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 Γ0 be a set of inferences. The quantitative algebraic theory U generated by Γ0 is the set of inferences derivable from the rules (refl) t1 =0 t1 (sym) t1 =ǫ t2 t2 =ǫ t1 (triang) t1 =ǫ1 t2 t2 =ǫ2 t3 t1 =ǫ1+ǫ2 t3 (wk) t1 =ǫ t2 t1 =ǫ+δ t2 (δ ≥ 0) (arch) t1 =ǫ+δ t2 | δ > 0 t1 =ǫ t2 (nexp) t1 =ǫ t′

1

. . . tn =ǫ t′

n

f(t1, . . . , tn) =ǫ f(t′

1, . . . , t′ n)

(subst) Γσ tσ =ǫ sσ ((Γ ⊢ t =ǫ s) ∈ U) (assn) φ (φ ∈ Γ0)

Paul Wild Pseudometric Graded Semantics

Quantitative algebra

A quantitative algebra is a triple A = (A, ΣA, dA), where (A, ΣA) is an algebra of type Σ dA is a metric on A such that all f/n ∈ Σ are nonexpansive: if dA(a1, b1) ≤ ǫ, . . . , dA(an, bn) ≤ ǫ, then dA(f(a1, . . . , an), f(b1, . . . , bn)) ≤ ǫ.

Paul Wild Pseudometric Graded Semantics

Satisfaction

A satisfies an inference Γ ⊢ t1 =ǫ t2 if for every assignment σ: Var → A such that dA(s1σ, s2σ) ≤ δ for all (s1 =δ s2) ∈ Γ, dA(t1σ, t2σ) ≤ ǫ. Quantitative algebras form a category QAΣ, 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 TX denote the set of Σ-terms over X. Given a theory U, we can define a pseudometric dU on TX: dU(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, USet(A)) ∼ = K(Σ, U)(T[X], A). So we get a monad TU 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 ΣX be the signature Σ, extended by constants x (x ∈ X). Let UX be the theory generated by U together with axioms ∅ ⊢ x = ǫy for each x, y ∈ X with d(x, y) ≤ ǫ. Interpret T[∅] ∈ K(ΣX, UX) as Td[X] ∈ K(Σ, U) by forgetting the additional constants. Td[X] satisfies the adjunction Met((X, d), UMet(A)) ∼ = K(Td[X], A) giving rise to a monad TU 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 t1 = t2. 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 Td[X] satisfying the adjunction PMet((X, d), UPMet(A)) ∼ = K(Td[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 ∈ TX. 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 (SC) x +p y = y +1−p x (SA) 0 < p, q < 1, r = q−pq

1−pq

x +p (y +q z) = (x +pq y) +r z (IB) x1 =ǫ1 y1, x2 =ǫ2 y2 x1 +p x2 =ǫ1+pǫ2 y1 +p y2

Paul Wild Pseudometric Graded Semantics

Convex algebras (2)

In K(Σ, U), the term algebra Td[X] is isomorphic to the space DfinX of finitely supported probability measures on (X, d), equipped with the Kantorovich metric Kd(µ, ν) = sup{|

• fdµ −
• fdν|}

where f ranges over nonexpansive functions (X, d) → ([0, 1], de).

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