Extensions of S1S and the Composition Method Giovanna DAgostino and - - PowerPoint PPT Presentation

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions

Extensions of S1S and the Composition Method

Giovanna D’Agostino and Angelo Montanari and Alberto Policriti

Department of Mathematics and Computer Science University of Udine, Italy {dagostin,montana,policrit}@dimi.uniud.it

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions

1

The composition method MSO logic over expansions of S1S The notion of ¯ k-type Composing ¯ k-types

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Expansions with unary predicates Ultimately type-periodic words Morphic words

3

Expansions with binary predicates Morphic pictures Linear morphic pictures

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions MSO logic over expansions of S1S

Let fix a signature Σ = {r1, ..., rn}, where each ri is a relational symbol with arity hi. Definition (Monadic Second-Order (MSO) Logic) We define (a variant) of MSO logic as follows: we have set-variables X, Y , Z, ... only (no first-order variables are used) atomic formulas are of the form

X1 ⊆ X2 ‘X1 is a subset of X2’ ri(X1, ..., Xhi) ‘ri(x1, ..., xhi) for some x1 ∈ X1, ..., xhi ∈ Xhi’ NE(X1, X2) ‘X1 intersects X2’ ALL(X1, ..., Xh) ‘the union of X1, ..., Xh is the universe’

more complex formulas are build up via

the Boolean connectives ∧ , ∨ , ¬ quantifications ∃X, ∀X over set-variables.

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions MSO logic over expansions of S1S

Hereafter, we assume that Σ contains at least a binary relation <. We shall evaluate a given MSO-formula ϕ(X1, ..., Xm) with m free variables (parameters) over a linearly ordered structure

• S, P1, ..., Pm
• where

Dom(S) is either N or an initial segment {0, ..., i} < is the usual ordering of the natural numbers Pi is an interpretation for the parameter Xi.

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions MSO logic over expansions of S1S

Hereafter, we assume that Σ contains at least a binary relation <. We shall evaluate a given MSO-formula ϕ(X1, ..., Xm) with m free variables (parameters) over a linearly ordered structure

• S, P1, ..., Pm
• where

Dom(S) is either N or an initial segment {0, ..., i} < is the usual ordering of the natural numbers Pi is an interpretation for the parameter Xi. Model checking problem We want to find linear structures (S, ¯ P) with decidable MSO-theories, namely, effective procedures to decide, for any given MSO-formula ϕ( ¯ X), whether (S, ¯ P)

• ϕ( ¯

X)

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions The notion of ¯ k-type

We now introduce ¯ k-types to tackle the model checking problem... Proposition Every MSO-formula ϕ(¯ X) can be written in prenex normal form as Qt ¯ Yt ... Q1 ¯ Y1 ψ( ¯ X, ¯ Y1, ..., ¯ Yt), where each Qi is either ∀ or ∃ and each ¯ Yi is a ki-tuple of MSO-variables. Definition (Complexity of an MSO-formula) We define the complexity of an MSO-formula in prenex normal form Qt ¯ Yt ... Q1 ¯ Y1 ψ( ¯ X, ¯ Y1, ..., ¯ Yt) as the tuple ¯ k = (k1, ..., kt). Definition (¯ k-formula) A ¯ k-formula is any MSO-formula with complexity ¯ k.

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions The notion of ¯ k-type

Let fix a number m of parameters and a complexity ¯ k. Definition (Type) The ¯ k-type of a structure (S, ¯ P) is the set of all ¯ k-formulas ϕ( ¯ X) that hold in (S, ¯ P) when Xi is interpreted as Pi.

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions The notion of ¯ k-type

Let fix a number m of parameters and a complexity ¯ k. Definition (Type) The ¯ k-type of a structure (S, ¯ P) is the set of all ¯ k-formulas ϕ( ¯ X) that hold in (S, ¯ P) when Xi is interpreted as Pi. We can inductively define a finite object

• (S, ¯

P)

• ¯

k representing

the ¯ k-type of a structure (S, ¯ P) as follows:

• (S, ¯

P)

• ε =
• ϕ(¯

X) atomic :

• S, ¯

P

• ϕ(¯

X)

• (S, ¯

P)

• ¯

k.k′ =

• (S, ¯

P, ¯ Q)

• ¯

k.k′ : ¯

Q ∈ P(Dom(S))k′

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions The notion of ¯ k-type

Lemma Given a ¯ k-formula ϕ( ¯ X) and the ¯ k-type

• (S, ¯

P)

• ¯

k of a structure

(S, ¯ P), one can effectively establish whether (S, ¯ P) ϕ( ¯ X). Proposition The MSO-theory of a structure (S, ¯ P) is decidable iff there is a computable function f that maps a complexity ¯ k to the corresponding ¯ k-type

• (S, ¯

P)

• ¯

k.

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions The notion of ¯ k-type

Lemma Given a ¯ k-formula ϕ( ¯ X) and the ¯ k-type

• (S, ¯

P)

• ¯

k of a structure

(S, ¯ P), one can effectively establish whether (S, ¯ P) ϕ( ¯ X). Proposition The MSO-theory of a structure (S, ¯ P) is decidable iff there is a computable function f that maps a complexity ¯ k to the corresponding ¯ k-type

• (S, ¯

P)

• ¯

k.

Note: computing ¯ k-types of finite structures is easy ... How about ¯ k-types of infinite (expanded) structures?

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Composing ¯ k-types

Idea: use composition method, which allows one to compute the type of a compound structure from the types of its components. Definition (Ordered sum) Let I be an index ordering (e.g., (N, <)) and let

• Si, ¯

Pi

• i∈I

be a sequence of expanded linear orders. The ordered sum

i∈I(Si, ¯

Pi) is the disjoint union of the structures (Si, ¯ Pi

• , where i ranges over I.

(the ordering relation of

i∈I(Si, ¯

Pi) is implicitly given by the

• rdering relations of I and (Si, ¯

Pi

• ).

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Composing ¯ k-types

Composition Theorem - Shelah ’75 Let ¯ k be a complexity and m a number of parameters. We denote by {σ1, ..., σs} the set of all ¯ k-types on m parameters. One can compute a complexity ¯ r such that for any index ordering I for any sequence

• Si, ¯

Pi

• i∈I of expanded linear orders

the ¯ k-type of the ordered sum

i∈I(Si, ¯

Pi) is uniquely determined by (and computable from) the ¯ r-type of the expanded index ordering (I, Q1, ..., Qs) s.t. i ∈ Qj iff [Si]¯

k = σj.

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Composing ¯ k-types

Composition Theorem - Shelah ’75 Let ¯ k be a complexity and m a number of parameters. We denote by {σ1, ..., σs} the set of all ¯ k-types on m parameters. One can compute a complexity ¯ r such that for any index ordering I for any sequence

• Si, ¯

Pi

• i∈I of expanded linear orders

the ¯ k-type of the ordered sum

i∈I(Si, ¯

Pi) is uniquely determined by (and computable from) the ¯ r-type of the expanded index ordering (I, Q1, ..., Qs) s.t. i ∈ Qj iff [Si]¯

k = σj.

Example The ¯ k-type of the ordered sum (S1, ¯ P1) + (S2, ¯ P2) can be computed from the ¯ k-types of the two structures (S1, ¯ P1) and (S2, ¯ P2).

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Ultimately type-periodic words

Now, we briefly present some decidability results for infinite linear

• rders expanded with unary predicates.

Hereafter, we identify a linear order expanded with m unary predicates with its characteristic word w over Bm = {0, 1}m.

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Ultimately type-periodic words

Now, we briefly present some decidability results for infinite linear

• rders expanded with unary predicates.

Hereafter, we identify a linear order expanded with m unary predicates with its characteristic word w over Bm = {0, 1}m. Definition (Ultimately type-periodic words) A sequence (wi)i∈N of finite words is ultimately type-periodic if, for every complexity ¯ k, one can compute p, q such that for all i ≥ p, [wi]¯

k = [wi+q]¯ k.

An infinite word w is ultimately type-periodic if there is an ultimately type-period factorization w = w0 · w1 · w2 · ....

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Ultimately type-periodic words

Theorem Any ultimately type-periodic word has a decidable MSO-theory.

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Ultimately type-periodic words

Theorem Any ultimately type-periodic word has a decidable MSO-theory. Proof Let w = w0 ·w1 ·w2 ·... and let p, q satisfy ∀ i ≥ p. [wi]¯

k = [wi+q]¯ k.

Each wi is a finite word ⇒ one can compute its ¯ k-type [wi]¯

k.

By Composition Theorem, one can also compute a suitable complexity ¯ r such that [w]¯

k is uniquely determined by the

¯ r-type of the expanded index ordering (N, <, Q1, ..., Qs), where Qj = {i ∈ N : [wi]¯

k is the j-th ¯

k-type σj}. Since (N, <, Q1, ..., Qs) is ultimately periodic, it has a decidable MSO-theory and its ¯ r-type can be effectively computed. Note: the class of ultimately type-periodic words coincides with the class of profinitely/residually ultimately periodic words (see Carton and Thomas 2002, Rabinovich 2006).

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Morphic words

Noticeable examples of ultimately type-periodic predicates are generated via repeated applications of morphisms.

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Morphic words

Noticeable examples of ultimately type-periodic predicates are generated via repeated applications of morphisms. Example Consider the morphism τ : B → B∗ defined by τ(0) = 01 τ(1) = 10 and the sequence of finite words τ 1(1) = 1 0, τ 2(1) = 10 01, τ 3(1) = 10 01 01 10, ... The sequence converges to an infinite word τ ω(1) = 100101100110... which encodes the ultimately type-periodic predicate PThue−Morse

• f all i whose binary expansions contain an even number of 1’s.

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Morphic words

Definition (Morphic word) Let τ : A → A∗ be a morphism and let a ∈ A satisfy τ(a) = a · u. The morphic word generated by τ from a is the limit of the infinite sequence of increasing words τ 0(a) = a τ 1(a) = a · u τ 2(a) = a · u · τ(u) τ 3(a) = a · u · τ(u) · τ 2(u) ⇒ τ ω(a) = a · u · τ(u) · τ 2(u) · ...

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Morphic words

Definition (Morphic word) Let τ : A → A∗ be a morphism and let a ∈ A satisfy τ(a) = a · u. The morphic word generated by τ from a is the limit of the infinite sequence of increasing words τ 0(a) = a τ 1(a) = a · u τ 2(a) = a · u · τ(u) τ 3(a) = a · u · τ(u) · τ 2(u) ⇒ τ ω(a) = a · u · τ(u) · τ 2(u) · ... Theorem (Carton and Thomas ’00) Any morphic word τ ω(a) is profinitely ultimately periodic (hence ultimately type-periodic).

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Morphic pictures

How about linear orders expanded with binary predicates? Note: a linear order expanded with a binary predicate is now identified with its characteristic picture.

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Morphic pictures

How about linear orders expanded with binary predicates? Note: a linear order expanded with a binary predicate is now identified with its characteristic picture. Example The expanded linear order (N, <, 2×) is encoded by 1 · · · 1 1 1 . . . ...

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Morphic pictures

We focus our attention on morphic pictures (see Maes 1999). (Here we adopt a simplified setting where picture morphisms are functions from A to A-squares of the same dimension.) Definition (Morphic picture) Let τ : A → Ad×d be a picture morphism and assume that for some a ∈ A, the symbol at top-left position of τ(a) is a. The morphic picture τ ω(a) generated by τ from a is the limit of the infinite sequence of increasing pictures τ 0(a), τ 1(a), τ 2(a), ...

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Morphic pictures

Example Consider the morphism τ : B → B2×2 defined by τ(0) = 0 τ(1) = 1 1 1 and the sequence of pictures τ 1(1) =

1 1 1 0 , τ 2(1) =

1 1 1 1 1 1 1 1 1

, τ 3(1) =

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

, ... The sequence converges to an infinite picture τ ω(1), which encodes the binary predicate PPascal−triangle ⊆ N2.

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Morphic pictures

Proposition (Korec 1997) The first-order theory of (N, +, ×) can be interpreted in the weak MSO-theory of (N, <, PPascal−triangle).

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Morphic pictures

Proposition (Korec 1997) The first-order theory of (N, +, ×) can be interpreted in the weak MSO-theory of (N, <, PPascal−triangle). ⇒ this shows that there are morphic pictures with undecidable MSO-theories (Maes results about decidability of first-order theories of morphic pictures do not lift trivially to MSO). We need to enforce stronger conditions to achieve decidability for MSO-theories of morphic pictures...

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Linear morphic pictures

Definition (Linearity conditions) Let τ : A → Ad×d be a picture morphism and let a1, ..., ad be the symbols on the diagonal of τ(a1). The morphic picture τ ω(a1) is said to be linear if for any complexity ¯ k, one can compute ¯ r such that the ¯ k-type of the morphic picture τ ω(a1) only depends on (and can be computed from) the ¯ r-type

• f the infinite word σ1,1 σ1,2...σ1,d σ2,2...σ2,d ...,

where σi,j =

• τ i(aj)
• ¯

k for all i > 0, 0 < j ≤ d

the ¯ k-type σi+1,j is uniquely determined by the ¯ k-type σi,j.

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Linear morphic pictures

Definition (Linearity conditions) Let τ : A → Ad×d be a picture morphism and let a1, ..., ad be the symbols on the diagonal of τ(a1). The morphic picture τ ω(a1) is said to be linear if for any complexity ¯ k, one can compute ¯ r such that the ¯ k-type of the morphic picture τ ω(a1) only depends on (and can be computed from) the ¯ r-type

• f the infinite word σ1,1 σ1,2...σ1,d σ2,2...σ2,d ...,

where σi,j =

• τ i(aj)
• ¯

k for all i > 0, 0 < j ≤ d

the ¯ k-type σi+1,j is uniquely determined by the ¯ k-type σi,j. Note: the second condition enforces that the infinite word σ1,1 σ1,2...σ1,d σ2,2...σ2,d ... is ultimately periodic ⇒ its ¯ r-type can be effectively computed given τ and ¯ k.

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Linear morphic pictures

Moreover, the first linearity condition implies that the ¯ k-type

• f the picture τ ω(a1) can be effectively computed from the ¯

r-type

• f the infinite word σ1,1 σ1,2...σ1,d σ2,2...σ2,d ...

Theorem Any linear morphic picture has a decidable MSO-theory.

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Linear morphic pictures

Moreover, the first linearity condition implies that the ¯ k-type

• f the picture τ ω(a1) can be effectively computed from the ¯

r-type

• f the infinite word σ1,1 σ1,2...σ1,d σ2,2...σ2,d ...

Theorem Any linear morphic picture has a decidable MSO-theory. What about examples of linear morphic pictures?

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Linear morphic pictures

Example Consider the morphism τ : {a, b, c} → {a, b, c}2×2 defined by τ(a) = a b c a τ(b) = b c c c τ(c) = c c c c and the sequence of pictures τ 1(a) =

a b c a , τ 2(a) =

a b b c c a c c c c a b c c c a

, τ 3(a) =

a b b c b c c c c a c c c c c c c c a b c c c c c c c a c c c c c c c c a b b c c c c c c a c c c c c c c c a b c c c c c c c a

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Linear morphic pictures

Example Consider the morphism τ : {a, b, c} → {a, b, c}2×2 defined by τ(a) = a b c a τ(b) = b c c c τ(c) = c c c c and the sequence of pictures τ 1(a) =

a b c a , τ 2(a) =

a b b c c a c c c c a b c c c a

, τ 3(a) =

a b b c b c c c c a c c c c c c c c a b c c c c c c c a c c c c c c c c a b b c c c c c c a c c c c c c c c a b c c c c c c c a

The limit of the sequence encodes the binary predicate Pflip ⊆ N2:

r r r r r r r r r r r r r r r r r ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ☛ ✟ ☛ ✟ ☛ ✟ ☛ ✟ ☛ ✟ ☛ ✟ ☛ ✟ ☛ ✟ ☛ ✟ ✛ ✛ ✛ ✛ ✗ ✔ ✗ ✔ ✗ ✔ ✗ ✔ ✛ ✛ ✬ ✩ ✬ ✩ ✛ ✬ ✩ ✛ ✬ ✩

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Linear morphic pictures

Theorem The binary predicate Pflip is encoded by a linear morphic picture.

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Linear morphic pictures

Theorem The binary predicate Pflip is encoded by a linear morphic picture. Proof idea As for the first linearity condition, the proof follows the same lines of the proof of the Composition Theorem (the word σ1,1 σ1,2...σ1,d σ2,2...σ2,d ... plays the role of an expanded index ordering) As for the second linearity condition, one shows that τ i+1(a) is obtained from τ i(a) by copying the square τ i(a) along the diagonal and by filling in the remaining positions with c’s, except for one b to be put in position (0, 2i).

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Linear morphic pictures

Theorem The binary predicate Pflip is encoded by a linear morphic picture. Proof idea As for the first linearity condition, the proof follows the same lines of the proof of the Composition Theorem (the word σ1,1 σ1,2...σ1,d σ2,2...σ2,d ... plays the role of an expanded index ordering) As for the second linearity condition, one shows that τ i+1(a) is obtained from τ i(a) by copying the square τ i(a) along the diagonal and by filling in the remaining positions with c’s, except for one b to be put in position (0, 2i). Corollary (Montanari, Peron, Policriti 1999) The MSO-theory of (N, <, Pflip) is decidable.

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions

Summing up: we started from Maes results involving the decidability

• f FO-theories of morphic pictures, trying to achieve

analogous results for MSO logic we showed that there exist morphic pictures (e.g., Pascal triangle) with undecidable MSO-theories we introduced a proper subclass of morphic pictures (linear morphic pictures) with decidable MSO-theories we provided a concrete example of linear morphic picture (the binary predicate Pflip). (The results can be generalized to predicates of arbitrary arity.)