Computing the size of a language

We next introduce a function cnt_{D,E : D → N ∪ { ω } which for each graph} (D, E) ⊆ (DA, EA) and profile d ∈ D, returns an estimate of the number of

elements in CanRI/E with the profile d. We show below that for each d ∈ D,



profile−1_D,E(d)≤ cnt_D,E(d) ≤ 

profile−1_D

A,EA(d)



. (3.13)

For correctness purposes, any value in the range is sufficient. The proof that our procedure always shows emptiness only requires that cnt_D,E(d) is at most the total number of elements in CanDA,EA with a profile d, while the proof that

our procedure always shows non-emptiness only requires that cnt_D,E(d) is at least the number of explored elements in Can_D,E with a profile d.

Before showing (3.13), we first must define cnt_D,E. In the definition, we use the choose function C : (N ∪ { ω }) × N → N ∪ { ω } which is partially extended to ω, i.e,

C(n, k) = n!/k!(n − k)!, C(ω, 0) = 1, and C(ω, k) = ω if k > 0. Definition 3.4.19. For each (D,_{E) ⊆ (D}A, EA), let cntD,E: D → N ∪ { ω } be

the function such that cnt_{D,E(d) = ω if d E}+_{d and otherwise}

• For each constant c ∈ Σ, cnt_D,E(c, P ) = 1. • For each free symbol f ∈ Σ with arity n > 0,

cnt_D,E(f, P ) = X (f1,P1),...,(fn,Pn) ∈ D (∀i∈[1,n]) (fi,Pi) E (f,P ) statesf(P1,...,Pn) = P n Y i=1 cnt_D,E(fi, Pi).

• For an AC symbol ◦ ∈ Σ that is idempotent in E or R, cnt_D,E(◦, P ) = ω if |M (ψ◦,P,I)| = ω, and otherwise,

cnt_D,E(◦, P ) = X u ∈ M (ψ◦,P,I) Y dE(◦,P ) C(cnt_D,E(d), ud) where ψ◦,P,I(x) = ψ◦,P(x) ∧ ^ d E (◦,P ) xd ≤ cntD,E(d) ∧ ^ d 6E (◦,P ) xd= 0.

cnt_D,E(+, P ) = ω if |M (ψ+,P,AC)| = ω, and otherwise, cnt_D,E(+, P ) = X u ∈ M (ψ+,P,AC) Y dE(+,P ) cnt_D,E(d)>0 C(cnt_D,E(d) + ud− 1, ud) where ψ+,P,AC(x) = ψ+,P(x) ∧ ^ d E (+,P ) cntD,E(d)=0 xd= 0 ∧ ^ d 6E (+,P ) xd= 0.

For proving the computability and correctness of cnt_D,E, we define the binary relation_{C ⊆ E as follows:}

d1C d2 ⇐⇒ d1E d2∧ ¬(d2E+d2).

The relation_C+ is irreflexive. Every irreflexive and transitive relation over a finite set is well-founded, and so it follows that_C+ _{is well-founded as well.}

Lemma 3.4.20. The function cnt_D,E is computable.

Proof. To show this, observe that if in evaluating cnt_D,E(d), we recursively call cnt_D,E(d0) for some d0 ∈ D, then d0

C d. Since C is well-founded, it follows that the chain of recursive calls is finite. Most of the other operations are straight- forward to implement. For representing elements of N ∪ { ω }, an abstract data type should be used that can represent any natural number as well as the con- stant ω. Each of the formulas ψ appearing an expression M (ψ) are formulas in Presburger arithmetic, and thus M (ψ) is effectively a semilinear set [62]. It fol- lows that one can easily decide whether |M (ψ)| = ω and enumerate the vectors if M (ψ) is finite.

Before we can prove the claim made in equation (3.13), we need to show how the edge relation_EA can be used to detect when CanRI/E contains an infinite

number of equivalence classes with a given profile. To show this, we first define the size of a term t ∈ TΣ, denoted size(t) to be the number of symbols in t.

Since the associativity and commutativity equations in EAC preserve the size of

a term, one can observe that if t =EAC u, then size(t) = size(u).

Lemma 3.4.21. If d1E+Ad2 for d1, d2∈ DA, then for all [t1] ∈ CanRI/E such

that profile_A/E([t1]E) = d1, there ∃[t2] ∈ CanRI/E such that profileA/E([t2]E) =

d2 and size(t2↓_Rˆ_I/EAC) > size(t1↓RˆI/EAC).

Proof. We prove this by induction on the length of the chain of inferences used to show d1E+_Ad2.

The inductive case is easier, and so we prove it first. In this case, we know there is a profile d0 ∈ DA such that d1E_A+ d0E+_Ad2. By our first induction

hypothesis we know that there is an equivalence class [t0] ∈ Can_Rˆ_I/EAC such that

then implies the existence of [t2] ∈ Can_Rˆ_I/EAC such that profileA/E([t2]E) = d2

and size(t1) < size(t0) < size(t2).

In the base case, we know that d1EAd2. By the definition of EA, there

must be equivalence classes [u], [v] ∈ CanRI/E such that profileA/E([u]) = d1,

profile_A/E([v]) = d2, and [u]Eflat[v]. Assuming profileA/E([t1]) = d1, we con-

struct t2 by analyzing why [u]Eflat[v]. There are two cases to consider:

• If v ↓_Rˆ_I/EAC=EAC f (v1, . . . , vn) where f is a free symbol and u =EAC vifor

some i ∈ [1, n], then we let

t2= f (v1, . . . , vi−1, t1, vi+1, . . . , vn).

Clearly, [t1] Eflat[t2]. We know that statesA/E([t1]) = statesA/E([u]),

and consequently statesA/E([t2]) = statesA/E([v]) by Lemma 3.4.7. As

profile_A/E([v]) = d2= (f, statesA/E([v]), it follows that profileA/E([t2]) =

d2. Finally,

size(t2↓_Rˆ_I/EAC) = 1 +

X

j∈[1,n]\{i}

size(vj) + size(t1↓_Rˆ_I/EAC),

and thus size(t2↓_Rˆ_I/EAC) > size(t1↓_Rˆ_I/EAC).

• Otherwise, v ↓_Rˆ_I/EAC=EAC v1+ · · · + vn with + an AC or ACI symbol,

n ≥ 2, root(vi) 6= + for all i ∈ [1, n], and u =EAC vi for some i ∈ [1, n]. If

t1 ↓_Rˆ_I/EAC=EAC vj for some j ∈ [1, n], then we let t2 = u and it trivially

follows that profile_A/E([t2]) = d2 and size(t2 ↓_Rˆ_I/EAC) > size(t1 ↓_Rˆ_I/EAC).

Otherwise, we let

t2= v1+ · · · + vi−1+ t1↓_Rˆ_I/EAC +vi+1+ · · · + vn.

We know that t1↓_Rˆ_I/EAC6=EAC vi for all i ∈ [1, n], and thus t2 is ˆRI/EAC-

irreducible. It follows that size(t2 ↓_Rˆ_I/EAC) > size(t1 ↓_Rˆ_I/EAC). Since

n > 2, root(u) = +, and thus profile_A/E([u]) = d1 = (+, P ) for some

P ⊆ Q. It follows by Lemma 3.4.14 that

#(profile_A/E([u1]), . . . , profileA/E([un])) ∈ M (ψ+,P).

As profile_A/E([t1]) = profileA/E([u1]), it follows profileA/E([t2]) = d2.

We can use the previous lemma to make the following observation: Corollary 3.4.22. For all d ∈ DA, if dE+_Ad, thenprofile−1DA,EA(d)

 = ω. Proof. For all d ∈ DA, there is a [t] ∈ CanRI/E such that profileA/E([t]) = d.

If d_E+_Ad, then we can use Lemma 3.4.21 to construct an infinite sequence [t1], [t2], · · · ∈ CanRI/E of equivalence classes each with profile d and where

size(ti ↓_Rˆ_I/EAC) < size(tj ↓RˆI/EAC) for i < j. It follows that for all distinct

i, j ∈ N, ti↓_Rˆ_I/EAC6=EAC tj ↓RˆI/EAC, and thus are also distinct modulo E . Conse-

quently, Can_RI/E contains an infinite number of equivalence classes with profile

d.

We are now ready to prove our previous claim in equation (3.13). Lemma 3.4.23. For all profile graphs (D,_{E) ⊆ (D}A, EA) and d ∈ D,



profile−1_D,E(d)≤ cnt_D,E(d) ≤ 

profile−1_DA,EA(d). (3.13)

Proof. We prove (3.13) for all d ∈ D by induction on d with respect to the well- founded relation_{C. In our inductive proof, there are four cases to consider:}

• If d E+ _{d, then cnt}

D,E(d) = ω, and thus



profile−1_D,E(d) ≤ cnt_D,E(d). On the other hand, as _{E is a subset of E}A, we know that dE+Ad. By

Cor. 3.4.22, it follows thatprofile−1_DA,EA(d)= ω.

• If d = (c, P ) with c a constant, then because D ⊆ DA, we know there

is an equivalence class [t] ∈ CanRI/E such that root(t ↓RˆI/EAC) = c and

statesA/E([t]) = P . However, root(t ↓_Rˆ_I/EAC) = c implies that t =E

c. As there is only one equivalence class containing c, it follows that 

profile−1_D,E(d)= 1 and 

profile−1_DA,EA(d)= 1.

• If d = (f, P ) with f a free symbol with arity n > 0, then by using Lemma 3.4.8 with both (D,_{E) and (D}A, EA), we can reduce (3.13) for

d to the problem of showing (3.13) for all d0_{E d. However, this follows} trivially by our induction hypothesis as d 6_E+_{d and d}0_{E d implies d}0_{C d.}

• If d = (◦, P ) with ◦ ∈ Σ idempotent in E or R, then we first note that for all d0∈ D, d0

Ed =⇒ d0Cd as d 6E+_{d. It follows that we may assume that}

equation (3.13) holds for each d0_{E d. This implies that M (ψ}◦,P,Can_D,E) ⊆

M (ψ◦,P,I) ⊆ M (ψ◦,P,Can_DA,EA), and consequently by using Lemma 3.4.17,

we can reduce the problem of showing (3.13) for all d ∈ D to two problems: (1) for all d0_{E d and k ≤}profile−1_D,E(d0),

C(profile−1_D,E(d0), k) ≤ C(cnt_D,E(d0), k), and (2) for all for all d0_{E d and k ≤ cntD,E}(d0),

C(cnt_D,E(d0), k) ≤ C(profile−1_D

A,EA(d

0₎ , k).

Both of these problems follow easily from our induction hypothesis and the definition of C.

Starting with the empty graph (D0, E0) = (∅, ∅), we freely apply either of the

rules below to construct (Di+1, Ei+1) from (Di, Ei) subject to the condition that

a rule may only be applied if the resulting graph (Di+1, Ei+1) is distinct from

(Di, Ei). The rules are applied until completion to obtain the graph (D∗, E∗).

choose free symbol f ∈ Σ and (f1, P1), . . . , (fn, Pn) ∈ Di

Di+1:= Di∪ { (f, statesf(P1, . . . , Pn)) }

Ei+1:= Ei∪ { ((fj, Pj), (f, statesf(P1, . . . , Pn))) | j ∈ [1, n] }

choose AC or ACI symbol + ∈ Σ and P ⊆ Q s.t. (∃x) ψ+,P,Di,Ei(x)

Di+1:= Di∪ { (f, P ) }

Ei+1:= Ei∪ { (d, (f, P )) | d ∈ Di∧ (∃x) ψ+,P,Di,Ei(x) ∧ xd> 0}

where for each AC symbol ◦ ∈ Σ that is idempotent in E or R and each AC symbol + ∈ Σ that is not idempotent in E or R, we let

ψ◦,P,Di,Ei(x) = ψ◦,P(x) ∧ ^ d ∈ D(◦)\Di xd= 0 ∧ ^ d ∈ D(◦) ∩ Di xd≤ cntDi,Ei(d) ψ+,P,Di,Ei(x) = ψ+,P(x) ∧ ^ d ∈ D(+)\Di xd= 0 ∧ ^ d ∈ D(+) ∩ Di cnt_Di,Ei(d) = 0 xd= 0.

Figure 3.4: Inference System for Constructing (D∗, E∗)

• Otherwise d = (+, P ) with + ∈ Σ an AC symbol that is not idempo- tent in R. We first note that for all d0 _{E d, d}0 _{C d as d 6 E}+d. It follows that we may assume (3.13) for each d0 _{E d. This implies that} M (ψ+,P,CanD,E) ⊆ M (ψ+,P,AC) ⊆ M (ψ+,P,CanDA,EA), and consequently

by using Lemma 3.4.18, we can reduce the problem of showing (3.13) for all d ∈ D to two problems: (1) for all d0 _{E d and k ∈ N where} 

profile−1_D,E(d0)> 0,

C(cnt_D,E(d0) + k − 1, k) ≤ C(profile−1_DA,EA(d0)+ k − 1, k), and (2) for all d0_{E d and k ∈ N where cntD,E}(d0) > 0,

C(cnt_D,E(d0) + k − 1, k) ≤ C(profile−1_DA,EA(d0)+ k − 1, k). Both of these problems follow easily from our induction hypothesis and the definition of the choose function C.

In document Decision Procedures for Equationally Based Reasoning (Page 87-91)