Behavioral Extensions of Institutions

1. Behavioral Extensions of Institutions Andrei Popescu GrigoreRoşu University of Illinois at Urbana-Champaign

2. Motivation Many algebraic formalisms have been enriched with behavioral or observational equivalence • Hidden algebra logics (Goguen et al.) • Observational logic (Bidoit, Hennicker et al.) • Swinging types (Padawits) These beh. logics build upon powerful formalisms Challenges • Can we capture abstractly the essence of behavioral equivalence and behavioral satisfaction of a property? • Provide logic-independent framework for these concepts Formal recipe to extend behaviorally existing formalisms

3. Our results Given institution I, build institutionIbeh • Capture visible signatures and sentences • Define (behavioral) satisfaction in Ibehas satisfaction in Iin appropriate quotient models • Deduction in Isound in Ibeh • Ibehexhibits many known relevant properties of particular behavioral logics • Satisfaction in Ibeh reduces to satisfaction in Iin the same model, via (abstraction of) experiments • Novel properties unexpectedly discovered

4. Overview • Basic notions • Institutions, behavioral equivalence • Behavioral extension of an institution • Logic-independent behavioral concepts and properties • Related work and conclusions

5. Institutions Set Sen Sign ╨ Catop Mod |= Mod(S) Sen(S)  φ Mod(φ) Sen(φ) |=’ ’ Mod(S’) Sen(S’)

6. Behavioral / hidden logicsHidden Signature • Standard algebraic signature in which sorts are split into visible and hidden Hidden signature • Tuple  := (V, H, ) • Sorts S = V  H • V = visible sorts (stay for data: integers, reals) • H = hidden sorts (stay for states, objects, etc.) • = S-sorted algebraic signature

7. Behavioral / hidden logicsHidden Algebra • Loose-data approach • Unconstrained models and morphisms • Fixed-data approach • Fix the “visible” signature ↾V, say Ѱ • Fix some Ѱ-algebra D (data algebra) • Hidden algebra.-algebra A with A↾Ѱ = D • Hidden morphism. h : A → B with h↾Ѱ = 1D Coalgerbraic nature of hidden algebra Under restrictions on  (one hidden argument), categ. of -algebras is a categ. of coalgebras

8. Behavioral / hidden logicsContexts and experiments Context = a term with a hidden “slot” Experiment = a context ofvisible result Visible sort if context is an experiment Operations in  z : h

9. Behavioral / hidden logicsBehavioral equivalence Behavioral equivalence on A • a ≡ a’iff Ac(a) = Ac(a’) for any experiment c Hidden congruenceon A: • congruence relation, identity on visible carriers a a’ Coinduction: ≡ is the largest hidden congruence However, final models may not exist!

10. Behavioral / hidden logics Behavioral satisfaction Abehaviorally satisfies(X) t = t’, written A |≡ (X) t = t’ iff θ(t) ≡ θ(t’) for any map θ : X → A • Other properties of behavioral logics will be recalled as they are “institutionalized” Equivalent definition: A |≡ e iff A↾≡ |= e

11. Behavioral Extension of an InstitutionFramework Framework • InstitutionI = (Sign, Sen, Mod, |=) • Fixed data: ѰSign, DMod(Ѱ) • Loose data under investigation; overall simpler • Quotient systems on model categories • Dual to inclusion systems; unique quotients • Directed colimits of models, and these colimits are preserved by model reducts

12. Signatures: morphisms φ : Ѱ Σ One can constrain these to inclusions, but not needed Sentences: precisely the -sentences of I Models: the fiber categoryMod(φ)-1(D) Behavioral Extension of an Institution Construction of Ibeh Ѱ D φ A↾φ= D A Modbeh(φ) 

13. Behavioral Extension of an Institution (Behavioral) Satisfaction in Ibeh Data-consistent quotient (φ : Ѱ Σ,D Mod(Ѱ)) A,B Mod(Σ),e : A  B quotient, e↾φ = 1D Intuitively, Ajgives the behavioral equivalence on A Proposition. The category of data-consistent quotients of A has a unique final objectA  Aj Definition. Call Aj the φ-quotient of A Satisfaction inIbeh : A |≡ ρ iff Aj |= ρin I

14. Behavioral Extension of an Institution Subtlety: Signature morphisms Definition of signature morphisms in Ibehis subtle Digression: Signature morphisms in hidden logics ξ : (V  H, Σ)  (V H’, Σ’) • ξ identity on V • ξ(H)  H’ • ’ ∊ ξ(Σ) for each ’ ∊ Σ’ with an argument in ξ(H) Faithful to encapsulation and yields institution Can we capture this intricate definition institutionally?

15. Behavioral Extension of an InstitutionSignature morphisms in Ibeh ξpreserves all the j’-quotients Answer: Yes, yet quite elegantly! ξ Σ’ Σ j’ j Ѱ One can show that in concrete situations this definition captures precisely the three conditions above

16. Important Result Theorem • Ibehis an institution • There is a natural morphismIbeh I • Takes φ : Ѱ Σin SignbehtoΣ in Sign • Takes A in Modbeh(φ)to AjinMod(Σ) • Keeps sentences unchanged

17. Logic-independent behavioral concepts and properties Deduction in I is sound in Ibeh E |= ρ implies E |≡ρ Strict and behavioral satisfaction coincide for sentences over visible signature: ( φ : Ѱ Σ, D Mod(Ѱ) ) if ρ∊Sen(Ѱ) then A |≡ φ(ρ) iff D |= ρ

18. Logic-independent behavioral concepts and properties (ii) Visible φ-sentences: strict and behavioral satisfaction coincide, i.e., A |= ρ iff A|≡ρ • Equivalently, preserved and reflected by data-consistent quotients Quasi-visible φ-sentences: behavioral satisfaction implies strict satisfaction • Equivalently, reflected by data-consistent quotients Definitions ( φ : Ѱ Σ, ρ ∊Sen(Ѱ) )

19. Stronger properties for restricted types of sentences • One cannot expect all properties of behavioral equational logics to hold in arbitrary institutions • E.g., if FOL is the starting logic (e.g., Bidoit & Henicker), then the following are not true: • behavioral satisfaction expressible as strict satisfaction of an (infinite) set of sentences • any sentence reflected by model-morphisms (just use negations to obtain simple counterexamples) • Fortunately, one can distinguish certain types of sentences abstractly, in institutions.

20. Institution-independent sentence constructs • Basic sentences (Diaconescu 2003) A |= ρ iff there exists Tρ A • In concrete situations, Tρ is a quotient of initial algebra • In FOL and EQL, ground and existential ground atoms are basic • φ-quantification (Tarlecki 1986): ( φ : Σ’  Σ, ρ∊Sen(Σ), A’∊Mod(Σ’) ) A’ |= (φ) ρ iff A |= ρ for all φ-expansions A of A’ (Similarly for the existental quantifier) • Logical connectives (, , ) defined in the obvious way • Positive sentences: obtained from basics by • connectives ,  • universal and existential φ-quantifications

21. Stronger properties for restricted types of sentences (ii) Proposition. Visible and quasi-visible sentences • preserved by signature morphisms • closed under positive connectives and under quantification (visible closed under negation too) • coincide if positive Proposition. Under Birkhoff-styleconditions (closure under subobjects and homomorphic images), sentences are behaviorally reflectedby model-morphisms: A  B and B |≡ρ imply A |≡ρ

22. Stronger properties for restricted types of sentences (iii) ( φ: Ѱ Σ, D Mod(Ѱ), AModbeh(φ) , ρ∊Sen(Σ) ) Proposition. Satisfaction of basic sentences equivalent to data-consistent factorizing: A |≡ ρ iff (A/ρ)↾φ = D ( A/ρ is “A factored by ρ”, formally A ∐Tρ )

23. Digression: behavioral versus strict satisfaction in behavioral logics • Behavioral satisfaction reducible to strict satisfaction without changing the model A |≡ (X) t = t’ iff A |= (X var(c)) c[t] = c[t’] for all experiments c

24. Stronger properties for restricted types of sentences (iv) Proposition. IfI has model-theoretic diagrams (Tarlecki 1986, Diaconescu 2004) and ρ is a universally quantified basic sentence, then there exists a set of sentences Eρ such that for any A A |≡ ρ iff A |= Eρ Specifically, Eρ={() |  quasi-visible, ρ |= ()} All sentences in Eρ are quasi-visible

25. Very related work Burstall & Diaconescu 1994 • institution-independent • morphism between (their) Ibeh and I Burstall & Diaconescu 1994 has several limitations • Does not cover the cases of hidden constants (e.g. formal automata) or non-monadic hidden operations • Assumes data from “outside” the original institution to guide the construction • Does not define signature morphisms; instead, they just assume just assume them • Does not prove any property of Ibeh

26. Related work • Sannella & Tarlecki 1987: Observational equivalence, sketch of an institutional approach • Bidoit & Tarlecki 1996: Quasi-abstract treatment of behavioral satisfaction (concrete model categories) • Hofmann & Sannella 1996: Behavioral satisfaction in higher-order logic • Bidoit & Henicker 2002: The institution of first-order observational logic

27. What we’ve done A construction I  Ibeh • Provided logic-independent concepts • behavioral equivalence • behavioral satisfaction • hidden signature morphism • visible sentence • Proved logic-independent results • soundness of strict deduction for behavioral logic • relation between strict and behavioral satisfaction • closure properties for visible sentences • relation between behavioral equivalence and data-consistent factoring • Captured several existing behavioral logics (including those with hidden constants and non-monadic ops)

28. Future plans • Cover the loose-data case too, possibly using Grothendieck constructions • Explore more deeply the consequences of our general results in concrete cases • our universally quantified basic sentences include second-order  - sentences • our assumptions about the institution accommodate infinitary logics too, etc. • Logic-independent relationship between behavioral abstraction and information hiding

29. Thank you This is joint work with Andrei Popescu