Quantitative Algebraic Reasoning (an Overview) Giorgio Bacci - PowerPoint PPT Presentation

Quantitative Algebraic Reasoning (an Overview) Giorgio Bacci Aalborg University, Denmark 
 (invited talk, EXPRESS/SOS 2020 ) based on joint work with R. Mardare, P . Panangaden, G. Plotkin

Why Algebraic Reasoning? Algebraic reasoning is extensively used in process algebras 
 ...actually, all theoretical computer science! • Representability: describing mathematical structures as an algebra of operations subject to equations 
 eg: Labelled transition systems as process algebras • E ff ectful Languages: understanding computational e ff ects as operations to be performed during execution 
 eg: Moggi's monadic e ff ects • Algorithms: compositional reasoning, up-to techniques, normal forms, etc...

An Application P 𝖾𝖿𝗀 Consider the recursively defined CCS process P = a | P = a ∣ P P ∼ P ∣ P not image finite! Bisimulation up-to technique is an elegant and way to prove it! ℛ P ∼ a ∣ P P ∣ P P x ∼ 0 ∣ x ( x ∣ y ) ∣ z ∼ x ∣ ( y ∣ z ) a a x ∣ y ∼ y ∣ x ℛ P ∣ P ∼ a p ∣ P ∣ 0 ∣ a q | P a n ∣ 0 ∣ P ∼ P

To understand how should be a quantitative generalisation 
 of the concept of equational logic we first need to understand Universal algebras (its categorical generalisation)

Historical Perspective • Lawvere'64: categorically axiomatised (the clone of) equational theories • Eilenberg-Moore'65, Kleisli'65: Every standard construction is induced by a pair of adjoint functors, and EM-algebras for are universal algebras • Linton'66: general connection between monads and Lawvere theories

Universal Algebras Lawvere'64, Linton’66 (the standard picture) Equational 
 operations & equations Theory U 𝒱 Lawvere Monad M Theory 𝓜 on Set Set partial converse (needs generalised form of 𝓜 ) Mod( 𝓜 , Set) = M -Algebras ≅ Eilenberg-Moore Algebras

Historical Perspective (continued) • ... • Moggi'88: How to incorporate e ff ects into denotational semantics? - Monads as notions of computations • Plotkin & Power'01: (most of the) Monads are given by operations and equations expressed by means of enriched Lawvere Theories - Generic Algebraic E ff ects

Universal Algebras Power (TAC’99) (the "enriched"picture) Enriched over a locally finitely presentable monoidal category V finite rank V-Lawvere V-Monad Theory 𝓜 M on C C exact converse! Missing a logical representation Mod( 𝓜 , C) = M -Algebras ≅ C

Historical Perspective (continued) • ... • Moggi'88: How to incorporate e ff ects into denotational semantics? - Monads as notions of computations • Plotkin & Power'01: (most of the) Monads are given by operations and equations expressed by means of enriched Lawvere Theories - Generic Algebraic E ff ects • Mardare, Panangaden, Plotkin (LICS'16): 
 Theory of e ff ects in a metric setting 
 - Quantitative Algebraic E ff ects (operations & quantitative equations give m onads on EMet )

The picture of the Talk operations & Quantitative Equational 
 quantitative equations Theory U 𝒱 EMet-Lawvere EMet Monad 
 Theory 𝓜 M on EMet EMet category of extended metric spaces and 
 non-expansive maps Mod( 𝓜 , C ) = M -Algebras ≅ EMet

Quantitative Equations s = ε t "s is approximately equal to t up to an error e" ε t s

Quantitative Equational Theory Mardare, Panangaden, Plotkin (LICS’16) A quantitative equational theory U of type M is a set of Σ 𝒱 conditional { t i = ε i s i ∣ i ∈ I } ⊢ t = ε s quantitative equations closed under substitution of variables, logical inference, 
 and the following "meta axioms" (Refl) ⊢ x = 0 x (Symm) x = ε y ⊢ y = ε x (Triang) x = ε y , y = δ z ⊢ x = ε + δ y (NExp) x 1 = ε y 1 , …, y n = ε y n ⊢ f ( x 1 , …, x n ) = ε f ( y 1 , …, y n ) − for f ∈ Σ (Max) x = ε y ⊢ x = ε + δ y − for δ > 0 (Inf) { x = δ y ∣ δ > ε } ⊢ x = ε y

Quantitative Algebras Mardare, Panangaden, Plotkin (LICS’16) The models of a quantitative equational theory U of type M are Σ 𝒱 Quantitative Σ -Algebras: 𝒝 = ( A , α : Σ A → A ) − Universal Σ -algebras on EMet Satisfying the all the conditional quantitative equations in 𝒱 Satisfiability 𝒝 ⊧ ( { t i = ε i s i ∣ i ∈ I } ⊢ t = ε s ) i ff for any assignment ι : X → A ( ∀ i ∈ I . d A ( ι ( t i ), ι ( s i )) ≤ ε i ) implies d A ( ι ( t ), ι ( s )) ≤ ε

Category of Models The collection of models for forms a category denoted by 𝒱 𝕃 ( Σ , 𝒱 ) with morphisms being the non-expansive homomorphisms α Σ A A non-expansive Σ h h S-homomorphism Σ β Σ B B

Quantitative Semilattices with ⊥ Are the quantitative algebras over the signature Σ 𝒯 = { 0 : 0, + : 2} bottom join satisfying the following conditional quantitative equations (S0) ⊢ x + 0 = 0 x (S1) ⊢ x + x = 0 x (S2) ⊢ x + y = 0 y + x (S3) ⊢ ( x + y ) + z = 0 x + ( y + z ) (S4) x = ϵ y , x ′ � = ϵ ′ � y ′ � ⊢ x + x ′ � = δ y + y ′ � , where δ = max{ ϵ , ϵ ′ � }

Example of models Unit interval with Euclidian distance and max as join ( 0 ) [0,1] = 0 ( + ) [0,1] = max ([0,1], d [0,1] ) Finite subsets with Hausdor ff distance , with union as join ( 0 ) 𝒬 fin = ∅ ( + ) 𝒬 fin = ∪ ( 𝒬 fin ( X ), ℋ ( d X )) Compact subsets with Hausdor ff distance , with union as join ( 0 ) 𝒟 = ∅ ( + ) 𝒟 = ∪ ( 𝒟 ( X ), ℋ ( d X ))

Interpolative Barycentric Algebras Are the quantitative algebras over the signature Σ ℬ = { + e : 2 ∣ e ∈ [0,1]} convex sum satisfying the following conditional quantitative equations (B1) ⊢ x + 1 y = 0 x (B2) ⊢ x + e x = 0 x (B3) ⊢ x + y = 0 y + x (SC) ⊢ x + e y = 0 y + 1 − e x (SA) 1 − ee ′ � z ) , for e , e ′ � ∈ (0,1) ⊢ ( x + e y ) + e ′ � z = 0 x + ee ′ � ( y + (1 − e ) e ′ � (IB) x = ϵ y , x ′ � = ϵ ′ � y ′ � ⊢ x + e x ′ � = δ y + e y ′ � , where δ = e ϵ + (1 − e ) ϵ ′ �

A geometric intuition (IB) x = ϵ y , x ′ � = ϵ ′ � y ′ � ⊢ x + e x ′ � = δ y + e y ′ � , where δ = e ϵ + (1 − e ) ϵ ′ � ϵ x y e δ = e ϵ + (1 − e ) ϵ ′ � x + e x ′ � y + e y ′ � 1 − e x ′ � y ′ � ϵ ′ �

Example of models Unit interval with Euclidian distance and convex combinators ( + e ) [0,1] ( a , b ) = ea + (1 − e ) b ([0,1], d [0,1] ) Finitely supported distributions with Kantorovich distance ( + e ) 𝒠 ( μ , ν ) = e μ + (1 − e ) ν ( 𝒠 ( X ), 𝒧 ( d X )) Borel probability measures with Kantorovich distance ( + ) Δ ( μ , ν ) = e μ + (1 − e ) ν ( Δ ( X ), 𝒧 ( d X ))

Completeness Mardare, Panangaden, Plotkin (LICS’16) For quantitative the equational logic we have an analogue of the usual completeness theorem Theorem ∀𝒝 ∈ 𝕃 ( Σ , 𝒱 ) . 𝒝 ⊧ ( Γ ⊢ t = ε s ) i ff Models of U 𝒱 ( Γ ⊢ t = ε s ) ∈ 𝒱

Free Quantitative Algebra Given U and a metric space M \in EMet, 𝒱 M ∈ EMet there exists a quantitative algebra* (TM, \p \STM -> TM) ( T M , ϕ M : Σ T M → T M ) and non-expansive map \n M -> TM such that η M : M → T M η M ϕ M Free-model Σ T M T M M for U 𝒱 Σ h h β ∈ 𝕃 ( Σ , 𝒱 ) Σ A A α (*)TM is the term algebra with distance dU(t,s) = inf {e | |- t =e s \in Um} d 𝒱 M ( t , s ) = inf{ ϵ ∣ ⊢ t = ϵ s ∈ 𝒱 M } T M

Free Monad on MET EMet 𝕃 ( Σ , 𝒱 ) Models of U over 𝒱 extended metric spaces ⊢ Free QA functor Forgetful functor (maps a extended metric (maps a quantitative space to the free model of ) algebra to its carrier) 𝒱 EMet T 𝒱 Monad induced by U on Met EMet 𝒱

U Models are TU-Algebras 𝒱 T 𝒱 Bacci, Mardare, Panangaden, Plotkin (LICS’18) Theorem For any basic quantitative equational theory U of type M Σ 𝒱 𝕃 ( Σ , 𝒱 ) ≅ T 𝒱 - Alg EM algebras for the monad TU T 𝒱 A quantitative equational theory U is basic if it can be axiomatised 𝒱 by a set of basic conditional quantitative equations { x i = ε i y i ∣ i ∈ I } ⊢ t = ε s basic quantitative equation

The picture of the Talk (...once again) Quantitative Equational 
 Theory U 𝒱 EMet-Lawvere EMet Monad 
 Theory 𝓜 M on EMet EMet Mod( 𝓜 , C ) = M -Algebras ≅ EMet

Examples of Monads 𝕃 ( Σ 𝒯 , 𝒯 ) T 𝒯 ≅ 𝒬 fin EMet ⊢ Quantitative semilattices Finite subsets with with bottom Hausdor ff metric 𝕃 ( Σ ℬ , ℬ ) T ℬ ≅ 𝒠 EMet ⊢ Interpolative Finitely supported probability barycentric algebras distributions with Kantorovich metric ...and many more: total variation , p-Wasserstein distance , ...

The Continuous Case (Complete Separable Metric Spaces) all Cauchy exists a countable sequences have limit dense subset

̂ Free Monads on CMet CEMet A quantitative equational theory is continuous if it can be axiomatised by a collection of continuous schemata of quantitative equations − for ε ≥ f ( ε 1 , …, ε n ) { x 1 = ε 1 y 1 , …, x n = ε n y n } ⊢ t = ε s continuous real-valued function ℂ Models of U 𝒱 𝕃 ( Σ , 𝒱 ) ℂ𝕃 ( Σ , 𝒱 ) over complete metric spaces ⊢ ⊢ ℂ CEMet EMet T 𝒱 ℂ T 𝒱