processos e concorr^encia 2015/16 bloco de acetatos 7alfa.di.uminho.pt/~madeira/uc/bloco7.pdf ·...

$: Processos e Concorr^encia 2015/16 Bloco de acetatos 7alfa.di.uminho.pt/~madeira/UC/Bloco7.pdf · the mCRL2 axiomatics can be understood as an algebraic speci cation for the \generic$
Processos e Concorrencia 2015/16Bloco de acetatos 7

Alexandre MadeiraHASLab INESC TEC, DI UMINHO

April 6, 2017

Algebraic specification is everywhere

Definition A (Monoid)

A monoid is an algebraic structure with a single associative binaryoperation and an identity element.

Definition B (Monoid)

A monoid is a model of the following specification:

Sorts s;

Op · : s × s → s

e :→ s

Ax (a · b) · c = a · (b · c)

a · e = a

e · a = a

Algebraic specification is everywhere

Specification of a computer memory

Sorts state;

id;

elt;

Op update: id × elt ×state → state;

lookup: id × state → elt;

Ax lookup(i , update(i , e, s)) = e

i 6= j → lookup(i , update(j , e, s)) = lookup(i , s)

Algebraic specification of data andprocesses

Algebraic specification

• rooted in universal algebra

• Original aims: modelling software systems as algebras

• specialised along the times: definition of ADT (abstractdata types)

Process algebra

• rooted in automata and languages theory

• aims to formal modelling and analysis of concurrent systems

Algebraic specification in this course

AS and PA are highly related:

Micro perspective

• actions can be parametrized by data – we need a rigorous wayto define new suitable ADT

Macro perspective

• a process algebra can be seen, itself, as an algebraicspecification

Revisiting mCRL2

e.g., the fragment

A1 x + y = y + x

A2 (x + y) + z = x + (y + z)

A3 x + x = x

A4 (x + y) · z = x · z + y · z

A5 (x · y) · z = x · (y · z)

A6 x + δ = x

A7 δ · x = δ

• actually mCRL2 can be seen as an algebraic specification• the mCRL2 axiomatics can be understood as an algebraic

specification for the “generic shape of processes”

Revisiting CCS

e.g., the fragment

(act)a.p

a−→ p

pa−→ p′

(sum − l)p + q

a−→ p′

qa−→ q′

(sum − r)p + q

a−→ q′

• the axiomatics of CCS is now given by a set of inference rules(against the equational axiomatization of mCRL2)

• the previous analogy remains valid

Universal Algebra

Many-sorted sets

Definition (Many-sorted set)

An S-sorted set is an S-family of sets X = (Xs)s∈S .

Given two S-sorted sets X = (Xs)s∈S and Y = (Ys)s∈S :

X ∪ Y = (Xs ∪ Ys)s∈S

X × Y = (Xs × Ys)s∈S

X ⊆ Y iff for each s ∈ S , Xs ⊆ Ys

. . .

Many-sorted sets

Definition (Many-sorted set function)

Let X = (Xs)s∈S and Y = (Ys)s∈S two S-sorted sets. An S-sortedfunction f : X → Y is an S-family of functions fs : Xs → Ys .

Many-sorted sets

Many-sorted relation

Let X = (Xs)s∈S and Y = (Ys)s∈S two S-sorted sets. An S-sortedrelation R ⊆ X × Y is an S-family of relations Rs ⊆ Xs × Ys .

Many-sorted equivalence relation

An S-sorted relation X ⊆ R ×R is an equivalence on X if, for eachs ∈ S :

• Rs is reflexive, i.e., for any x ∈ Xs , (x , x) ∈ Rs ;• Rs is simetric, i.e., for any x , y ∈ Xs , (x , y) ∈ Rs implies

(y , x) ∈ Rs ;• Rs is transitive, i.e., for any x , y , z ∈ Xs , (x , y) ∈ Rs and

(y , z) ∈ Rs implies (x , z) ∈ Rs ;.

Many-sorted sets

Quotient setGiven an S-sorted equivalence relation R ⊆ X × X :

• the equivalence class of x ∈ Xs modulo R:[x ]Rs = y ∈ Xs | (x , y) ∈ Rs,

• the quotient of X modulo R: X/R = (Xs/Rs)s∈S , whereXs/Rs = [x ]Rs |x ∈ Xs.

Many-sorted sets

Exercise E1

1 Show that, in any S-sorted equivalence relation, [x ]≡s = [y ]≡s iffx ≡s y .

2 Let consider the sorted set X = (Xs)s∈s1,s2, with Xs1 = 1, 2 andXs2 = a, b, c. Identify, in the following relations, the equivalencerelations. Define, for these cases, the respective quotient set.

a) R1 = (1, 1), (2, 2) and R2 = (a, a), (b, b), (c , c)b) S1 = (1, 1), (2, 2), (1, 2), (2, 1) and

S2 = (a, a), (b, b), (c , c)c) T1 = (1, 1), (2, 2) and

T2 = (a, a), (b, b), (b, c), (c , b), (c , c)d) Q1 = (1, 2), (2, 1) and

Q2 = (a, a), (b, b), (b, c), (c , b), (c , c)

Signatures and Algebras

Definition (Many-sorted signature)

A signature is a pair Σ = (S ,Ω), where:

• S is a set of sort names

• Ω is a (S∗ × S)-sorted set of operation names

Notation - as usual we may denote

• operation names f ∈ Ωs1...sn,s by f : s1 × · · · × sn → s ∈ Σ

• (in particular) constants names c ∈ Ωε,s by c : s ∈ Σ


Signature example: Σ = (S ,Ω), where

• S = s1, s2• Ωε,s1 = c1, Ωε,s2 = c2• Ωs1,s1 = f , Ωs2s1,s1 = g• Ωω,s = ∅ for other ω ∈ S∗, s ∈ S

A more standard presentation

sorts s1

s2

op c1 :→ s1

c2 :→ s2

f : s1 → s1

g : s2 × s1 → s1


Exercise E2

1 define a signature for groups

2 define a signature for natural numbers

3 define a signature for stacks of elements of a set A

4 define a signature for boolean algebra

5 a signature for a simple process algebra?• e.g., for the fragment of CCS

P := 0 | a.P | P + P | P|P

for a fixed set of actions A


Exercise E2





5 a signature for a simple process algebra?

• e.g., for the fragment of CCS

P := 0 | a.P | P + P | P|P



Exercise E2





5 a signature for a simple process algebra?• e.g., for the fragment of CCS

P := 0 | a.P | P + P | P|P



Definition (Many-sorted algebra)

Let Σ = (S ,Ω) be a many-sorted signature. A Σ-algebra Aconsists of:

• an S-sorted set |A|, i.e., for each s ∈ S , |A|s is a set; and

• for each f : s1 × · · · × sn → s ∈ Σ, a functionf A : |A|s1 × · · · × |A|s1 → |A|s .


Let consider the signature Σ:

sorts s1, s2

op c1 :→ s1, c2 :→ s2

f : s1 → s1

g : s2 × s1 → s1

Two example of Σ-algebras:

|A|s1 = a, b, |A|s2 = 1, 2, 3cA

1 = a cA2 = 3

f A(a) = a, f A(b) = a

gA = (1, a) 7→ 1, (1, b) 7→1, (2, a) 7→ 2, (2, b) 7→ 2, (3, a) 7→3, (3, b) 7→ 3

|B|s1 = •, |B|s2 = ♥,♠cB

1 = •, cB2 = ♠

f B(•) = •gB = (♥, •) 7→ ♥, (♠, •) 7→ ♠


Exercise E3Define two distinct algebras for each signature introduced inExercise E2.

Morphisms and Congruences

Morphism

Let A and B two Σ-algebras. A Σ-morphism h : A→ B is anS-sorted function h : |A| → |B| such that, for eachf : s1, . . . , sn → s ∈ Σ, and for any a1 ∈ |A|s1 , . . . , an ∈ |A|sn ,

hs(f A(a1, . . . , an)) = f B(hs1(a1), . . . , hsn(an))

|A|s1 × · · · × |A|snhs1×···×hsn //

f A

|B|s1 × · · · × |B|snf B

|A|s

hs// |B|s


Exercise E4Let consider the algebras A and B of previous example. Define, ifpossible, two morphisms h : A→ B and h′ : B → A.


Definition (Congruence)

Let A be a Σ-algebra and ≡⊆ |A| × |A| an equivalence relation onA. The relation ≡ is a congruence if

• for all a1, a′1 ∈ |A|s1 , . . . , an, a

′n ∈ |A|s1 , if as1 ≡ a′s1

and · · ·and asn ≡ a′sn , then,

f A(a1, . . . , an) ≡ f A(a′1, . . . , a′n)


Definition (Quotient Algebra)

Let A be a Σ-algebra and ≡ ⊆ |A| × |A| a congruence on A. Thequotient algebra A modulo ≡ is the Σ-algebra A/ ≡ defined by:

• |A/ ≡ | = |A|/ ≡, and

• for each f : s1 × · · · × s2 → s ∈ Σ,f A/≡([a1]≡s1

, . . . , [an]≡sn) = [f A(a1, . . . , an)]≡s for all

a1 ∈ |A|s1 , . . . , an ∈ |A|sn .


Exercise E5Given a S-function f : X → Y , let us consider the S-relationKer(f ), defined for each s ∈ S as

Ker(fs) = (x , y)|x , y ∈ Xs and fs(x) = fs(y)

1 show that Ker(f ) is an equivalence relation

2 let us assume an homomorphism h : A→ B. Show thatKer(h) is a congruence.

3 Using the morphism h : A→ B of exercise E4, define thealgebra A/Ker(h).

Term Algebras

Definition (Σ-terms)

Let Σ be a signature and X = (Xs)s∈S a S-sorted set of variablesfor Σ. The set Σ-terms over X is the smallest S-set T(Σ,X )such that:

• Xs ⊆ T(Σ,X )s ;

• Ωε,s ⊆ T(Σ,X )s ;

• For any f : s1 × · · · × sn → s ∈ Σ andt1 ∈ T(Σ,X )s1 , . . . , tn ∈ T(Σ,X )sn , f (t1, . . . , tn) ∈ T(Σ,X )s ;

ExerciseEnumerate the terms of each signature of Exercise E2.

Term Algebras

Ground terms

• The set of terms T(Σ, ∅) is called the set of ground terms.

Term Algebras

Definition (Term Algebra)

If T(Σ,X ) is non empty, the term algebra over X is the Σ-algebraT (Σ,X ) such that

• |T (Σ,X )| = T(Σ,X )

• for any f : s1 × sn → s ∈ Σ and everyt1 ∈ T(Σ,X )s1 , . . . , tn ∈ T(Σ,X )sn ,

f T (Σ,X )(t1, . . . , tn) := f (t1, . . . , tn)

Term algebra

Fact.For any Σ-algebra A and for any S-function v : X → |A|, there isexactly one Σ-morphism v # : T(Σ,X )→ A that extends v , i.e.,such that v #(ιX (x)) = v(x), where ιX : X → T(Σ,X ) maps eachvariable to its corresponding term.

XιX //

v##

T(Σ,X )

v#

|A|

Definition (Term evaluation)

The interpretation of a term t ∈ T(Σ,X )s in a Σ-algebra A underthe valuation v : X → |A|, denoted by tA(v), is v #(t).

Exercise

Exercise E6 - Consider the following signature:

sorts nat

op 0 : nat

suc : nat → nat

+ : nat × nat → nat

1 Consider its set of terms;

2 Consider its set of ground terms;

3 Consider 2 models for this signature;

Equational Specification

Equations and satisfaction

Equations

A Σ-equation (of sort s) is an expression t1 = t2, fort1, t2 ∈ T(Σ,X )s .

SatisfactionLet A be a Σ-algebra and v : X → |A| be a valuation. Then

A, v |= t1 = t2 iff tA1 (v) = tA2 (v)

Notation

• A |= t1 = t2 iff for any v : X → |A|, A |= t1 = t2.

• A |= Φ iff for any t1 = t2 ∈ Φ, A |= t1 = t2.

• K |= t1 = t2 iff for any A ∈ K , A |= t1 = t2.

• Φ |= t1 = t2 iff, for any Σ-algebra A, if A |= Φ, thenA |= t1 = t2

Specifications

(Flat) Specification

A flat specification consists of a pair SP = (Σ,Φ), where

• Σ is a signature

• Φ is a set of Σ-equations

Specification models

For SP = (Σ,Φ),

Mod[SP] = A|A is Σ-algebra and A |= Φ

Exercise

Exercise E7Consider the signature of exercise E6.

• Suggest an axiomatisation for natural numbers (at least 5equations)

• Identify 3 models for your specification and a counter-example(in the same signature)

Some Examples of thebase data types of mCRL2

printed from the tool’s reference:Jan Friso Groote, Mohammad Reza Mousavi. Modeling andAnalysis of Communicating Systems. MIT Press, 2008.

Example: generic useful auxiliary stuff

Example: Booleans

Example: positive naturals

Example: lists

Example: sets

Theories

Definition(Theories)

Let Φ be a set of Σ-equations and K be a class of Σ-algebras.Then, we define:

• Mod(Φ) := A|A is Σ-algebra and A |= Φ• ThΣ(K ) = t1 = t2|for any A ∈ K ,A |= t1 = t2

Galois connections

1 Φ ⊆ Ψ implies Mod(Φ) ⊇ Mod(Ψ);

2 K ⊆ K ′ implies ThΣ(K ) ⊇ ThΣ(K ′);

3 Φ ⊆ ThΣ(Mod(Φ)) and K ⊆ Mod(ThΣ(K )).

Equational Calculus

(reflexivity)Φ `Σ t = t

, t1 = t2 ∈ Φ (axioms)Φ `Σ t1 = t2

Φ `Σ t1 = t2(symmetry)

Φ `Σ t2 = t1

Φ `Σ t1 = t2 Φ `Σ t2 = t3(transitivity)

Φ `Σ t1 = t3

Φ `Σ t1 = t ′1 · · · Φ `Σ tn = t ′n, f : s1 × · · · × sn → s ∈ Σ (congruence)

Φ `Σ f (t1, · · · , tn) = f (t ′1, · · · , t ′n)

Φ `Σ t1 = t2, σ : X → T(Σ,X )(replacement)

Φ `Σ σ(t1) = σ(t2)

Exercise

Sort bool

Op true : bool

false : bool

¬ : bool → bool

∧ : bool × bool → bool

AxBool • ¬true = false• ¬false = true• p ∧ true = p• p ∧ false = false• p ∧ ¬p = false

Using the equational calculus prove that:

1 AxBool ` ¬¬true = true

2 AxBool ` ¬true ∧ ¬false = false

3 AxBool ` (p ∧ ¬true) ∧ false = ¬true

Equational Calculus

DefinitionLet ≡Φ⊆ T(Σ,X )× T(Σ,X ) the relation defined by

≡Φ= (t1, t2)|Φ ` t1 = t2

i.e.t1 ≡Φ t2 iff Φ ` t1 = t2

Equational Calculus

Lemma≡Φ is a congruence in T (Σ,X )

Equational Calculus

LemmaΦ ` t1 = t2 iff T (Σ,X )/ ≡Φ|= t1 = t2

Equational Calculus

Theorem (Soundness and completeness of equational calculus)

For any set of Σ-equations Φ and for any equation t1 = t2,

Φ ` t1 = t2 iff Φ |= t1 = t2

Hints for the proof:

soundness, i.e., for implication ⇒, use induction over theequational calculus rules

completness, i.e., for implication ⇐, use the previous Lemma

Initial Models

• The class of Σ-algebras given by loose semantics of a Σ -specification contains too many algebras to be useful inpractices. E.g.,

• if Σ has no constants, the empty algebra is a model of anyΣ-specification

• if Σ has constants, an algebra carried by an S-family ofsingletons is a model of any Σ-specification

Initial Models

Let A ∈ Mod(Σ,Φ). A contains

• Junk, if A is not reachable, i.e., there is an a ∈ |A|s suchthat, there is no a ground term t ∈ T(Σ,X ) such that tA = a

• Confusion, if A satisfies some ground equation t1 = t2 suchthat Φ 6` t1 = t2

What should be a good model for a specification?

Exercise

Sort bool

Op true : bool

false : bool

¬ : bool → bool

∧ : bool × bool → bool

Ax • ¬true = false• ¬false = true• p ∧ true = p• p ∧ q = q ∧ p• p ∧ false = false• p ∧ ¬p = false

1 Present 3 finite models with 1, 2 and 3 elements.

2 Classify the models with respect to “junk” and “confusion”.

3 Build the algebra T (ΣBool)/ ≡Φ, where Φ is the set ofequations of the specification.

Initial Models

LemmaThe model T (Σ)/ ≡Φ has no junk neither confusion

Initial Model

Initial modelAn algebra A is initial in a class K if for any algebra B ∈ K , thereis an unique morphism h : A→ B.

LemmaInitial models are unique up to isomorphism

Initial Model

TheoremT (Σ)/ ≡Φ is an initial model of Mod(〈Σ,Φ〉)

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