The C 2 -theory of the subtrace order Dietrich Kuske Technische - - PowerPoint PPT Presentation

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The C 2 -theory of the subtrace order Dietrich Kuske Technische - - PowerPoint PPT Presentation

Dec 11, 2022 173 likes •406 views

The C 2 -theory of the subtrace order Dietrich Kuske Technische Universit at Ilmenau 1 / 14 Definition u is a subword of v iff u results from v by dropping some letters. We write u v and call v a superword of u . Examples bbabb

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The C 2-theory of the subtrace order

Dietrich Kuske

Technische Universit¨ at Ilmenau

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Definition

u is a subword of v iff u results from v by dropping some letters. We write u Ď v and call v a superword of u.

Examples

bbabb “ ✁

❆

ab✁

❆

aba✁

❆

abb Ď ababaabb, u Ď u, and ε “✚

❩

u Ď u

Relevance

‚ term rewriting (special form of lexicographic path order) ‚ verification of infinite state systems (lossy channel systems, asynchronous cellular machines)

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ε a b aa ab ba bb aaa aab aba baa abb bab bba bbb

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ε a b aa ab ba bb aaa aab aba baa abb bab bba bbb no semilattice, wqo (Higman ’52), arbitrarily long finite antichains

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ε a b aa ab ba bb aaa aab aba baa abb bab bba bbb @y : x Ď y satisfied ð ñ x “ ε

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ε a b aa ab ba bb aaa aab aba baa abb bab bba bbb @y, z : y, z Ď x Ñ y Ď z _ z Ď y satisfied ð ñ x P a˚ Y b˚

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ε a b aa ab ba bb aaa aab aba baa abb bab bba bbb x Ĺ y “ px Ď y ^ y Ď xq satisfied ð ñ x is proper subword of y

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ε a b aa ab ba bb aaa aab aba baa abb bab bba bbb x ¨ z “ px Ĺ z ^ Dy : x Ĺ y Ĺ zq satisfied ð ñ z is upper cover/neighbor of x

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ε a b aa ab ba bb aaa aab aba baa abb bab bba bbb b Ď x ^ Dx2 : px2 Ĺ x ^ Dx1 : x2 Ĺ x1 Ĺ xq ^ x2 P paaq˚ satisfied ð ñ x contains precisely one b and an even number of a

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ε a b aa ab ba bb aaa aab aba baa abb bab bba bbb The words a and b satisfy the same formulas

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ε a b aa ab ba bb aaa aab aba baa abb bab bba bbb The words a and b satisfy the same formulas (similarly for ab vs. ba and aba vs. bab etc.)

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General question

What classes of properties of the subword order are decidable?

The structure

pA˚, Ď, REGq consists of ‚ universe A˚, ‚ binary relation Ď, and ‚ a predicate K for every regular language K Ď A˚.

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General question

What classes of properties of the subword order are decidable?

formulas of FO

(x1, x2 variables, K Ď A˚ regular language) ‚ x1 Ď x2, x1 P K ‚ ϕ, ϕ _ ψ, etc. ‚ Dx1 : ϕ, @x1 : ϕ Σ1: Dy1 . . . Dyn : ψ with ψ quantifier-free Σ2: Dx1 . . . Dxn : ψ with ψ P Σ1 FOk: only k variable names Ck: only k variable names + quantifiers Děn

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Some results

  • 1. Σ1-theory of pA˚, Ďq decidable (K ’06)
  • 2. Σ1-theory of pA˚, Ď, SINGLETONq undecidable

(Halfon, Schnoebelen, Zetzsche ’17)

  • 3. Σ2- and FO3-theory of pA˚, Ďq undecidable

(Karandikar, Schnoebelen ’15)

  • 4. FO2-theory of pA˚, Ď, REGq decidable

(Karandikar, Schnoebelen ’15)

  • 5. C2-theory of pA˚, Ď, REGq decidable

(K, Zetzsche ’19)

  • 6. C2-theory of pA˚, Ď, PTq in 2EXPSPACE

(K, Schwarz ’20)

main ingredient in proofs: preservation of regularity under rational transductions Posets, semilattices, lattices, graphs etc. have been considered under substructure relation. In these settings, rational transductions are not available. Here: Mazurkiewicz traces

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Mazurkiewicz traces

words model execution of a single process (alphabet A = permitted actions) now: fixed finite set of resources actions (= letters) use associated resources

Example

letter a b c d resources r1, r2 r2, r3 r3 r1 a b c b a d i.e., traces are (certain) labeled directed acyclic graphs words = traces with a single resource

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Subtrace = induced subgraph

Example

a b c b a d a b d a c b d

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Idea for solution

Instead of regular languages and rational relations, base proofs on logical descriptions and interpretations. These logical descriptions and interpretations use the logic MSO that talks about the internal structure of a trace and is equally expressive as finite automata for traces (i.a., “REG “ MSO”).

Example

a b c b a d satisfies ‚ Dv : λpvq “ a ‚ @v : λpvq “ d Ñ Dw : λpwq “ a ^ Epv, wq ‚ @v : λpvq “ c Ñ Dw : λpwq “ a ^ E ˚pv, wq

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C2-theory of pMpA, Dq, Ď, MSOq

‚ 2 variables x and y ‚ quantifiers Děk for k P N8 ‚ atomic formulas x Ď y and x | ù ϕ for ϕ P MSO

Theorem

The C2-theory of pMpA, Dq, Ď, MSOq has effective quantifier elimination. Proof idea: by structural induction on ϕ (as usual) central task: Eliminate Děk from ϕ “ Děk y : ψpx, yq where ψpx, yq is a Boolean combination of formulas x Ď y, y Ď x, x | ù α, and y | ù α with α MSO-formula!

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deMorgan laws, basic arithmetic etc: ϕ is effectively equivalent to a Boolean combination of formulas x | ù β Děℓy : x Ĺ y ^ y | ù β Děℓy : : x Ľ y ^ y | ù β Děℓy : px Ď y _ y Ď xq ^ y | ù β with ℓ P N8 and β MSO-formula. remaining task: from β P MSO and ℓ P N construct γ P MSO such that t | ù γ ð ñ Děℓs : s | ù β ^ t Ĺ s t | ù γ ð ñ Děℓs : s | ù β ^ t Ľ s t | ù γ ð ñ Děℓs : s | ù β ^ t Ę s ^ |t| ă |s| t | ù γ ð ñ Děℓs : s | ù β ^ t Ğ s ^ |t| ě |s|

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Let β be some sentence from MSO. (1) formula for “there exists a trace s Ď t with s | ù β”: DX : βæX (2) formula for “there exist distinct traces s1, s2 Ď t with si | ù β”: DX1, X2 : βæX1 ^ βæX2 ^ X1 ‰ X2 ^ X1 and X2 are ”leftmost” a b c b a d

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(3) formula for “there exists a trace s Ě t with s | ù β” uses standard results from theory of recognizable trace languages and their relation to MSO (4) this approach can be extended to threshold counting

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(5) formula for “there exists a trace s t with s | ù β” cases |s| ď |t| and |s| ą |t| handled separately, iff there are traces s1, s2 s.t.

(a) s1 Ď t, (b) @b P minps2q: s1b Ę t, (c) |s2| ą |t| ´ |s1|, and (d) s1s2 | ù β.

iff there are traces s1, s2 s.t. (a-c) and

(d’) there is 1 ď i ď n with s1 | ù µi and s2 | ù νi for some computable family pµi, νiq1ďiďn (Shelah ’79)

iff for some 1 ď i ď n and B Ď A: “there is s1 Ď t s.t. s1 | ù µi and s1b Ę t f.a. b P B” and νi ^ min “ B is satisfiable by some “long” trace (decidable by W. Thomas ’90). (6) this approach can be modified for other case and extended to threshold counting

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Theorem

The C2-theory of pMpA, Dq, Ď, MSOq “ pMpA, Dq, Ď, REGq has effective quantifier elimination (uniformly in pA, Dq with non-elementary size increase) and is therefore decidable.

Open questions

‚ modulo counting for traces, in particular: incomparable traces (for words see K & Zetzsche ’19) ‚ complexity (K & Schwarz ’20: for pA˚, Ď, PTq in 2EXPSPACE) ‚ other structures (graphs, trees, message sequence charts, . . .) where rational relations are not available some proof ideas explained: http://eiche.theoinf.tu-ilmenau.de/kuske/csr20-proofs

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