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The word problem for ΣΠ-categories, i.e. properties of free ΣΠ-categories (II) Robin Cockett Department of Computer Science University of Calgary CANADA * Luigi Santocanale * LIF, Marseille e Universit´ de Provence FRANCE Cape Town, Category Theory 2009 1/26 Outline Introductory remarks ΣΠ-categories and their theory Context, some motivations The free ΣΠ-category “En route” towards a decision procedure The CT2006 results Softness, cardinals, and (some hints on) a decision procedure Understanding softness Pushouts-pullbacks, bouncing 2/26 Outline Introductory remarks ΣΠ-categories and their theory Context, some motivations The free ΣΠ-category “En route” towards a decision procedure The CT2006 results Softness, cardinals, and (some hints on) a decision procedure Understanding softness Pushouts-pullbacks, bouncing 3/26 ΣΠ-categories and their theory A ΣΠ-category : a category with (chosen) ﬁnite products and ﬁnite coproducts. Binary products: , [X,A] × [X,B ] −− −→ [X,A × B ] πi − [ Xi , A ] −→ [ X0 × X1 , A ] such that πi ( f0 , f1 ) = fi , π0 (f ), π1 (f ) = f . Binary coproducts: dual nats { , } and σj , j = 0, 1, . . . . . . and dual equations. 4/26 ΣΠ-categories and their theory A ΣΠ-category : a category with (chosen) ﬁnite products and ﬁnite coproducts. Binary products: , [X,A] × [X,B ] −− −→ [X,A × B ] πi − [ Xi , A ] −→ [ X0 × X1 , A ] such that πi ( f0 , f1 ) = fi , π0 (f ), π1 (f ) = f . Binary coproducts: dual nats { , } and σj , j = 0, 1, . . . . . . and dual equations. 4/26 ΣΠ-categories and their theory A ΣΠ-category : a category with (chosen) ﬁnite products and ﬁnite coproducts. Binary products: , [X,A] × [X,B ] −− −→ [X,A × B ] πi − [ Xi , A ] −→ [ X0 × X1 , A ] such that πi ( f0 , f1 ) = fi , π0 (f ), π1 (f ) = f . Binary coproducts: dual nats { , } and σj , j = 0, 1, . . . . . . and dual equations. 4/26 Units, empty products and coproducts, terminal and initial objects The units: [ X , 1 ] = { !X } [ 0, A ] = { ?A } Some derived equations: {!X , !Y } = !X +Y → :X +Y − 1 !0 = ? 1 → :0− 1 5/26 Units, empty products and coproducts, terminal and initial objects The units: [ X , 1 ] = { !X } [ 0, A ] = { ?A } Some derived equations: {!X , !Y } = !X +Y → :X +Y − 1 !0 = ? 1 → :0− 1 5/26 The word problem for ΣΠ-categories? Elementary but important mathematical theory, Ties to linear logic (additives), A challenge for category theorists and logicians: Joyal & Hu 96, Hu 98, Dosen 99, Dosen & Petric 2009, Hughes 2000, Hughes & van Glabbek 03, Joyal 95, Cockett & Seely 2001, From Whitman’s condition to softess: mimicking the theory of free lattices, Joyal’s theory of communication: free ΣΠ-categories are semantics for communication protocols, we tackle equivalence of these protocols. 6/26 The word problem for ΣΠ-categories? Elementary but important mathematical theory, Ties to linear logic (additives), A challenge for category theorists and logicians: Joyal & Hu 96, Hu 98, Dosen 99, Dosen & Petric 2009, Hughes 2000, Hughes & van Glabbek 03, Joyal 95, Cockett & Seely 2001, From Whitman’s condition to softess: mimicking the theory of free lattices, Joyal’s theory of communication: free ΣΠ-categories are semantics for communication protocols, we tackle equivalence of these protocols. 6/26 The word problem for ΣΠ-categories? Elementary but important mathematical theory, Ties to linear logic (additives), A challenge for category theorists and logicians: Joyal & Hu 96, Hu 98, Dosen 99, Dosen & Petric 2009, Hughes 2000, Hughes & van Glabbek 03, Joyal 95, Cockett & Seely 2001, From Whitman’s condition to softess: mimicking the theory of free lattices, Joyal’s theory of communication: free ΣΠ-categories are semantics for communication protocols, we tackle equivalence of these protocols. 6/26 The word problem for ΣΠ-categories? Elementary but important mathematical theory, Ties to linear logic (additives), A challenge for category theorists and logicians: Joyal & Hu 96, Hu 98, Dosen 99, Dosen & Petric 2009, Hughes 2000, Hughes & van Glabbek 03, Joyal 95, Cockett & Seely 2001, From Whitman’s condition to softess: mimicking the theory of free lattices, Joyal’s theory of communication: free ΣΠ-categories are semantics for communication protocols, we tackle equivalence of these protocols. 6/26 ΣΠs: a theory of communication Objects of ΣΠ(A) are channels, arrows of ΣΠ(A) are mediating protocols. G M GH mediating protocol 1st channel 2nd channel !"# !"# # /// '&%$ '&%$ !"# '&%$ !"# '&%$ / / M /× // : G //: y r : H //: !"# '&%$ !"# '&%$ !"# '&%$ !"# '&%$ q p hooks left user right user 7/26 ΣΠ(A), the free ΣΠ category on A free ΣΠ-category generated by A: η A GG G ΣΠ(A) GG GG GG GGF GG GG ∃!F o GG ˜ ΣΠ-functor GG GG 5 o C ΣΠ-category ΣΠ-functor: preserves ﬁnite products and ﬁnite coproducts, 8/26 ΣΠ(A), the free ΣΠ category on A free ΣΠ-category generated by A: η A GG G ΣΠ(A) GG GG GG GGF GG GG ∃!F o GG ˜ ΣΠ-functor GG GG 5 o C ΣΠ-category ΣΠ-functor: preserves ﬁnite products and ﬁnite coproducts, 8/26 CS01 : a Lambek-style construction of ΣΠ(A) f → x− y η(f ) −→ η(x) − − η(x) − ! R1 → X − 1 f f g → Xi − A → X − A → X − B Li × R× πi (f ) f ,g −− X0 × X1 − − → A −− X −−→ A × B − ? L0 → 0− A f g f X − A → Y − A → → X − Aj L+ Rj + {f ,g } σj (f ) X +Y −−→ A −− −− X − − → A0 + A1 9/26 CS01: conﬂuence modulo equations Proposition The cut-elimination procedure is conﬂuent modulo the equations: πi ( f , g ) = πi (f ), πi (g ) σj ({f , g }) = {σj (f ), σj (g )} πi (σj (f )) = σj (πi (f )) { f11 , f12 , f21 , f22 } = {f11 , f21 }, {f12 , f22 } πi (!) = ! σj (?) = ? {!, !} = ! ?, ? = ? !0 =?1 10/26 Abstract characterization of free ΣΠ-cats [CS01, J95] → 1. The functor η : A − ΣΠ(A) is full and faithful. 2. Generators are atomic: → [ η(a), Yj ] − [ η(a), Yj ] j j → [ Xi , η(b) ] − [ Xi , η(b) ] i i are isomorphisms. 3. ΣΠ(A) is soft: i,j [ Xi , Yj ] G j[ i Xi , Yj ] i [ Xi , j Yj ] G[ i Xi , j Yj ] is a pushout. 11/26 Abstract characterization of free ΣΠ-cats [CS01, J95] → 1. The functor η : A − ΣΠ(A) is full and faithful. 2. Generators are atomic: → [ η(a), Yj ] − [ η(a), Yj ] j j → [ Xi , η(b) ] − [ Xi , η(b) ] i i are isomorphisms. 3. ΣΠ(A) is soft: i,j [ Xi , Yj ] G j[ i Xi , Yj ] i [ Xi , j Yj ] G[ i Xi , j Yj ] is a pushout. 11/26 Abstract characterization of free ΣΠ-cats [CS01, J95] → 1. The functor η : A − ΣΠ(A) is full and faithful. 2. Generators are atomic: → [ η(a), Yj ] − [ η(a), Yj ] j j → [ Xi , η(b) ] − [ Xi , η(b) ] i i are isomorphisms. 3. ΣΠ(A) is soft: i,j [ Xi , Yj ] G j[ i Xi , Yj ] i [ Xi , j Yj ] G[ i Xi , j Yj ] is a pushout. 11/26 Characterization of free ΣΠ-categories (cont.) Theorem The pair (η, ΣΠ(A)) satisﬁes 1,2,3. → If C is a ΣΠ-category “generated” by A, F : A − C, and (F , C) satisﬁes 1,2,3, ˆ → then the extension F : ΣΠ(A) − C is an equivalence. 12/26 Outline Introductory remarks ΣΠ-categories and their theory Context, some motivations The free ΣΠ-category “En route” towards a decision procedure The CT2006 results Softness, cardinals, and (some hints on) a decision procedure Understanding softness Pushouts-pullbacks, bouncing 13/26 A result presented at CT06 Theorem In ΣΠ(A) coproducts are weakly disjoint: g X G Y1 f σ1 σ0 Y0 G Y0 + Y1 14/26 A result presented at CT06 Theorem In ΣΠ(A) coproducts are weakly disjoint: g X jGR Y1 ∃h j jjjj ' jj?Y1 j jjjj 0 f σ1 ? Y0 × σ0 Y0 G Y0 + Y1 14/26 . . . and some Corollaries decide in linear time whether an object of ΣΠ(A) is isomorphic to 0 or 1, an arrow of ΣΠ(A) factors through 0 and or 1. a simple characterization of monic coproduct injections: → σ0 : A − A + B is monic iﬀ either B is not pointed or A is pointed. 15/26 . . . and some Corollaries decide in linear time whether an object of ΣΠ(A) is isomorphic to 0 or 1, an arrow of ΣΠ(A) factors through 0 and or 1. a simple characterization of monic coproduct injections: → σ0 : A − A + B is monic iﬀ either B is not pointed or A is pointed. 15/26 . . . and some Corollaries decide in linear time whether an object of ΣΠ(A) is isomorphic to 0 or 1, an arrow of ΣΠ(A) factors through 0 and or 1. a simple characterization of monic coproduct injections: → σ0 : A − A + B is monic iﬀ either B is not pointed or A is pointed. 15/26 . . . and some Corollaries decide in linear time whether an object of ΣΠ(A) is isomorphic to 0 or 1, an arrow of ΣΠ(A) factors through 0 and or 1. a simple characterization of monic coproduct injections: → σ0 : A − A + B is monic iﬀ either B is not pointed or A is pointed. 15/26 Outline Introductory remarks ΣΠ-categories and their theory Context, some motivations The free ΣΠ-category “En route” towards a decision procedure The CT2006 results Softness, cardinals, and (some hints on) a decision procedure Understanding softness Pushouts-pullbacks, bouncing 16/26 Softness: the focus of a decision procedure A decision procedure focuses on the homset [ X × Y , A + B ]. For example: → l e t equal f g : X − A × B = let f = f1 , f2 and g = g1 , g2 in e q u a l f1 g1 && e q u a l f2 g2 17/26 Softness: the focus of a decision procedure A decision procedure focuses on the homset [ X × Y , A + B ]. For example: → l e t equal f g : X − A × B = let f = f1 , f2 and g = g1 , g2 in e q u a l f1 g1 && e q u a l f2 g2 17/26 Understanding softness The homset [ X × Y , A + B ] is . . . 18/26 Understanding softness The homset [ X × Y , A + B ] is . . . the pushout [X,A] + [X,B ] + [Y,A] + [Y,B ] G [X × Y,A] + [X × Y,B ] [X,A + B ] + [Y,A + B ] G [X × Y,A + B ] 18/26 Understanding softness The homset [ X × Y , A + B ] is . . . the colimit of the “diagram of cardinals”: π0 σ0 [X × Y,A] o [X,A] G [X,A + B ] y y π1 σ1 [Y,A] [X,B ] σ0 π0 σ1 π1 [Y,A + B ] o [Y,B ] G [X × Y,B ] 18/26 Understanding softness The homset [ X × Y , A + B ] is . . . the quotient of [X,A + B ] + [Y,A + B ] + [X × Y,A] + [X × Y,B ] under the equivalence relation generated by elementary pairs (f , g ): h ∈ [X,A] mm QQQ π0 mmm QQQσ0 mmmm QQQ QQQ vmmm @ f ∈ [X × Y,A] g ∈ [X,A + B ] f = π0 (h) σ0 (h) = g 18/26 (In)deﬁnite arrows Deﬁnition An arrow is indeﬁnite if it factors through 0 or through 1. Otherwise, it is deﬁnite. A simple decision procedure for indeﬁnite maps: l e t equal f g = (∗f g f a c t o r through 0∗) let f = f ;? and g = g ;? in i f cod f i s p o i n t e d t h e n t r u e else f =g 19/26 (In)deﬁnite arrows Deﬁnition An arrow is indeﬁnite if it factors through 0 or through 1. Otherwise, it is deﬁnite. A simple decision procedure for indeﬁnite maps: l e t equal f g = (∗f g f a c t o r through 0∗) let f = f ;? and g = g ;? in i f cod f i s p o i n t e d t h e n t r u e else f =g 19/26 A useful Lemma If π : f = f0 g1 f2 . . . gn = g is a path of elementary pairs crossing a corner, then [f ] = [g ] ∈ [ X × Y , A + B ] is indeﬁnite. Consider . . . fi−1 = π0 (hi ) σ0 (hi ) = gi gi = σ1 (hi+1 ) π0 (hi+1 ) = fi+1 . . . then hi and hi+1 are copointed and [g ] = [f ] as well. Lemma ∗ → If [f ] ∈ [ X × Y , A + B ] is deﬁnite and π : f − g is a path of elementary pairs, then π “bounces” along one side of this diagram. 20/26 A useful Lemma If π : f = f0 g1 f2 . . . gn = g is a path of elementary pairs crossing a corner, then [f ] = [g ] ∈ [ X × Y , A + B ] is indeﬁnite. Consider . . . fi−1 = π0 (hi ) σ0 (hi ) = gi gi = σ1 (hi+1 ) π0 (hi+1 ) = fi+1 . . . then hi and hi+1 are copointed and [g ] = [f ] as well. Lemma ∗ → If [f ] ∈ [ X × Y , A + B ] is deﬁnite and π : f − g is a path of elementary pairs, then π “bounces” along one side of this diagram. 20/26 A useful Lemma If π : f = f0 g1 f2 . . . gn = g is a path of elementary pairs crossing a corner, then [f ] = [g ] ∈ [ X × Y , A + B ] is indeﬁnite. Consider ... σ0 (hi ) = gi gi = σ1 (hi+1 ) ... then hi and hi+1 are copointed and [g ] = [f ] as well. Lemma ∗ → If [f ] ∈ [ X × Y , A + B ] is deﬁnite and π : f − g is a path of elementary pairs, then π “bounces” along one side of this diagram. 20/26 A useful Lemma If π : f = f0 g1 f2 . . . gn = g is a path of elementary pairs crossing a corner, then [f ] = [g ] ∈ [ X × Y , A + B ] is indeﬁnite. Consider ... σ0 (hi ) = gi gi = σ1 (hi+1 ) ... then hi and hi+1 are copointed and [g ] = [f ] as well. Lemma ∗ → If [f ] ∈ [ X × Y , A + B ] is deﬁnite and π : f − g is a path of elementary pairs, then π “bounces” along one side of this diagram. 20/26 A useful Lemma If π : f = f0 g1 f2 . . . gn = g is a path of elementary pairs crossing a corner, then [f ] = [g ] ∈ [ X × Y , A + B ] is indeﬁnite. Consider ... σ0 (hi ) = gi gi = σ1 (hi+1 ) ... then hi and hi+1 are copointed and [g ] = [f ] as well. Lemma ∗ → If [f ] ∈ [ X × Y , A + B ] is deﬁnite and π : f − g is a path of elementary pairs, then π “bounces” along one side of this diagram. 20/26 Softness for deﬁnite maps The previous Lemma transforms – for deﬁnite maps – the cardinal diagram from π0 σ0 [X × Y,A] o [X,A] G [X,A + B ] y y π1 σ1 [Y,A] [X,B ] σ0 π0 σ1 π1 [Y,A + B ] o [Y,B ] G [X × Y,B ] 21/26 Softness for deﬁnite maps The previous Lemma transforms – for deﬁnite maps – the cardinal diagram to π0 σ0 [X × Y,A] o [X,A] G [X,A + B ] [X × Y,A] [X,A + B ] y y π1 σ1 [Y,A] [X,B ] σ0 π0 [Y,A + B ] [X × Y,B ] σ1 π1 [Y,A + B ] o [Y,B ] G [X × Y,B ] 21/26 A new main result Theorem A pushout diagram [X,A] q MMM π0 qqq MMσ0 qqq MMM xqqq MM8 [X × Y,A] [X,A + B ] 8 x Po is also a pullback. ∗ → That is: given π : f − g a path “bouncing” on the upper row, there exists a unique h such that f = π0 (h) , σ0 (h) = g . 22/26 A new main result Theorem A pushout diagram [X,A] q MMM π0 qqq MMσ0 qqq MMM xqqq MM8 [X × Y,A] [X,A + B ] 8 x Po is also a pullback. ∗ → That is: given π : f − g a path “bouncing” on the upper row, there exists a unique h such that f = π0 (h) , σ0 (h) = g . 22/26 Deciding whether [f ] = [g ] in [ X0 × X1 , A0 + A1 ] let equivalent f g = (∗ f g a r e both d e f i n i t e → → f : X × Y − A , g : X − A + B ∗) find h s . t . f = σ0 (h) && π0 (h) = g Uniqueness of such h “makes it is easy” to ﬁnd it. Theorem There exists an algorithm to decide whether two parallel arrow-terms f , g of ΣΠ(A) are equal in [ X , A ]. The procedure runs in time polynomial in 23/26 Deciding whether [f ] = [g ] in [ X0 × X1 , A0 + A1 ] let equivalent f g = (∗ f g a r e both d e f i n i t e → → f : X × Y − A , g : X − A + B ∗) find h s . t . f = σ0 (h) && π0 (h) = g Uniqueness of such h “makes it is easy” to ﬁnd it. Theorem There exists an algorithm to decide whether two parallel arrow-terms f , g of ΣΠ(A) are equal in [ X , A ]. The procedure runs in time polynomial in 23/26 Deciding whether [f ] = [g ] in [ X0 × X1 , A0 + A1 ] let equivalent f g = (∗ f g a r e both d e f i n i t e → → f : X × Y − A , g : X − A + B ∗) find h s . t . f = σ0 (h) && π0 (h) = g Uniqueness of such h “makes it is easy” to ﬁnd it. Theorem There exists an algorithm to decide whether two parallel arrow-terms f , g of ΣΠ(A) are equal in [ X , A ]. The procedure runs in time polynomial in size(X ) · size(A) . 23/26 Deciding whether [f ] = [g ] in [ X0 × X1 , A0 + A1 ] let equivalent f g = (∗ f g a r e both d e f i n i t e → → f : X × Y − A , g : X − A + B ∗) find h s . t . f = σ0 (h) && π0 (h) = g Uniqueness of such h “makes it is easy” to ﬁnd it. Theorem There exists an algorithm to decide whether two parallel arrow-terms f , g of ΣΠ(∅) are equal in [ X , A ]. The procedure runs in time linear in (hgt(X ) + hgt(A)) · size(X ) · size(A) . 23/26 Background on bouncers Lemma Consider a pushout in Set A AA } AA g }} AA }} f } AA ~}} 2 B C 2 ~ Po TFAE: the diagram is a weak pullback, Ker (f ) and Ker (g ) commute. A bouncer for (a0 , a2 ) ∈ A is a1 ∈ A such that (a0 , a1 ) ∈ Ker (f ) , (a1 , a2 ) ∈ Ker (g ) . 24/26 Background on bouncers Lemma Consider a pushout in Set A AA } AA g }} AA }} f } AA ~}} 2 B C 2 ~ Po TFAE: the diagram is a weak pullback, Ker (f ) and Ker (g ) commute. A bouncer for (a0 , a2 ) ∈ A is a1 ∈ A such that (a0 , a1 ) ∈ Ker (f ) , (a1 , a2 ) ∈ Ker (g ) . 24/26 Bouncers within ΣΠ(A) A bouncer for (f , g ) ∈ [ X , A ] is an h ∈ [ X , A ] such that f GA wY X FF π0 www FF σ w FF 0 ww FF ww F4 X ×Y h A` + B GG x GG xx GG xx π0 GGG xσ 5 # xxx 0 X GA g 25/26 Bouncers within ΣΠ(A) A bouncer for (f , g ) ∈ [ X , A ] is an h ∈ [ X , A ] such that f GA x` X EE π0 xxx EE σ x EE 0 xx EE xx E4 X ×0 h A+1 FF y` FF yy FF yy π0 FFF yσ 4 # yyy 0 X GA g 25/26 A minimal nontrivial bouncer In ΣΠ(∅) all bouncer are trivial. This is not the case in general: Proposition In ΣΠ(A) (f , g ) have a bouncer if and only if (g , f ) have a bouncer. Proposition In ΣΠ(A) there exists at most one bouncer for (f , g ). 26/26 A minimal nontrivial bouncer In ΣΠ(∅) all bouncer are trivial. This is not the case in general: z π0 (?) f = σ0 (!) idx : (0 × 0) + x G (1 + 1) × x z π1 (?) g= σ1 (!) idx Proposition In ΣΠ(A) (f , g ) have a bouncer if and only if (g , f ) have a bouncer. Proposition In ΣΠ(A) there exists at most one bouncer for (f , g ). 26/26 A minimal nontrivial bouncer In ΣΠ(∅) all bouncer are trivial. This is not the case in general: z π0 (?) f = σ0 (!) idx z π1 (?) G (1 + 1) × x h= : (0 × 0) + x σ0 (!) idx z π1 (?) g= σ1 (!) idx Proposition In ΣΠ(A) (f , g ) have a bouncer if and only if (g , f ) have a bouncer. Proposition In ΣΠ(A) there exists at most one bouncer for (f , g ). 26/26 A minimal nontrivial bouncer In ΣΠ(∅) all bouncer are trivial. This is not the case in general: z π0 (?) f = σ0 (!) idx z π0 (?) G (1 + 1) × x h = : (0 × 0) + x σ1 (!) idx z π1 (?) g= σ1 (!) idx Proposition In ΣΠ(A) (f , g ) have a bouncer if and only if (g , f ) have a bouncer. Proposition In ΣΠ(A) there exists at most one bouncer for (f , g ). 26/26 A minimal nontrivial bouncer In ΣΠ(∅) all bouncer are trivial. This is not the case in general: z π0 (?) f = σ0 (!) idx z π0 (?) G (1 + 1) × x h = : (0 × 0) + x σ1 (!) idx z π1 (?) g= σ1 (!) idx Proposition In ΣΠ(A) (f , g ) have a bouncer if and only if (g , f ) have a bouncer. Proposition In ΣΠ(A) there exists at most one bouncer for (f , g ). 26/26 A minimal nontrivial bouncer In ΣΠ(∅) all bouncer are trivial. This is not the case in general: z π0 (?) f = σ0 (!) idx z π0 (?) G (1 + 1) × x h = : (0 × 0) + x σ1 (!) idx z π1 (?) g= σ1 (!) idx Proposition In ΣΠ(A) (f , g ) have a bouncer if and only if (g , f ) have a bouncer. Proposition In ΣΠ(A) there exists at most one bouncer for (f , g ). 26/26

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