Download Sub-Birkhoff

Sub-Birkhoff
restriction
(progressive)
none
no (symmetry)
also no (reflexivity)
also no (transitivity)
just no (axiom)
logic
(Definition 1)
equational
rewrite(ii)
transitive(iii)
step
empty
model relation
(Definition 3)
equivalence
quasi-order
transitive
relation
identity
quotient model
(Lemma 7)
equality(i)
partial order(ii)
strict order(iii)
acyclic
identity
rewrite
(Lemma 8)
↔∗
→∗
→+
→
≡(iv)
Definition 1 A subequational logic L is a system hS,Ii consisting of a specification S and a set
of inferences I ⊆ {(reflexivity), (symmetry), (transitivity), (congruence), (axiom)}. A specification
S is a system hΣ,Ai consisting of a signature Σ and an axiom system A.
• A signature Σ is a system hF ,αi consisting of symbols f ∈ F having an arity α(f ) ∈ N.
• An axiom system A is a system hX,N ,%i consisting of variables x ∈ X, names ν ∈ N and
a rule %. The rule % maps each ν to a pair of terms over Σ ] X, with α(x) = 0.
The combination of a symbol with its arity is called a function symbol. The combination of a
name with its rule is called an axiom. Subequational logics generate subequational theories.
Definition 2 For a subequational logic L = hS,Ii its theory L is generated by the following
inference rules, where an inference rule (i) only applies if i ∈ I. s, t and r range over terms.
`sLs
(reflexivity)
`sLt
`tLs
` ~s ≡[L] ~t
` f (~s) L f (~t)
(symmetry)
(congruence)
`sLt
` l(~s) L r(~s)
`tLr
`sLr
(transitivity)
if %(ν) = hl,ri (axiom)
The sequences in the (axiom)-clause correspond to the sequence of variables occurring in the axiom.
Two remarks on notation used in the definition: ≡[L] expresses that corresponding components
are identical (as terms) except for one index where they are related by L. The notation s(~t)
expresses that the sequence of terms ~t is substituted for the corresponding sequence of variables
in s. Elements of the theory are called the theorems. Logics may have relational models.
Definition 3 A relational model M of subequational logic L = hhΣ,Ai,Ii is a system hA,Ri where
A is a Σ-algebra and R a relation on the carrier of A validating I for A.
• A Σ-algebra of a signature Σ = hF ,αi is a system hC,FA i, where C is the carrier set and
FA contains for each n-ary function f ∈ F a function fA : C n → C.
• A relation R on the carrier of the Σ-algebra A validates I for A = hX,N ,%i if for each i ∈ I,
the inference rule (i) below holds, where a, b and c range over C.
|= a M a
(reflexivity)
|= a M b
|= b M a
|= ~a =[M] ~b
|= fA (~a) M fA (~b)
(symmetry)
(congruence)
1
|= a M b
|= l M r
|= b M c
|= a M c
(transitivity)
if %(ν) = hl,ri (axiom)
Here |= a M b holds if a R b and |= s M t holds if |= sM]v M tM]v , for any valuation
v : X → C. Here sM]v ∈ C is obtained from s via the homomorphic extension of the algebra
of M and the valuation v to terms. Write |= s L t if |= s M t for all models M of L.
Theorem 4 ` s L t iff |= s L t.
Proof Define the term model ML as having the terms for carrier, mapping function symbols to
themselves (fML = f ), and having L for relation. We show ` s L t iff |= s ML t. by induction on
` s L t, matching occurrences of inference rules (i), for i ∈ I, in the logic and the model and using
the Substitution Lemma in the (axiom)-case noting that sML ]id = s, where id is the identity. 2
Lemma 5 (Substitution) s(~t)M]v = sM]v~t , where v~t assigns tiM]v to xi , for all terms s, ~t.
The class of models may be restricted by quotienting models by strongly connected components.
Definition 6 Let M = hhA,Ri,Ii be a model and ∼
= ⊆ R∗ be a congruent equivalence relation on
1
∼
∼
∼
A. Define M/= = hhA/=,R/=i,Ii by:
• For an algebra A = hC,FA i, its quotient A/∼
= is hC/∼
=,FA/∼
= i. The carrier consists of
~
∼
a)].
=-equivalence classes of C. Denoting the equivalence class of a by [a], fA/∼
= ([a]) = [fA (~
• Define [a] R/∼
= [b] if a ∼
=;R;∼
= b, where ; denotes relation composition.
Lemma 7 Under the assumptions of Definition 6, M/∼
= is a model.
∼
Proof Note that fA/∼
= and R/= are well-defined, i.e. defined independent of the choice of representative, by the assumption that it is a congruent equivalence. To show that R/∼
= validates I
for the axioms, one uses ∼
= ⊆ R∗ for (transitivity) and the Substitution Lemma for (axiom). 2
∼R as the largest (it exists!) congruent equivalence contained in R∗ for a relation R,
Defining =
∼L is a model by Lemma 7. Note that always id ⊆ ∼
ML /=
=L and in case (congruence) ∈ I, even
∼
=L = L∗ ∩ (L−1 )∗ . It is well-known (e.g. [1, Thm. 3.1.12]) that rewriting is logical in the sense
that s is equal to t in equational logic iff s is convertible to t in the rewrite system induced by the
specification. This extends to subequational logics (cf. [3, Lem. 2.6] for rewrite logic).
Lemma 8 For a subequational logic L = hS,Ii with (axiom),(congruence) ∈ I we have ` s L t iff
s →IS t. Here →IS is the closure of the rewrite relation →S induced by S, under all elements of I.
The numbers of the remarks below correspond to the numbers of the summarizing table above.
i. Completeness for equational logic is due to [2]. Equality models are known as models.
ii. Rewrite logic is due to [3], where categorical models are used to model concurrency. Its
partial-order models are known as quasi-models in term rewriting [6].
iii. Variations can be easily obtained. For instance, reduction logic could be defined as transitive
logic having a terminating specification (as term rewrite system). It is sound and complete
for inversely well-founded strict order models, known as well-founded monotone algebras ([5]).
iv. Not insisting on (axiom),(congruence) ∈ I in Lemma 8, rewriting (applying rules in contexts)
has to be restricted in its statement, e.g. as in the table or to head-steps as in SOS [4].
References
[1]
[2]
[3]
[4]
[5]
[6]
F. Baader and T. Nipkow. Term Rewriting and All That. Cambridge University Press, 1998.
G. Birkhoff. On the structure of abstract algebras. Proc. Cambridge Phil. Soc., 31:433–454, 1935.
J. Meseguer. Conditional rewriting logic as a unified model of concurrency. TCS, 96:73–155, 1992.
G. Plotkin. A structural approach to operational semantics. Technical Report FN-19, Aarhus, 1981.
H. Zantema. Termination of term rewriting: interpretation and type elimination. JSC, 17:23–50, 1994.
H. Zantema. Termination of term rewriting by semantic labelling. Fund. Inf., 24:89–105, 1995.
1 More
general conditions, like R ; ∼
=⊆∼
= ; R∗ , could be formulated, but ∼
= ⊆ R∗ suffices for our purposes.
2