Interacting Hopf algebras - the theory of linear systems

Interacting Hopf Algebras The Theory of Linear Systems Fabio Zanasi 5 October 2015 1 / 28 Introduction 2 / 28 Graphical formalisms Diagrammatic notations play an important role in various branches of science. x -1 2 x 3 / 28 Formal approaches to network diagrams monolithic approach compositional approach influenced by physics influenced by computer science 4 / 28 In this thesis A compositional theory of linear systems 5 / 28 In this thesis A compositional theory of linear systems • (diagrammatic) syntax • denotational semantics • sound and complete axiomatisation – the theory of interacting Hopf algebras 5 / 28 In this thesis A compositional theory of linear systems • (diagrammatic) syntax • denotational semantics leading example: signal flow diagrams • sound and complete axiomatisation – the theory of interacting Hopf algebras 5 / 28 In this thesis A compositional theory of linear systems • (diagrammatic) syntax leading example: signal flow diagrams • structural operational semantics • denotational semantics • sound and complete axiomatisation – the theory of interacting Hopf algebras • full abstraction • realisability 5 / 28 Syntax and Semantics 6 / 28 Signal Flow Graphs • Signal Flow Graphs are stream processing circuits studied in Control Theory since the 1950s. • Constructed combining four kinds of gate k k∈k x 7 / 28 Signal Flow Graphs • Signal Flow Graphs are stream processing circuits studied in Control Theory since the 1950s. • Constructed combining four kinds of gate k k k∈k x 7 / 28 Signal Flow Graphs • Signal Flow Graphs are stream processing circuits studied in Control Theory since the 1950s. • Constructed combining four kinds of gate k k k k k∈k x 7 / 28 Signal Flow Graphs • Signal Flow Graphs are stream processing circuits studied in Control Theory since the 1950s. • Constructed combining four kinds of gate k k k k k∈k k l x 7 / 28 Signal Flow Graphs • Signal Flow Graphs are stream processing circuits studied in Control Theory since the 1950s. • Constructed combining four kinds of gate k k k k k∈k k l k+ l x 7 / 28 Signal Flow Graphs • Signal Flow Graphs are stream processing circuits studied in Control Theory since the 1950s. • Constructed combining four kinds of gate k k k l k k∈k k l k+ l x 7 / 28 Signal Flow Graphs • Signal Flow Graphs are stream processing circuits studied in Control Theory since the 1950s. • Constructed combining four kinds of gate k k k l k kl k∈k k l k+ l x 7 / 28 Signal Flow Graphs • Signal Flow Graphs are stream processing circuits studied in Control Theory since the 1950s. • Constructed combining four kinds of gate k k k l k kl k∈k k l k+ l l 7 / 28 Signal Flow Graphs • Signal Flow Graphs are stream processing circuits studied in Control Theory since the 1950s. • Constructed combining four kinds of gate k k k l k kl k∈k k l k+ l k l 7 / 28 Signal Flow Graphs • Signal Flow Graphs are stream processing circuits studied in Control Theory since the 1950s. • Constructed combining four kinds of gate k k k l k kl k∈k k l k+ l k k l 7 / 28 Signal flow graphs An example: x x 8 / 28 Signal flow graphs An example: x x 8 / 28 Signal flow graphs An example: x x 1 0 0 8 / 28 Signal flow graphs An example: 0 0 0 1 0 0 8 / 28 Signal flow graphs An example: 1 0 0 1 0 0 0 8 / 28 Signal flow graphs An example: 1 0 0 1 0 0 0 0 1 8 / 28 Signal flow graphs An example: 1 1 0 0 1 0 0 0 8 / 28 Signal flow graphs An example: 1 0 0 1 1 0 0 0 1 8 / 28 Signal flow graphs An example: 1 1 1 0 1 0 0 0 8 / 28 Signal flow graphs An example: 1 1 1 0 0 0 1 1 8 / 28 Signal flow graphs An example: 1 1 1 0 0 0 1 1 8 / 28 Signal flow graphs An example: 2 1 0 0 0 0 1 1 1 8 / 28 Signal flow graphs An example: 2 1 1 0 0 0 0 1 0 1 1 8 / 28 Signal flow graphs An example: 2 1 0 0 0 0 1 1 1 1 8 / 28 Signal flow graphs An example: 2 1 1 1 0 0 0 0 1 1 1 1 8 / 28 Signal flow graphs An example: 2 1 0 0 0 0 1 2 1 1 8 / 28 Signal flow graphs An example: 2 1 0 0 0 0 1 2 2 1 8 / 28 Signal flow graphs An example: 2 1 0 0 0 0 1 2 1 2 8 / 28 Signal flow graphs An example: 3 1 0 0 0 0 0 1 3 1 2 3 8 / 28 Signal flow graphs An example: 3 1 0 0 0 0 0 1 3 1 2 3 Input 1000 . . . produces 1234 . . . . 8 / 28 Syntax diagrams are generated by the grammar | | k | | | k | | | x | | | x c, d :: = | | | c | c d | d 9 / 28 Syntax diagrams are generated by the grammar | | k | | | k | | | x | | | x c, d :: = | | | c | c d | d We can represent (orthodox) signal flow graphs as diagrams: x x ! x x 9 / 28 Syntax diagrams are generated by the grammar | | k | | | k | | | x | | | x c, d :: = | | | c | c d | d We can represent (orthodox) signal flow graphs as diagrams: x x ! x x 9 / 28 Structural Operational Semantics x l k  k l  kl  k k k k   k kl l  k  k k l  l k x k l k k l l k  x k k  x k k k k+l k k+l    k l 0  l c c d d → − → − k l → − − m → n → − → − k l  → − → − l m  c0 d0 c0 d0  c d → − → − k m  c0 d0 0  c d − → − → l k − → − n m →  c0 d0 10 / 28 Example 0 x 0 x 1 1  x 1 x 1 x 2 0 2  x 1 0 3  .. . 11 / 28 Denotational Semantics The semantics hh·ii maps a diagram to a linear relation between stream vectors (i.e. a subspace over streams). σ σ σ σ σ σ σ σ σ τ τ σ σ k σ x k·σ k·σ k σ x·σ ! − σ c ! − ρ ! − ρ ! − σ c d x σ σ d ! − τ x·σ σ τ σ σ+τ ! − τ − ! σ 1 σ+τ σ τ c − ! σ 1 − ! σ 2 0 0 ! − ! τ1 − σ 2 c d d ! − τ2 ! − τ1 ! − τ 2 12 / 28 Interacting Hopf Algebras 13 / 28 The equational theory IH = = = 1 = = = = = = = = p = p p p1 p2 = p1 p 2 p p p p = p = = = p2 p = -1 1 = = p p1+ p2 = p = = p = p1 0 = = = = = = = = p = = = p1 p2 p = p p p = = p p p = p1 = = p1 p2 -1 0 = p2 = = = p1+ p2 = p 6= 0 Theorem (soundness and completeness) For any diagrams c, d, IH c=d ⇔ hhcii = hhdii. 14 / 28 The equational theory IH interaction of x k the theory HAk[x] of k[x]-Hopf algebras = = = 1 = = = = = = = = p = p p p1 p2 = p 1 p2 p p p p = p = p1 0 = = p2 p1+ p2 = p = -1 = = = = = = = = p = = = p = 1 = = p = p = p1 p 2 p = = = p p p p1 p2 -1 = = p p p = p1 0 = p2 = = = p1+ p2 = p 6= 0 Theorem (soundness and completeness) For any diagrams c, d, IH c=d ⇔ hhcii = hhdii. 14 / 28 Modular construction of IH interaction of x k the theory HAk[x] of k[x]-Hopf algebras = = = = p = = = = = = p = p p p1 p2 = p1 p2 p = p = = p p p1 1 = 0 = p2 = p1+ p2 15 / 28 Modular construction of IH interaction of x k the theory HAk[x] of k[x]-Hopf algebras = = = = p = = = = = = p = p p p1 p2 = p1 p2 p = p = = p p p1 1 = 0 = p2 = p1+ p2 15 / 28 Modular construction of IH interaction of x k the theory HAk[x] of k[x]-Hopf algebras = = = = p = = = = = = p = p p p1 p2 = p1 p2 p = p = = p p p1 1 = 0 = p2 = p1+ p2 15 / 28 Modular construction of IH interaction of x k the theory HAk[x] of k[x]-Hopf algebras = = = = p 1 = = = = = = = p = p p p1 p2 = p1 p2 p = p = = p p p1 0 = p2 = p1+ p2 15 / 28 Modular construction of IH interaction of x k the theory HAk[x] of k[x]-Hopf algebras = = = = = = = p 1 = = = = p = p p p1 p2 = p1 p2 p = p = = p p p1 0 = the diagram p expresses a polynomial p = k0 + k1 x + · · · + kn xn p2 p := ... = p1+ p2 kn x x ... = k2 x x k1 x p k=0 x ... 15 / 28 Modular construction of IH interaction of x k the theory HAk[x] of k[x]-Hopf algebras = = = = = = = p 1 = = = = p = p p p1 p2 = p1 p2 p = p = = p p p1 0 = the diagram p expresses a polynomial p = k0 + k1 x + · · · + kn xn p2 p := ... = p1+ p2 kn x x ... = k2 x x k1 x p k=0 x ... 15 / 28 The equational theory IH interaction of x k the theory HAk[x] of k[x]-Hopf algebras = = = 1 = = = = = = = = p = p p p1 p2 = p 1 p2 p p p p = p = p1 0 = = p2 p1+ p2 = p = -1 = = = = = = = = p = = = p = 1 = = p = p = p1 p 2 p = = = p p p p1 p2 -1 = = p p p = p1 0 = p2 = = = p1+ p2 = p 6= 0 Theorem (soundness and completeness) For any diagrams c, d, IH c=d ⇔ hhcii = hhdii. 16 / 28 The equational theory IH x interaction of k the theory HAk[x] of k[x]-Hopf algebras the theory HA = = = p 1 = = = = = = = = p = p p p1 p2 = p 1 p2 p p p p = p = = = p2 -1 = = p = 1 = = p p1+ p2 = p = = p = p1 0 of “co”k[x]-Hopf algebras = = = = k = = op k[x] = = = x interaction of p1 p2 p = p p p = = p p p = p1 = = p1 p2 -1 0 = p2 = = = p1+ p2 = p 6= 0 Theorem (soundness and completeness) For any diagrams c, d, IH c=d ⇔ hhcii = hhdii. 16 / 28 The equational theory IH interaction of k the theory HAk[x] of k[x]-Hopf algebras the theory HA = = = p 1 = = = = = = = = p = p p p1 p2 = p1 p2 p p p p = = p = p = = = p2 p1+ p2 = p p = -1 = p p p = = p p p = p1 p1 0 = = = = = = = = k of “co”k[x]-Hopf algebras = = = op k[x] = 1 = = p p1 p 2 p = = p1 p2 0 = p2 = -1 = = p1+ p2 = p 6= 0 interaction of x k with x x x interaction of k Theorem (soundness and completeness) For any diagrams c, d, IH c=d ⇔ hhcii = hhdii. 16 / 28 The equational theory IH interaction of k the theory HAk[x] of k[x]-Hopf algebras the theory HA = = = p 1 = = = = = = = = p = p p p1 p2 = p1 p2 p p p p = = = p2 p p = -1 p1+ p2 = = p = p1 0 = = = = = = = = k of “co”k[x]-Hopf algebras = = = op k[x] = 1 = = p p = p1 p 2 p = p p p = = p p p = p1 = = p1 p2 0 = p2 = -1 = = p1+ p2 = p 6= 0 interaction of x k with x x x interaction of k Theorem (soundness and completeness) For any diagrams c, d, IH c=d ⇔ hhcii = hhdii. 16 / 28 The equational theory IH interaction of k the theory HAk[x] of k[x]-Hopf algebras the theory HA = = = p 1 = = = = = = = = p = p p p1 p2 = p1 p2 p p p p = = p = p = p1 0 = = p2 p1+ p2 = p p = -1 = = = = = = = = k of “co”k[x]-Hopf algebras = = = op k[x] = 1 = = p p1 p 2 = p p p = = p p p = p1 = p = p1 p2 0 = p2 = -1 = = p1+ p2 = p 6= 0 interaction of x k with x x x interaction of k Theorem (soundness and completeness) For any diagrams c, d, IH c=d ⇔ hhcii = hhdii. 16 / 28 The equational theory IH interaction of k the theory HAk[x] of k[x]-Hopf algebras the theory HA = = = p 1 = = = = = = = = p = p p p1 p2 = p 1 p2 p p p p = = p = p = p1 0 = = p2 p1+ p2 = p p = -1 = = = = = = = = k of “co”k[x]-Hopf algebras = = = op k[x] = 1 = = p p1 p 2 = p p p = = p p p = p1 = p = p1 p2 0 = p2 = -1 = = p1+ p2 = p 6= 0 interaction of x k with x x x interaction of k Theorem (soundness and completeness) For any diagrams c, d, IH c=d ⇔ hhcii = hhdii. 16 / 28 The equational theory IH interaction of k the theory HAk[x] of k[x]-Hopf algebras the theory HA = = = p 1 = = = = = = = = p = p p p1 p2 = p1 p 2 p p p p = = p = p = p1 0 = = p2 -1 = = = = = = = = k of “co”k[x]-Hopf algebras = = = op k[x] p p = 1 = = p p1+ p2 = = p1 p2 p = p p p = = p p p = p1 = = p 1 p2 0 = p2 = -1 = = p1+ p2 = p 6= 0 interaction of x k with x x x interaction of k Theorem (soundness and completeness) For any diagrams c, d, IH c=d ⇔ hhcii = hhdii. 16 / 28 Modular construction of IH HAk[x] HAop k[x] 17 / 28 Modular construction of IH HAk[x] ∼ =  Matk[x] HAop k[x] ∼ =  op Matk[x] 17 / 28 Modular construction of IH HAk[x] + HAop k[x] ∼ =  op Matk[x] + Matk[x] 17 / 28 Modular construction of IH HAk[x] + HAop k[x] ∼ = IHk[x]  op Matk[x] + Matk[x] 17 / 28 Modular construction of IH HAk[x] + HAop k[x] / HAop ; HAk[x] k[x] t HAk[x] ; HAop k[x] ∼ = IHk[x]  op Matk[x] + Matk[x] 17 / 28 Modular construction of IH distributive law = distributive law = HAk[x] + HAop k[x] / HAop ; HAk[x] k[x] t HAk[x] ; HAop k[x] ∼ = IHk[x]  op Matk[x] + Matk[x] 17 / 28 Modular construction of IH distributive law = distributive law = HAk[x] + HAop k[x] ∼ = t HAk[x] ; HAop k[x] / HAop ; HAk[x] k[x] / IHk[x] v  op Matk[x] + Matk[x] = = = -1 = p p = p p 6= 0 p = = -1 = = = 17 / 28 Modular construction of IH distributive law = distributive law = HAk[x] + HAop k[x] ∼ = t HAk[x] ; HAop k[x] / HAop ; HAk[x] k[x] / IHk[x] v  op Matk[x] + Matk[x] = = = -1 = p p = p p 6= 0 p = = -1 = = = 17 / 28 Modular construction of IH HAk[x] + HAop k[x] t HAk[x] ; HAop k[x] ∼ = ∼ =  op Matk[x] + Matk[x] / HAop ; HAk[x] k[x] / IHk[x] v ∼ =  / Matop ; Matk[x] k[x]  t op Matk[x] ; Matk[x] 17 / 28 Modular construction of IH HAk[x] + HAop k[x] t HAk[x] ; HAop k[x] ∼ =  t op Matk[x] ; Matk[x] ∼ = / HAop ; HAk[x] k[x] / IHk[x] v ∼ =  / Matop ; Matk[x] k[x]  op Matk[x] + Matk[x] / SVk(x) v 17 / 28 Modular construction of IH HAk[x] + HAop k[x] t HAk[x] ; HAop k[x] ∼ =  t op Matk[x] ; Matk[x] ∼ = / HAop ; HAk[x] k[x] / IHk[x] v ∼ =  / Matop ; Matk[x] k[x]  op Matk[x] + Matk[x] / SVk(x) v the field of fractions of polynomials 17 / 28 Modular construction of IH HAk[x] + HAop k[x] t HAk[x] ; HAop k[x] ∼ =  t op Matk[x] ; Matk[x] ∼ =  op Matk[x] + Matk[x] / HAop ; HAk[x] k[x] / IHk[x] v ∼ = ∼ =  / Matop ; Matk[x] k[x]  v / SVk(x) the field of fractions of polynomials 17 / 28 Modular construction of IH HAk[x] + HAop k[x] t HAk[x] ; HAop k[x] ∼ = ∼ =  op Matk[x] + Matk[x] t  op Matk[x] ; Matk[x] / HAop ; HAk[x] k[x] / IHk[x] u ∼ =  / Matop ; Matk[x] k[x]  u / SVk(x)  op Matk[[x]] + Matk[[x]]  t op Matk[[x]] ; Matk[[x]] ∼ =  / Matop ; Matk[[x]] k[[x]]  u / SVk((x)) 17 / 28 Modular construction of IH HAk[x] + HAop k[x] t HAk[x] ; HAop k[x] ∼ = ∼ =  op Matk[x] + Matk[x] t  op Matk[x] ; Matk[x] / HAop ; HAk[x] k[x] / IHk[x] u ∼ =  / Matop ; Matk[x] k[x]  u / SVk(x)  op Matk[[x]] + Matk[[x]]  t op Matk[[x]] ; Matk[[x]] ∼ =  / Matop ; Matk[[x]] k[[x]]  u / SVk((x)) the field of formal Laurent series (streams) 17 / 28 Modular construction of IH HAk[x] + HAop k[x] t HAk[x] ; HAop k[x] ∼ = ∼ = / HAop ; HAk[x] k[x] / IHk[x]  op Matk[x] + Matk[x] t  op Matk[x] ; Matk[x] u ∼ =  u / SVk(x)   / Matop ; Matk[x] k[x] hh·ii  / Matop ; Matk[[x]] k[[x]] op Matk[[x]] + Matk[[x]]  t op Matk[[x]] ; Matk[[x]] ∼ = 
u / SVk((x)) the field of formal Laurent series (streams) 17 / 28 Modular construction of IH HAk[x] + HAop k[x] ∼ = t HAk[x] ; HAop k[x] / HAop ; HAk[x] k[x] / IHk[x] ∼ =  op Matk[x] + Matk[x] ∼ = t  op Matk[x] ; Matk[x] u  u / SVk(x)   / Matop ; Matk[x] k[x] hh·ii  / Matop ; Matk[[x]] k[[x]] op Matk[[x]] + Matk[[x]]  t op Matk[[x]] ; Matk[[x]] ∼ = 
u / SVk((x)) the field of formal Laurent series (streams) Theorem (soundness and completeness) For any diagrams c, d , IH c=d ⇔ hhcii = hhdii. 17 / 28 One cube to rule them all / HAopk[x] ; HAk[x] op HAk[x] + HAk[x] u ∼ = op / IHk[x] HAk[x] ; HAk[x]  ∼ = ∼ = op ∼ = w / Matk[x] + Matk[x] u  op Matk[x] ; Matk[x] /   op Matk[x] ; Matk[x] w SVk(x) 18 / 28 One cube to rule them all / HAopk[x] ; HAk[x] op HAk[x] + HAk[x] u ∼ = op / IHk[x] HAk[x] ; HAk[x]  ∼ = ∼ = op ∼ = w / Matk[x] + Matk[x] u  op Matk[x] ; Matk[x] /  SVk(x) w  op Matk[x] ; Matk[x] / op u op HAR ; HAR HAR + HAR ∼ = / IHR  op ∼ =  u op MatR ; MatR ∼ = MatR + MatR op HAR ; HAR x / ∼ =  op MatR ; MatR  / SVk x 18 / 28 One cube to rule them all principal ideal domain / HAopk[x] ; HAk[x] op HAk[x] + HAk[x] u ∼ = op / IHk[x] HAk[x] ; HAk[x]  ∼ = ∼ = op ∼ = w / Matk[x] + Matk[x] u  op Matk[x] ; Matk[x] /  SVk(x) w  op Matk[x] ; Matk[x] / op u op HAR ; HAR HAR + HAR ∼ = / IHR  op ∼ =  u op MatR ; MatR ∼ = MatR + MatR op HAR ; HAR x / ∼ =  op MatR ; MatR  / SVk x the field of fractions of R 18 / 28 One cube to rule them all case k[x]: calculus of signal flow diagrams (control theory) principal ideal domain / HAopk[x] ; HAk[x] op HAk[x] + HAk[x] u ∼ = op / IHk[x] HAk[x] ; HAk[x]  ∼ = ∼ = op ∼ = w / Matk[x] + Matk[x] u  op Matk[x] ; Matk[x] /  SVk(x) w  op Matk[x] ; Matk[x] / op u op HAR ; HAR HAR + HAR ∼ = / IHR  op ∼ =  u op MatR ; MatR ∼ = MatR + MatR op HAR ; HAR x / ∼ =  op MatR ; MatR  / SVk x the field of fractions of R 18 / 28 One cube to rule them all case k[x]: calculus of signal flow diagrams (control theory) principal ideal domain / HAopk[x] ; HAk[x] op HAk[x] + HAk[x] u ∼ = op / IHk[x] HAk[x] ; HAk[x]  op ∼ = / Matk[x] + Matk[x] u  / op Matk[x] ; Matk[x]  / op op ∼ = HAZ ; HAZ /  op ∼ = u  op MatZ ; MatZ  w op x ∼ = MatZ + MatZ  / SVQ x /  / HAZ2 ; HAZ  op MatZ ; MatZ / op 2 ∼ = op 2 ∼ = 2 op 2 MatZ ; MatZ 2 x ∼ = op 2 / op 2 HAZ ; HAZ2 ∼ = IHZ2 MatZ + MatZ v  op MatR ; MatR the field of fractions of R HAZ2 + HAZ v   / SVk x op HAZ ; HAZ ∼ = / MatR + MatR u MatR ; MatR IHZ ∼ = ∼ = op  op HAR ; HAR x / IHR  ∼ = op Matk[x] ; Matk[x] SVk(x) HAZ + HAZ u op ∼ = HAR ; HAR ∼ = ∼ = / op HAR + HAR u w /  op 2 MatZ ; MatZ 2  x SVZ 2 18 / 28 One cube to rule them all case k[x]: calculus of signal flow diagrams (control theory) principal ideal domain / HAopk[x] ; HAk[x] op HAk[x] + HAk[x] u ∼ = op / IHk[x] HAk[x] ; HAk[x]  op ∼ = / Matk[x] + Matk[x] u  / op Matk[x] ; Matk[x]  / op op ∼ = HAZ ; HAZ /  op ∼ = u  op MatZ ; MatZ  w op x ∼ = MatZ + MatZ  / SVQ x /  / HAZ2 ; HAZ  op MatZ ; MatZ / op 2 ∼ = op 2 ∼ = 2 op 2 MatZ ; MatZ 2 x ∼ = op 2 / op 2 HAZ ; HAZ2 ∼ = IHZ2 MatZ + MatZ v  op MatR ; MatR the field of fractions of R HAZ2 + HAZ v   / SVk x op HAZ ; HAZ ∼ = / MatR + MatR u MatR ; MatR IHZ ∼ = ∼ = op  op HAR ; HAR x / IHR  ∼ = op Matk[x] ; Matk[x] SVk(x) HAZ + HAZ u op ∼ = HAR ; HAR ∼ = ∼ = / op HAR + HAR u w /  op 2 MatZ ; MatZ 2  x SVZ 2 case Z: a graphical syntax for rational subspaces 18 / 28 One cube to rule them all case k[x]: calculus of signal flow diagrams (control theory) principal ideal domain / HAopk[x] ; HAk[x] op HAk[x] + HAk[x] u ∼ = op / IHk[x] HAk[x] ; HAk[x]  op ∼ = / Matk[x] + Matk[x] u  / op Matk[x] ; Matk[x]  / op op ∼ = HAZ ; HAZ /  op ∼ = u  op MatZ ; MatZ  w op x ∼ = MatZ + MatZ  / SVQ x /   ∼ = 2 x ∼ = op 2 op 2 MatZ ; MatZ / op 2 HAZ ; HAZ2 ∼ = IHZ2 MatZ + MatZ v  2 case Z: a graphical syntax for rational subspaces / HAZ2 ; HAZ op MatZ ; MatZ / op 2 ∼ = op 2 op MatR ; MatR the field of fractions of R HAZ2 + HAZ v   / SVk x op HAZ ; HAZ ∼ = / MatR + MatR u MatR ; MatR IHZ ∼ = ∼ = op  op HAR ; HAR x / IHR  ∼ = op Matk[x] ; Matk[x] SVk(x) HAZ + HAZ u op ∼ = HAR ; HAR ∼ = ∼ = / op HAR + HAR u w /  op 2 MatZ ; MatZ 2  x SVZ 2 case Z2 phase-free ZX-calculus (categorical quantum mechanics) tweak of the calculus of stateless connectors (concurrency theory) 18 / 28 Full Abstraction and Realisability 19 / 28 Structural Operational Semantics x l k  k l  kl  k k k k   k kl l  k  k k l  l k x k l k k l l k  x k k  x k k k k+l k k+l    k l 0  l c c d d → − → − k l → − − m → n → − → − k l  → − → − l m  c0 d0 c0 d0  c d → − → − k m  c0 d0 0  c d − → − → l k − → − n m →  c0 d0 20 / 28 Full Abstraction The observable behaviour hci of a diagram c is the set of all traces starting from an initial state for c (i.e. one where all the registers are labeled with 0). 21 / 28 Full Abstraction The observable behaviour hci of a diagram c is the set of all traces starting from an initial state for c (i.e. one where all the registers are labeled with 0). Theorem (?) For any diagrams c and d hhcii = hhdii ⇐⇒ hci = hdi 21 / 28 Full Abstraction The observable behaviour hci of a diagram c is the set of all traces starting from an initial state for c (i.e. one where all the registers are labeled with 0). Theorem (?) For any diagrams c and d hhcii = hhdii ⇐⇒ hci = hdi Not true in general. The denotational semantics is coarser than the operational semantics. 21 / 28 Full Abstraction A counterexample x x h x ii = hh ii = hh x i(h i(h x x x ii x hh i 22 / 28 Full Abstraction A counterexample ii = hh x i(h i(h x ii i x x x ii = hh x x x h x x x hh 22 / 28 x Full Abstraction A counterexample x ii = hh x i(h i(h x x x ii i x x x h ii = hh x x x hh k k  l l  m m . . . 22 / 28 x Full Abstraction A counterexample x x h x ii = hh ii = hh x i(h i(h x x x x ii x hh i 0 0 k k  l l  m m . . . 22 / 28 x Full Abstraction A counterexample x x h x ii = hh ii = hh x i(h i(h x x x x ii x hh i 0 k k  0 kl k l l l  m m . . . 22 / 28 x Full Abstraction A counterexample x ii = hh ii = hh x i(h i(h x x 0 x x h 0 x ii x hh i 0 k k  0 kl k l l l  m m . . . 22 / 28 Full Abstraction A counterexample x ii = hh ii = hh x i(h i(h x x 0 00 x x h 0 x ii x hh i 0 k k  k k 0 kl k l l l  m m . . . 22 / 28 Full Abstraction A counterexample x ii = hh ii = hh x i(h i(h x x 0 00 l ii i 0 k k  k k kk x x h 0 x x hh 0 kl k l l l  l l l m m . . . . . . 22 / 28 Full Abstraction A counterexample x ii = hh ii = hh x i(h i(h x x x x h 0 x 0 k k 00 0 kl  k kk l l l  l m m . . . . . . x x has deadlocks and x l l We say that i 0 k k l ii x hh x needs initialisation. 22 / 28 Full Abstraction Theorem For any diagrams c, d deadlock and initialisation free hhcii = hhdii ⇐⇒ hci = hdi 23 / 28 Realisability In presence of deadlocks or initialisation, we cannot determine directionality of the flow. x x x x A trace for these diagrams cannot be thought as the execution of a state-machine. 24 / 28 Realisability In presence of deadlocks or initialisation, we cannot determine directionality of the flow. x x x x A trace for these diagrams cannot be thought as the execution of a state-machine. However, all the diagrams can be put into an executable form using the IH equational theory = . Realisability Theorem For any diagram c there exists IH d deadlock and initialisation free such that c = d. 24 / 28 Realisation via IH-rewriting Implementing the Fibonacci circuit x -1 x -1 x x -1 x x x x -1 x x x x x x -1 x x x x x 25 / 28 Realisation via IH-rewriting Implementing the Fibonacci circuit x -1 x -1 x x -1 x x x x -1 x x x x x x -1 x x x x x 25 / 28 Realisation via IH-rewriting Implementing the Fibonacci circuit x -1 x -1 x x -1 x x x x -1 x x x x x x -1 x x x x x 25 / 28 Realisation via IH-rewriting Implementing the Fibonacci circuit x -1 x -1 x x -1 x x x x -1 x x x x x x -1 x x x x x 25 / 28 Realisation via IH-rewriting Implementing the Fibonacci circuit x -1 x -1 x x -1 x x x x x = x x -1 x x x -1 x x x x x 25 / 28 Realisation via IH-rewriting Implementing the Fibonacci circuit x -1 x x -1 x x x x -1 x x x x x x = = x -1 -1 x x x x x 25 / 28 Realisation via IH-rewriting Implementing the Fibonacci circuit x -1 x -1 x x -1 x x -1 x x x x = x x x -1 x=x -1 x x x x x 25 / 28 Realisation via IH-rewriting Implementing the Fibonacci circuit x -1 x -1 x x -1 x x -1 x x = x x x x x -1 -1 x -1 x x x x x 25 / 28 Realisation via IH-rewriting Implementing the Fibonacci circuit x -1 x -1 x x -1 = x x x x -1 x -1 x x x x x x -1 x x x x x 25 / 28 Realisation via IH-rewriting Implementing the Fibonacci circuit x -1 x -1 x x -1 x x x x -1 x x x x x x -1 x x x x x 25 / 28 Conclusions 26 / 28 Conclusions ◦ We gave a compositional account of linear systems, whose main feature is the equational theory IH . 27 / 28 Conclusions ◦ We gave a compositional account of linear systems, whose main feature is the equational theory IH . ◦ IH exhibits the interplay of Hopf algebras, which gives raise to Frobenius algebras. 27 / 28 Conclusions ◦ We gave a compositional account of linear systems, whose main feature is the equational theory IH . ◦ IH exhibits the interplay of Hopf algebras, which gives raise to Frobenius algebras. ◦ The construction of IH is based on modular techniques for composing algebraic theories via distributive laws, which are developed in the thesis extending recent work by S. Lack and E. Cheng. 27 / 28 Conclusions ◦ We gave a compositional account of linear systems, whose main feature is the equational theory IH . ◦ IH exhibits the interplay of Hopf algebras, which gives raise to Frobenius algebras. ◦ The construction of IH is based on modular techniques for composing algebraic theories via distributive laws, which are developed in the thesis extending recent work by S. Lack and E. Cheng. ◦ Interesting instances of IH are • IHZ — graphical linear algebra over rational subspaces • IHZ2 — relevant for categorical quantum mechanics and concurrency theory • IHk[x] — calculus of signal flow diagrams, presented in this talk 27 / 28 Conclusions ◦ In studying the case of signal flow diagrams, we explored the operational ramifications of our approach. 28 / 28 Conclusions ◦ In studying the case of signal flow diagrams, we explored the operational ramifications of our approach. ◦ We argue that a formal theory of signal flow does not need to be endowed with a primitive notion of causality. 28 / 28 Conclusions ◦ In studying the case of signal flow diagrams, we explored the operational ramifications of our approach. ◦ We argue that a formal theory of signal flow does not need to be endowed with a primitive notion of causality. flow graphs differ from electrical network graphs in that their branches are directed. In accounting for branch directions it is necessary to take an entirely different line of approach from that adopted in electrical network topology. (S. J. Mason - 1953) 28 / 28 Conclusions ◦ In studying the case of signal flow diagrams, we explored the operational ramifications of our approach. ◦ We argue that a formal theory of signal flow does not need to be endowed with a primitive notion of causality. flow graphs differ from electrical network graphs in that their branches are directed. In accounting for branch directions it is necessary to take an entirely different line of approach from that adopted in electrical network topology. (S. J. Mason - 1953) Adding a signal flow direction is often a figment of one’s imagination, [which] needlessly complicates matters, mathematically and conceptually. A good theory of systems takes the behavior as the basic notion [...] and switches back and forth between a wide variety of convenient representations. (J. C. Willems - 2009) 28 / 28