Towards Pseudometric Graded Semantics Paul Wild - PowerPoint PPT Presentation

Towards Pseudometric Graded Semantics Paul Wild Friedrich-Alexander-Universität Erlangen-Nürnberg Paul Wild Pseudometric Graded Semantics

Quantitative algebraic reasoning Introduced by Mardare, Panangaden, Plotkin [2]. Basic idea: build an algebraic theory consisting of equations of the form x = ǫ y with the intended meaning that x and y differ by at most ǫ . Paul Wild Pseudometric Graded Semantics

Quantitative equational theory Let Σ be an algebraic signature and let Γ 0 be a set of inferences. The quantitative algebraic theory U generated by Γ 0 is the set of inferences derivable from the rules ( sym ) t 1 = ǫ t 2 ( triang ) t 1 = ǫ 1 t 2 t 2 = ǫ 2 t 3 ( refl ) t 1 = 0 t 1 t 2 = ǫ t 1 t 1 = ǫ 1 + ǫ 2 t 3 t 1 = ǫ t 2 ( arch ) t 1 = ǫ + δ t 2 | δ > 0 ( δ ≥ 0) ( wk ) t 1 = ǫ + δ t 2 t 1 = ǫ t 2 t 1 = ǫ t ′ t n = ǫ t ′ . . . n 1 ( nexp ) f ( t 1 , . . . , t n ) = ǫ f ( t ′ 1 , . . . , t ′ n ) Γ σ tσ = ǫ sσ ((Γ ⊢ t = ǫ s ) ∈ U ) ( assn ) φ ( φ ∈ Γ 0 ) ( subst ) Paul Wild Pseudometric Graded Semantics

Quantitative algebra A quantitative algebra is a triple A = ( A, Σ A , d A ) , where ( A, Σ A ) is an algebra of type Σ d A is a metric on A such that all f/n ∈ Σ are nonexpansive: if d A ( a 1 , b 1 ) ≤ ǫ, . . . , d A ( a n , b n ) ≤ ǫ , then d A ( f ( a 1 , . . . , a n ) , f ( b 1 , . . . , b n )) ≤ ǫ . Paul Wild Pseudometric Graded Semantics

Satisfaction A satisfies an inference Γ ⊢ t 1 = ǫ t 2 if for every assignment σ : Var → A such that d A ( s 1 σ, s 2 σ ) ≤ δ for all ( s 1 = δ s 2 ) ∈ Γ , d A ( t 1 σ, t 2 σ ) ≤ ǫ . Quantitative algebras form a category QA Σ , where the morphisms are nonexpansive homomorphisms of Σ -algebras. Given a quantitative equational theory U , write K (Σ , U ) for the category of quantitative algebras satisfiying all of U . Paul Wild Pseudometric Graded Semantics

Term algebra Let T X denote the set of Σ -terms over X . Given a theory U , we can define a pseudometric d U on T X : d U ( s, t ) = inf { ǫ | U ⇒ ∅ ⊢ s = ǫ t } . Quotient out terms of distance 0 to get a metric space ( T [ X ] , d ∼ = ) . Due to nonexpansivity, T [ X ] is also a quantitative algebra. Paul Wild Pseudometric Graded Semantics

Term monad The term algebra ( T [ X ] , Σ , d ∼ = ) is a functorial construction and gives rise to the following adjunction: Set ( X, U Set ( A )) ∼ = K (Σ , U )( T [ X ] , A ) . So we get a monad T U on Set, mapping elements of X to T [ X ] . Paul Wild Pseudometric Graded Semantics

Metric term monad Let ( X, d ) be a metric space. Let Σ X be the signature Σ , extended by constants x ( x ∈ X ). Let U X be the theory generated by U together with axioms ∅ ⊢ x = ǫy for each x, y ∈ X with d ( x, y ) ≤ ǫ . Interpret T [ ∅ ] ∈ K (Σ X , U X ) as T d [ X ] ∈ K (Σ , U ) by forgetting the additional constants. T d [ X ] satisfies the adjunction Met (( X, d ) , U Met ( A )) ∼ = K ( T d [ X ] , A ) giving rise to a monad T U on Met. Paul Wild Pseudometric Graded Semantics

Pseudometric term monad As we are interested in behavioural distances and these are usually pseudometrics, we extend the framework by explicit equalities t 1 = t 2 . When constructing the term monad T [ X ] , we instead quotient by provable equality: s ∼ = t : ⇔ U ⇒ ∅ ⊢ s = t. We then get a pseudometric term monad T d [ X ] satisfying the adjunction PMet (( X, d ) , U PMet ( A )) ∼ = K ( T d [ X ] , A ) . Paul Wild Pseudometric Graded Semantics

Graded term monad We can also extend the framework to graded theories: To each operator f ∈ Σ , assign a depth d ( f ) ∈ N . Extend depth to terms t ∈ T X . Restrict to equations of uniform depth . The depth n fragments of the term algebra form the steps of a graded monad. Paul Wild Pseudometric Graded Semantics

Convex algebras (1) Let Σ = { + p / 2 | p ∈ [0 , 1] } and U generated by the axioms ( B1 ) x + 0 y = y ( B2 ) x + p x = x 0 < p, q < 1 , r = q − pq 1 − pq ( SC ) x + p y = y + 1 − p x ( SA ) x + p ( y + q z ) = ( x + pq y ) + r z x 1 = ǫ 1 y 1 , x 2 = ǫ 2 y 2 ( IB ) x 1 + p x 2 = ǫ 1 + p ǫ 2 y 1 + p y 2 Paul Wild Pseudometric Graded Semantics

Convex algebras (2) In K (Σ , U ) , the term algebra T d [ X ] is isomorphic to the space D fin X of finitely supported probability measures on ( X, d ) , equipped with the Kantorovich metric � � K d ( µ, ν ) = sup {| f d µ − f d ν |} where f ranges over nonexpansive functions ( X, d ) → ([0 , 1] , d e ) . Paul Wild Pseudometric Graded Semantics

Future Work Further development of graded semantics (cf. Dorsch, Milius, Schröder [1]) in a quantitative setting: What is the exact relationship between pseudometric graded monads and pseudometric graded theories? Depth-1 quantitative graded monads Find more examples and establish a quantitative version of the linear-time/branching-time spectrum. Paul Wild Pseudometric Graded Semantics

References U. Dorsch, S. Milius, and L. Schröder. Graded monads and graded logics for the linear time – branching time spectrum. CoRR , abs/1812.01317, 2018. R. Mardare, P. Panangaden, and G. Plotkin. Quantitative algebraic reasoning. In M. Grohe, E. Koskinen, and N. Shankar, eds., Logic in Computer Science, LICS 2016 , pp. 700–709. ACM, 2016. Paul Wild Pseudometric Graded Semantics