Term rewriting in partial algebras Norbert Dojer 20.06.2014 Term - - PowerPoint PPT Presentation

Term rewriting in partial algebras

Norbert Dojer 20.06.2014

Term rewriting – motivating example

How to check whether two polynomial expressions are equal? (x + 2)(x − 1) + 3

?

= = (x + 1)2 − x  

• 



• x(x − 1) + 2(x − 1) + 3

x2 + 2x + 1 − x  

• 



• x2 − x + 2x − 2 + 3

− − − − → x2 + x − 1 Generally:

◮ each class of equivalent expressions contains a unique normal

form

◮ normal form is calculated by reducing expressions (rewriting) ◮ rewriting rules are derived from equational axioms

Abstract reduction system (ARS)

ARS is a pair (X, − →), where − →⊆ X 2

◮ ∗

← → is an equivalence relation defined by − →, i.e. symmetric-reflexive-transitive closure of − →

◮ ∗

− → is a reflexive-transitive closure of − →

◮ x is reducible iff ∃yx−

→y

◮ y is a normal form of x (y = x↓) iff x ∗

− → y and y is irreducible

◮ x and y are joinable (x ↓ y) iff ∃zx ∗

− → z

∗

← − y

ARS – properties

ARS is:

◮ terminating iff there is no infinite reduction sequence

x1− →x2− →x3− → . . .

◮ confluent iff ∀x,y,zx ∗

← − z

∗

− → y = ⇒ x ↓ y

◮ locally confluent iff ∀x,y,zx←

−z− →y = ⇒ x ↓ y

Theorem

Let (X, − →) be an ARS. Then:

• 1. If (X, −

→) is terminating, each x ∈ X contains a normal form that is computable with a finite number of reductions.

• 2. If (X, −

→) is confluent, each equivalence class of

∗

← → contains at most one normal form.

• 3. If (X, −

→) is terminating, it is confluent iff it is locally confluent.

Term rewriting system (TRS)

TRS is a set R of rewriting rules l → r for l, r ∈ TF(X) TRS R defines an ARS (TF(X), − →R), where − →R is a closure of R under substitutions and operation compatibility. TRS R is terminating, (locally) confluent iff (TF(X), − →R) does, respectively.

Theorem

Let R be a (locally) confluent and terminating TRS. Then:

• 1. Every term t ∈ TF(X) has a unique normal form t↓R that is

computable with a finite number of reductions.

• 2. For all t, s ∈ TF(X)

(t ≈ s) ∈ Th(Eq(R)) ⇐ ⇒ t↓R= s↓R where Eq(R) = { (l ≈ r) | (l → r) ∈ R }.

TRS – criteria for termination

Definition A well-founded order on the set of terms, closed under substitutions and compatible with operations is called reduction order.

Theorem

A TRS R is terminating iff R ⊆> for some reduction order .

TRS – criteria for confluence

Definition Assume that two rules l → r and l′ → r′ have overlapping left-hand sides, i.e. terms l|ω and l′ are unifiable for some non-variable position ω in l. The term σ(l), where σ is a most general unifier of l|ω and l′, may be reduced to:

◮ σ(r) with rule l → r ◮ σ(l[ω ←

֓ r′]) with rule l′ → r′ The pair of terms (σ(r), σ(l[ω ← ֓ r′])) is called critical pair of rules l → r and l′ → r′.

Critical Pair Lemma

TRS is locally confluent iff all its critical pairs are joinable.

Equations in partial algebras

Does an equation hold in an algebra when one or both sides are undefined for some valuations? existence equation: for all valuations both sides must be defined and equal Existential equational theories are

◮ partial equivalence relations on the set of terms, ◮ closed under substitutions of terms from their domains, ◮ compatible with operations (on their domains).

Existential equational theories

Theorem (Burmeister ’86)

A set of equations forms an existential equational theory if and

• nly if it is closed under the following inference rules:

⊢ x ≈ x if x is a variable t ≈ t′ ⊢ t ≈ t t ≈ t′ ⊢ t′ ≈ t t ≈ t′ , t′ ≈ t′′ ⊢ t ≈ t′′ f (t1, . . . , tn) ≈ f (t1, . . . , tn) ⊢ ti ≈ ti if i ∈ {1, . . . , n}

• ti ≈ t′

i

• 1in , f (t1, . . . , tn) ≈ f (t1, . . . , tn)

⊢ f (t1, . . . , tn) ≈ f (t′

1, . . . , t′ n)

t ≈ t′ ,

• σ(x) ≈ σ(x)
• x∈Var(t,t′)

⊢ σ(t) ≈ σ(t′)

Partial reduction system (PRedS)

PRedS is a triple (X, − →, N), where (X, − →) is an ARS and N ⊆ X. XN =

• x∈N

[x]

∗

← →

– subset of X indicated by N

N

← → =

∗

← → ∩ (XN)2 – partial equivalence relation defined by PRedS (X, − →, N) PRedS (X, − →, N) is:

◮ terminating iff (X, −

→) is terminating

◮ (locally) confluent iff (XN, −

→) is (locally) confluent

◮ reduced iff for all x ∈ N if x ∗

− → x′ then x′

∗

− → x′′ for some x′′ ∈ N

Partial reduction systems

Definition PRedS (X, − →, N) presents partial equivalence relation

N

← → iff the following conditions are satisfied:

• 1. Every x ∈ X has a normal form that is computable with a

finite number of reductions.

• 2. Every x ∈ XN has a unique normal form.
• 3. For all x, y ∈ X

x

N

← → y ⇐ ⇒ x↓= y↓∈ N

Theorem

Let (X, − →, N) be a terminating, (locally) confluent and reduced

• PRedS. Then it presents

N

← →.

Partial rewriting system (PRS)

PRS is a pair (R, D), where R is a set of rewriting rules and D is a set of terms. PRS (R, D) defines PRedS (TF(X), − →(R,D), ND), where:

◮ ND is a set of all terms ,,composed” of D-terms and variables ◮ −

→(R,D) is the closure of R-rules under substitutions of ND-terms and compatibility with operations PRS (R, D) is terminating, (locally) confluent, reduced iff (TF(X), − →(R,D), ND) does, respectively. PRS (R, D) presents

ND

← →(R,D) iff (TF(X), − →(R,D), ND) does.

PRSs and existential equational theories

Definition PRS (R, D) is subterm-closed iff a subset of TF(X) indicated by ND is subterm-closed.

Theorem

A PRS (R, D) is subterm-closed iff

ND

← →(R,D) is existential equational theory.

PRSs and existential equational theories

Definition PRS (R, D) is existential iff a subset of TF(X) indicated by ND contains all terms from R-rules.

Theorem

A PRS (R, D) is subterm-closed and existential iff

ND

← →(R,D) = ThE(Eq(R, D)), where Eq(R, D) = { (l ≈ r) | (l → r) ∈ R } ∪ { (d ≈ d) | d ∈ D }

PRS – presenting equational theories

Definition PRS (R, D) presents existential equational theory E iff the corresponding PRedS does.

Corollary

If PRS (R, D) is terminating, (locally) confluent, reduced, subterm-closed and existential, then it presents the existential equational theory ThE(Eq(R, D)).

PRS – criteria for presenting equational theory

Definition Assume that a term d overlaps with a left-hand side of a rule l → r, i.e. terms d|ω and l are unifiable for some non-variable position ω in d. Then the term σ(d[ω ← ֓ r]), where σ is a most general unifier of d|ω and l, is called critical term of d and (l → r).

Theorem

A terminating (R, D) presents ThE(Eq(R, D)) iff the following conditions are satisfied:

• 1. ∀(p,q)∈CP(R) p↓(R,D)= q↓(R,D)
• 2. ∀(l→r)∈R r↓(R,D)∈ ND
• 3. ∀(f (l1,...,ln)→r)∈R∀i∈{1,...,n} li↓(R,D)∈ ND
• 4. ∀f (d1,...,dn)∈D∀i∈{1,...,n} di↓(R,D)∈ ND
• 5. ∀t∈CT(R,D) t↓(R,D)∈ ND