C B ⊘ c b c a ⋆ A J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  73. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions a a a ǫ ǫ S ǫ SBC | ǫ BC @ A @ ǫ A @ ǫ @ @ X @ X ǫ b b b ǫ ǫ SCA | ǫ CA ǫ S @ B @ ǫ B @ ǫ @ X @ ǫ X @ c c c ǫ S ǫ SAB | ǫ AB @ C @ ǫ C @ ǫ @ moves! with @ ∈ { A , B , C , ǫ } and X ∈ { A , B , C } . b a b c c a b c a The derivation of can take the form: A C C B c A B b a ⊘ � b a B B S S S C B ⊘ c b c a ⋆ A J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  74. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions a a a ǫ ǫ S ǫ SBC | ǫ BC @ A @ ǫ A @ ǫ @ @ X @ X ǫ b b b ǫ ǫ SCA | ǫ CA ǫ S @ B @ ǫ B @ ǫ @ X @ ǫ X @ c c c ǫ S ǫ SAB | ǫ AB @ C @ ǫ C @ ǫ @ moves! with @ ∈ { A , B , C , ǫ } and X ∈ { A , B , C } . b a b c c a b c a The derivation of can take the form: A C C B c A B b a ⊘ � b ⋆ a B B S S S C B ⊘ c b c a A J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  75. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions a a a ǫ ǫ S ǫ SBC | ǫ BC @ A @ ǫ A @ ǫ @ @ X @ X ǫ b b b ǫ ǫ SCA | ǫ CA ǫ S @ B @ ǫ B @ ǫ @ X @ ǫ X @ c c c ǫ S ǫ SAB | ǫ AB @ C @ ǫ C @ ǫ @ moves! with @ ∈ { A , B , C , ǫ } and X ∈ { A , B , C } . b a b c c a b c a The derivation of can take the form: A C C B ⋆ c A B b a ⊘ � b a B B S S S C B ⊘ c b c a A J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  76. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions a a a ǫ ǫ S ǫ SBC | ǫ BC @ A @ ǫ A @ ǫ @ @ X @ X ǫ b b b ǫ ǫ SCA | ǫ CA ǫ S @ B @ ǫ B @ ǫ @ X @ ǫ X @ c c c ǫ S ǫ SAB | ǫ AB @ C @ ǫ C @ ǫ @ moves! with @ ∈ { A , B , C , ǫ } and X ∈ { A , B , C } . b a b c c a b c a The derivation of can take the form: A C C B c ⋆ A B b a ⊘ � b a B B S S S C B ⊘ c b c a A J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  77. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions a a a ǫ ǫ S ǫ SBC | ǫ BC @ A @ ǫ A @ ǫ @ @ X @ X ǫ b b b ǫ ǫ SCA | ǫ CA ǫ S @ B @ ǫ B @ ǫ @ X @ ǫ X @ c c c ǫ S ǫ SAB | ǫ AB @ C @ ǫ C @ ǫ @ moves! with @ ∈ { A , B , C , ǫ } and X ∈ { A , B , C } . b a b c c a b c a The derivation of can take the form: A C C B c A B b a ⊘ � b ⋆ a B B S S S C B ⊘ c b c a A J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  78. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions a a a ǫ ǫ S ǫ SBC | ǫ BC @ A @ ǫ A @ ǫ @ @ X @ X ǫ b b b ǫ ǫ SCA | ǫ CA ǫ S @ B @ ǫ B @ ǫ @ X @ ǫ X @ c c c ǫ S ǫ SAB | ǫ AB @ C @ ǫ C @ ǫ @ moves! with @ ∈ { A , B , C , ǫ } and X ∈ { A , B , C } . b a b c c a b c a The derivation of can take the form: A C C B c A B b a ⊘ � b a B B S S S C B ⊘ c b c a ⋆ A J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  79. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions a a a ǫ ǫ S ǫ SBC | ǫ BC @ A @ ǫ A @ ǫ @ @ X @ X ǫ b b b ǫ ǫ SCA | ǫ CA ǫ S @ B @ ǫ B @ ǫ @ X @ ǫ X @ c c c ǫ S ǫ SAB | ǫ AB @ C @ ǫ C @ ǫ @ moves! with @ ∈ { A , B , C , ǫ } and X ∈ { A , B , C } . b a b c c a b c a The derivation of can take the form: A C C B c A B b a ⊘ � b a B B S S S C B ⊘ c b c a ⋆ A J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  80. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions a a a ǫ ǫ S ǫ SBC | ǫ BC @ A @ ǫ A @ ǫ @ @ X @ X ǫ b b b ǫ ǫ SCA | ǫ CA ǫ S @ B @ ǫ B @ ǫ @ X @ ǫ X @ c c c ǫ S ǫ SAB | ǫ AB @ C @ ǫ C @ ǫ @ moves! with @ ∈ { A , B , C , ǫ } and X ∈ { A , B , C } . b a b c c a b c a The derivation of can take the form: A C C B c A B b a ⊘ � b a B B S S S C B ⊘ c b c a ⋆ A J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  81. Example: MIX = { w ∈ { a , b , c } ∗ : | w | a = | w | b = | w | c } When using the initial stack ǫ S and the following transitions a a a ǫ ǫ S ǫ SBC | ǫ BC @ A @ ǫ A @ ǫ @ @ X @ X ǫ b b b ǫ ǫ SCA | ǫ CA ǫ S @ B @ ǫ B @ ǫ @ X @ ǫ X @ c c c ǫ S ǫ SAB | ǫ AB @ C @ ǫ C @ ǫ @ moves! with @ ∈ { A , B , C , ǫ } and X ∈ { A , B , C } . b a b c c a b c a The derivation of can take the form: A C C B c A B b a ⊘ � b a B B S S S C B ⊘ c b c a A J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 11 / 17

  82. What’s wrong with this picture? As the diagram above is not built from cm-edges of the proposed cm-graph G , we need to re-interpret its components, e.g. , by splitting them up. E.g. , J¨ urgen Koslowski (TU-BS) A categorical model for 2-PDAs with states cmat14, Coimbra 12 / 17

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