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

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 Ď v and call v a superword of u . Examples bbabb “ ✁ abb Ď ababaabb , u Ď u , and ε “ ✚ ❩ ❆ ab ✁ ❆ aba ✁ ❆ u Ď u Relevance ‚ term rewriting (special form of lexicographic path order) ‚ verification of infinite state systems (lossy channel systems, asynchronous cellular machines) 2 / 14

aaa aab aba baa abb bab bba bbb aa ab ba bb a b ε 3 / 14

aaa aab aba baa abb bab bba bbb aa ab ba bb a b ε no semilattice, wqo (Higman ’52), arbitrarily long finite antichains 3 / 14

aaa aab aba baa abb bab bba bbb aa ab ba bb a b ε @ y : x Ď y satisfied ð ñ x “ ε 3 / 14

aaa aab aba baa abb bab bba bbb aa ab ba bb a b ε ñ x P a ˚ Y b ˚ @ y , z : y , z Ď x Ñ y Ď z _ z Ď y satisfied ð 3 / 14

aaa aab aba baa abb bab bba bbb aa ab ba bb a b ε x Ĺ y “ p x Ď y ^ � y Ď x q satisfied ð ñ x is proper subword of y 3 / 14

aaa aab aba baa abb bab bba bbb aa ab ba bb a b ε x � ¨ z “ p x Ĺ z ^ �D y : x Ĺ y Ĺ z q satisfied ð ñ z is upper cover/neighbor of x 3 / 14

aaa aab aba baa abb bab bba bbb aa ab ba bb a b ε b Ď x ^ D x 2 : p x 2 Ĺ x ^ �D x 1 : x 2 Ĺ x 1 Ĺ x q ^ x 2 P p aa q ˚ satisfied ð ñ x contains precisely one b and an even number of a 3 / 14

aaa aab aba baa abb bab bba bbb aa ab ba bb a b ε The words a and b satisfy the same formulas 3 / 14

aaa aab aba baa abb bab bba bbb aa ab ba bb a b ε The words a and b satisfy the same formulas (similarly for ab vs. ba and aba vs. bab etc.) 3 / 14

General question What classes of properties of the subword order are decidable? The structure p A ˚ , Ď , REG q consists of ‚ universe A ˚ , ‚ binary relation Ď , and ‚ a predicate K for every regular language K Ď A ˚ . 4 / 14

General question What classes of properties of the subword order are decidable? formulas of FO ( x 1 , x 2 variables, Σ 1 : D y 1 . . . D y n : ψ with ψ quantifier-free K Ď A ˚ regular language) Σ 2 : D x 1 . . . D x n : � ψ with ψ P Σ 1 FO k : only k variable names ‚ x 1 Ď x 2 , x 1 P K C k : only k variable names ‚ � ϕ , ϕ _ ψ , etc. + quantifiers D ě n ‚ D x 1 : ϕ , @ x 1 : ϕ 4 / 14

Some results 1. Σ 1 -theory of p A ˚ , Ď q decidable (K ’06) 2. Σ 1 -theory of p A ˚ , Ď , SINGLETON q undecidable (Halfon, Schnoebelen, Zetzsche ’17) 3. Σ 2 - and FO 3 -theory of p A ˚ , Ď q undecidable (Karandikar, Schnoebelen ’15) 4. FO 2 -theory of p A ˚ , Ď , REG q decidable (Karandikar, Schnoebelen ’15) 5. C 2 -theory of p A ˚ , Ď , REG q decidable (K, Zetzsche ’19) 6. C 2 -theory of p A ˚ , Ď , PT q 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 5 / 14

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 r 1 , r 2 r 2 , r 3 r 3 r 1 c b b a a d i.e., traces are (certain) labeled directed acyclic graphs words = traces with a single resource 6 / 14

Subtrace = induced subgraph Example c b b a a d c b b a a d d 7 / 14

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 c b b a a d satisfies ‚ D v : λ p v q “ a ‚ @ v : λ p v q “ d Ñ D w : λ p w q “ a ^ E p v , w q ‚ @ v : λ p v q “ c Ñ D w : λ p w q “ a ^ E ˚ p v , w q 8 / 14

C 2 -theory of p M p A , D q , Ď , MSO q ‚ 2 variables x and y ‚ quantifiers D ě k for k P N 8 ‚ atomic formulas x Ď y and x | ù ϕ for ϕ P MSO Theorem The C 2 -theory of p M p A , D q , Ď , MSO q has effective quantifier elimination. Proof idea: by structural induction on ϕ (as usual) central task: Eliminate D ě k from ϕ “ D ě k y : ψ p x , y q where ψ p x , y q is a Boolean combination of formulas x Ď y , y Ď x , x | ù α , and y | ù α with α MSO-formula! 9 / 14

deMorgan laws, basic arithmetic etc: ϕ is effectively equivalent to a Boolean combination of formulas D ě ℓ y : x Ĺ y ^ y | x | ù β ù β D ě ℓ y : : x Ľ y ^ y | D ě ℓ y : �p x Ď y _ y Ď x q ^ y | ù β ù β with ℓ P N 8 and β MSO-formula. remaining task: from β P MSO and ℓ P N construct γ P MSO such that D ě ℓ s : s | t | ù γ ð ñ ù β ^ t Ĺ s D ě ℓ s : s | t | ù γ ð ñ ù β ^ t Ľ s D ě ℓ s : s | t | ù γ ð ñ ù β ^ t Ę s ^ | t | ă | s | D ě ℓ s : s | t | ù γ ð ñ ù β ^ t Ğ s ^ | t | ě | s | result 10 / 14

Let β be some sentence from MSO. (1) formula for “there exists a trace s Ď t with s | ù β ”: D X : β æ X (2) formula for “there exist distinct traces s 1 , s 2 Ď t with s i | ù β ”: D X 1 , X 2 : β æ X 1 ^ β æ X 2 ^ X 1 ‰ X 2 ^ X 1 and X 2 are ”leftmost” c b b a a d 11 / 14

(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 12 / 14

(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 s 1 , s 2 s.t. (a) s 1 Ď t , (b) @ b P min p s 2 q : s 1 b Ę t , (c) | s 2 | ą | t | ´ | s 1 | , and (d) s 1 s 2 | ù β . iff there are traces s 1 , s 2 s.t. (a-c) and (d’) there is 1 ď i ď n with s 1 | ù µ i and s 2 | ù ν i for some computable family p µ i , ν i q 1 ď i ď n (Shelah ’79) iff for some 1 ď i ď n and B Ď A : “there is s 1 Ď t s.t. s 1 | ù µ i and s 1 b Ę t f.a. b P B ” and ν i ^ 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 13 / 14

Theorem The C 2 -theory of p M p A , D q , Ď , MSO q “ p M p A , D q , Ď , REG q has effective quantifier elimination (uniformly in p A , D q 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 p A ˚ , Ď , PT q 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 14 / 14