Composition for Orders with an Extra Binary Relation Wolfgang - - PowerPoint PPT Presentation

Composition for Orders with an Extra Binary Relation

Wolfgang Thomas Bruno’s Workshop, Bordeaux, June 2012

Two Traditions in Effective Logic

Orderings with unary predicates MSO-logic Automata, Composition method Graphs Courcelle theory (for MSO) Hanf and Gaifman theorems (for FO) Here: Study of FO-theory of orderings expanded by graphs Structures: (N, <, R) with binary R Restriction: R is finite valency

R ⊆ A × A is of finite valency if for each a ∈ A there are only

finitely many b ∈ A with R(a, b) or R(b, a). Injective functions f : N → N provide examples.

Wolfgang Thomas

Plan

• 1. MSO-Th(N, <, P) for unary P
• 2. Homogeneity of colorings is first-order definable
• 3. Orderings (N, <, R) with binary R
• 4. Conclusion

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Structures (N, <, P) with unary P

Identify P ⊆ N with 0-1-word α(P) Consequence of B¨ uchi’s analysis of MSO-Th(N, <): MSO-Th(N, <, P) is decidable iff the following decision problem is decidable: Given a B¨ uchi automaton A, decide whether A accepts α(P). So one only needs to decide whether the word α(P) can be cut into pieces u0, u1, . . . such that

A : q0

u0

→ q and A : q

ui

→ q for i = 1, 2, . . ., with q final.

The composition method allows this reduction to periodicity directly, without reference to automata.

Wolfgang Thomas

m-Types (for FO and MSO)

Given quantifier-depth m define for two words u, v (finite or infinite!):

u ≡m v :⇐

⇒

u and v satisfy the same sentences of quantifier-depth m

Facts:

≡m is an equivalence relation of finite index;

call the equivalence classes m-types. An m-type τ is definable by a sentence ϕτ of quantifier-depth m. Each sentence ψ of quantifier-depth m is equivalent to a disjunction of sentences ϕτ.

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Composition

• 1. From the m-types of u and v one can compute the m-type
• f uv.
• 2. From the m-type of u one can compute the m-type of

uuu . . ..

Consequence: Given α = uvvv . . ., the m-type ̺ of α is determined by the

m-types σ of u and τ of v;

we write ̺ = σ + ∑ω τ Ramsey’s Theorem guarantees such a decomposition for arbitrary α

Wolfgang Thomas

Finite Colorings

Given a finite set Col = {c1, . . . , cr} of colors. A coloring over N with Col is a map

C : {(m, n) | m < n} → Col C is additive if from C(ℓ, m) = C(ℓ′, m′) and C(m, n) = C(m′, n′) we can infer C(ℓ, n) = C(ℓ′, n′).

For colors c, d we may write c + d. Example: For quantifier-depth m and ω-word α define

Cm

α (i, j) = m-type of α[i, j − 1]

(either for FO or MSO)

Wolfgang Thomas

Ramsey’s Theorem

For any finite additive coloring C there is a ”homogeneous set” H = {h0 < h1 < h2 < . . . } such that all colors C(hi, hj) (where i < j) coincide. Consequence: Then there are two colors c, d such that

C(0, h0) = c and C(hi, hi+1) = d

Call a color pair (c, d) good for C if there is

H = {h0 < h1 < . . .} such that C(0, h0) = c and C(hi, hj) = d for i < j,

in particular, C(hi, hi+1) = d, and d = d + d.

Wolfgang Thomas

Back to MSOTh(N, <, P)

MSOTh(N, <, P) is decidable iff for each m we can compute the m-type ̺ of α(P) iff for each m and the associated coloring Cm

α(P) we can

compute those pairs (σ, τ) of m-types which are good for

Cm

α(P).

In other words, for any P: A sentence ψ of quantifier-depth m is effectively equivalent

• ver (N, <, P) to a disjunction of statements

”(σ, τ) is good for Cm

α(P)”

[Compare with the automata theoretic periodicity condition.]

Wolfgang Thomas

Defining to be Good

Let C be the tuple of binary predicates ”C(i, j) = c”. Consider the associated structure (N, <, C). Remark: There is an MSO-sentence ϕc,d saying in (N, <, C) that (c, d) is good for C:

∃X (X is infinite ∧C(0, x) = c for the smallest element x of X ∧ C(x, y) = d for any x, y ∈ X with x < y)

We show that an FO-sentence suffices. This will also give a proof of Ramsey’s Theorem.

Wolfgang Thomas

McNaughton’s Merge-Relation

Given α and an additive coloring C.

m, n merge at k (short m ∼C n(k)) if C(m, k) = C(n, k)

If m, n merge at k then also at each k′ > k.

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Lemma 1

(c, d) is good for C iff (∗) ∃n[C(0, n) = c ∧ ∀m∃k > m(C(n, k) = d ∧ n ∼C k)]

Show ⇐: Take n0 as the smallest n according to (∗). Assume n0, . . . , ni are defined, with n0 ∼C nj for j = 1, . . . , i. Let n0, . . . , ni merge at m. Define ni+1 as the smallest number k > m guaranteed by (∗), namely with C(n0, ni+1) = d and n0 ∼ nj for all

j = 1, . . . , i + 1.

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Consequence: Ramsey’s Theorem

Let M be an infinite ∼C-equivalence class Let n0 be its smallest element and set c = C(0, n0). For some d infinitely many n in M exist with C(n0, n) = d Then (∗) is satisfied. Hence (c, d) is good for C.

Wolfgang Thomas

Reducing Quantifier Alternation

(∗) ∃n[C(0, n) = c ∧ ∀m∃k > m(C(n, k) = d ∧ n ∼C k)]

is a Σ3-condition (w.r.t. unbounded quantifiers). Show that it can be written as a Boolean combination of

Σ2-conditions.

Define a set Mℓ,c(x): Consider the ℓ-tuples of distinct numbers n1, . . . , nℓ ≤ x such that C(0, ni) = c and any two of the nj do not merge at x. If such an ℓ tuple exists let Mℓ,c(x) contain the elements of the smallest such tuple (in lexicographical ordering)

• therwise let Mℓ,c(x) = {x}.

Wolfgang Thomas

Lemma 2

Define

gℓ,c(x) = max Mℓ,c(x) fℓ,c,d(x) = the greatest y < x such that for some z ∈ Mℓ,c C(z, y) = d and C(y, x) = d and C(z, x) = d

(take value 0 if such y does not exist) Then

(c, d) is good for C iff

r

ℓ=1(gℓ,c is bounded and fℓ,c,d unbounded).

Wolfgang Thomas

The ∀∃ ∧ ∃∀-Lemma

Let C be an additive finite coloring and C be the tuple of relations C(i, j) = c. There are bounded formulas ϕc,ℓ(y) and ψℓ,c,d(y) such that

(c, d) is good for C iff (N, <, C) | = |C|

ℓ=1(∃x∀y > x ϕc,ℓ(y) ∧ ∀x∃y > x ψc,d,ℓ(y))

Application: McNaughton’s Theorem Any B¨ uchi automaton can be converted into a deterministic Muller automaton. Use ∼A-classes as colors: u ∼A v iff for any states p, q

A : p

u

→ q [passing F] ⇔ A : p

v

→ q [passing F]

Wolfgang Thomas

Other Applications

• 1. For any P ⊆ N: MSO-Th(N, <, P) is decidable iff

WMSO-Th(N, <, P) is.

• 2. Any FO-definable ω-language can be recognized by a

counter-free Muller automaton.

Wolfgang Thomas

Binary relations and composition

Consider structures (N, <, R) with binary R The m-types of two segments ([ℓ, m), <, R|[ℓ,m)) and

([m, n), <, R|[m,n)) are not sufficient to determine the m-type

• f ([ℓ, n), <, R|[ℓ,n))

But we can do composition if enough interface information is provided.

Wolfgang Thomas

Finite Valency

Let R ⊆ N × N be of finite valency: For any a there are at most finitely many b with R(a, b) or R(b, a). Call [a, b] an R-segment if R(a, b) or R(b, a). An R-segment is maximal if it is not properly contained in another R-segment. Remark: If R is of finite valency then each R-segment is contained in a maximal one.

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m-Admissible Segments

Define for each b a sequence b(0) > b(1) > . . . as follows:

b(0) = b b(i + 1) =

biggest c which is below all maximal R-segments [k, ℓ] intersecting [b(i), ∞), if such c exists,

0 otherwise

The segment [a, b] is m-admissible if b(2m) > a Write b∗ for b(2m) if m is clear. Denote by

b the sequence (b(0), . . . , b(2m)).

For any k there is exist admissible segments [a, b] above k.

Wolfgang Thomas

T- and D-Types

Let [a, b] be m-admissible, a0, . . . , ar−1 ∈ [a, b]. let

Tm

R [a, b](a0, . . . , ar−1) be the FO-m type of the restriction

• f N to [a, b]

Dm

R[a, b](a0, . . . , ar−1) := Tm R [a∗, b](

a, b, a0, . . . , ar−1) Dm

R defines an almost total coloring:

For each a there are only finitely many b ≥ a such that [a, b] is not m-admissible.

Wolfgang Thomas

Composition Lemma

• 1. Given m-admissible segments [a, b] and [b, c],

Dm

R[a, b] and Dm R[b, c] determine effectively the type

Dm

R[a, c]

• 2. Given a sequence a0, a1, . . . such that [ai, ai+1] is

m-admissible and Dm

R[ai, ai+1] = τ for some m-type τ,

Dm

R[a0, ∞) is determined effectively by τ.

If Dm

R[0, a0] = σ we may write

Dm

R[0, ∞) = σ + τ + τ + . . .

Wolfgang Thomas

Nondefinability of + and ·

Theorem: In a structure (N, <, R) with R of finite valency, neither addition nor multiplication is FO-definable. Lemma: Let f : N2 → N be FO-definable in (N, <, R) where R is of finite valency. Then one of the following two sets is finite:

X f := {x ∈ N | λyf(x, y) is injective} Yf := {y ∈ N | λx f(x, y) is injective}.

Note: X+, Y+, X·, Y· are all infinite.

Wolfgang Thomas

Proof of Lemma

Assume f is FO-definable by ϕ(x, y, z) of quantifier-depth m Consider the coloring Cm+1

R

and a homogeneous set H Pick x0 ∈ X f, y0 ∈ Yf in distant H-segments:

x0 ∈ [hi, hi+1], y0 ∈ [hj, hj+1]

Let z0 = f(x0, y0), assume z0 is far from x0 Pick x′ ∈ [hi+1, hi+2] of same m-type as x0 in [hi, hi+1] Then f(x0, y0) = f(x′, y0) = z0

λyf(x0, y0) not injective, so y0 ∈ Yf

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A Normal Form

Over (N, < R) each sentence ϕ is equivalent to a sentence

n

i=1(∀x∃y∀z ϕi(x, y, z) ∧ ∃x∀y∃z ψi(x, y, z)

where the ϕi, ψi are bounded in z. Use the ∀∃ ∧ ∃∀-Lemma for additive colorings. As color formulas C(x, y) = c we take Dm

R [x, y] = τ

We can define Dm

R[x, y] = τ relative to a bound z by a formula

Dm

R[x, y](z) = τ bounded in z.

Then:

Dm

R[x, y] = τ

⇔ ∃ωz Dm

R[x, y](z) = τ

⇔ ∀ωz Dm

R[x, y](z) = τ

Wolfgang Thomas

Degree of FO-Th(N, <, R)

For R ⊆ N2 of finite valency, FO-Th(N, <, R) ≤T R′′′, and R′′′ cannot in general be replaced by R′′ Proof of lower bound:

V3 := {m | ∀k∃ℓ > k(ℓ ∈ Wm ∧ Wℓ = O)} (is Π3-complete)

So:

m ∈ V3

⇔

Wm contains infinitely many indices of the empty r.e. set

Find recursive R of finite valency s.t. a V3 ≤T FO-Th(N, <, R) Let (ℓi, mi) the i-th pair in enumeration of all (ℓ, m) with

ℓ ∈ Wm

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Explaining the Reduction

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Reduction

Recall: (ℓi, mi) is the i-th pair in enumeration of all (ℓ, m) with

ℓ ∈ Wm

At stage i include (i, i + mi + 1) in R, Using ℓi include (i, i) if up to stage i an element in Wℓi was found, At later stage j include (j, i) if then for the first time an element in Wℓi is found.

R is recursive and of finite valency.

∃ωℓ(ℓ ∈ Wm ∧ Wℓ = O)

iff (N, <, R) |

= ∃ωx (R(x, x + m + 1) ∧ ∀z¬R(z, x))

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Special Case

R is effectively of finite valency

if the finite sets Na = {b | R(a, b)} and Nb = {a | R(a, b)} are computable from a, b, respectively. In this case the bound R′′′ can be replaced by R′′.

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Strongly Monotone Functions

For a monotone function f : N → N define ∆ f by

∆ f(n) = f(n + 1) − f(n).

Call f strongly monotone if f and ∆ f are monotone (in the

≤-sense).

Theorem If f is strongly monotone and recursive, the first-order theory

• f (N, <, f) is decidable.

In contrast: Let fP(i) = i-th prime. Is FO-Th(N, <, fP) decidable? Is FO-Th(N, <, P) decidable?

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Conclusion

• 1. Variations:

Signatures <, R1, . . . , Rn rather than <, R. Finitely many exceptions of finite valency do not matter. Adaptation to relations of higher arity is possible. Injective function f allows definability of + and ·. In FO-Th(ω2, <, R) with R of finite valency + and · are definable.

• 2. Questions:

Are there natural examples of recursive R with FO-Th(N, <, R)? What about FO-Th(N, <, fP), FO-Th(, N, <, ⊥), FO-Th(N, +, ⊥) ? Are there applications in model-checking?

Wolfgang Thomas

Summary

Composition of FO-types is possible for segments of N if the order < is expanded by a binary relation R of finite valency. In this case, the structure (N, <, R) is ultimately periodic when analyzing it w.r.t. fixed quantifier-depth. This excludes definability of + and ·. Any FO-sentence is expressible as a Boolean combination

• f Σ3-sentences (with bounded kernel).

Wolfgang Thomas