Behavioral Extensions of Institutions

Andrei PopescuGrigore Roşu

University of Illinois at Urbana-Champaign

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MotivationMany 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

1. Can we capture abstractly the essence of behavioral equivalence and behavioral satisfaction of a property?

2. Provide logic-independent framework for these concepts Formal recipe to extend behaviorally existing formalisms

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Our resultsGiven institution I, build institution Ibeh

– Capture visible signatures and sentences– Define (behavioral) satisfaction in Ibeh as

satisfaction in I in appropriate quotient models – Deduction in I sound in Ibeh

– Ibeh exhibits many known relevant properties of particular behavioral logics• Satisfaction in Ibeh reduces to satisfaction in I in the

same model, via (abstraction of) experiments– Novel properties unexpectedly discovered

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Overview

• Basic notions– Institutions, behavioral equivalence

• Behavioral extension of an institution• Logic-independent behavioral concepts

and properties• Related work and conclusions

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Institutions

SetSign

Catop

Sen

Mod╨

’

φ

Mod()

Mod(’)

Sen()

Sen(’)

Mod(φ) Sen(φ)

|=

|=’

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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

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• 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

Behavioral / hidden logicsHidden Algebra

Coalgerbraic nature of hidden algebraUnder restrictions on (one hidden argument),categ. of -algebras is a categ. of coalgebras

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Behavioral / hidden logicsContexts and experiments

Context = a term with a hidden “slot”Experiment = a context of visible result

z : h

Operations in

Visible sort if contextis an experiment

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Behavioral / hidden logicsBehavioral equivalence

Behavioral equivalence on A– a ≡ a’ iff Ac(a) = Ac(a’) for any experiment c

Hidden congruence on A: – congruence relation, identity on visible carriers

a a’

Coinduction: ≡ is the largest hidden congruenceHowever, final models may not exist!

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Behavioral / hidden logics Behavioral satisfaction

A behaviorally 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

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Behavioral Extension of an InstitutionFramework

Framework– Institution I = (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

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Behavioral Extension of an Institution Construction of Ibeh

Signatures: morphisms φ : Ѱ Σ – One can constrain these to inclusions, but not needed

Sentences: precisely the -sentences of IModels: the fiber category Mod(φ)-1(D)

Ѱ

φ

Modbeh(φ)

D

A A↾φ = D

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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, A gives the behavioral equivalence on A

Proposition. The category of data-consistentquotients of A has a unique final object A A

Definition. Call A the φ-quotient of A

Satisfaction in Ibeh : A |≡ ρ iff A |= ρ in I

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Behavioral Extension of an Institution Subtlety: Signature morphisms

Definition of signature morphisms in Ibeh is 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?

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Behavioral Extension of an Institution Signature morphisms in Ibeh

ξ preserves all the ’-quotients

Σ

Ѱ

ξ

’

Answer: Yes, yet quite elegantly!

Σ’

One can show that in concrete situations this definitioncaptures precisely the three conditions above

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Important Result

Theorem1. Ibehis an institution

2. There is a natural morphism Ibeh I– Takes φ : Ѱ Σ in Signbeh to Σ in Sign– Takes A in Modbeh(φ) to A in Mod(Σ) – Keeps sentences unchanged

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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

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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) )

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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.

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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

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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-style conditions (closure under subobjects and homomorphic images), sentences are behaviorally reflected by model-morphisms:

A B and B |≡ ρ imply A |≡ ρ

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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 )

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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

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Stronger properties for restricted types of sentences (iv)

Proposition. If I 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

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Very related workBurstall & 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

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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

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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)

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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

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Thank you

This is joint work with

Andrei Popescu