Modulo Counting on Words and Trees (joint work with Witold Charatonik) Bartosz Bednarczyk bartosz.bednarczyk@ens-paris-saclay.fr ´ Ecole normale sup´ erieure Paris-Saclay and University of Wrocław FSTTCS 2017 Kanpur, December 13, 2017
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Agenda � Classical results on FO 2 and related logics � Logics on restricted classes of structures (words and trees) � The main results of the paper � namely the exact complexity of nice family of tree logics � able to handle modulo constraints (like parity) � with relatively small complexity blowup � Proof ideas � Our current research and open problems 2 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Historical results 3 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Facts about SAT and FO 2 on arbitrary structures � We are interested in finite satisfiability problems � Models = purely relational structures, no constants, no functions 4 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Facts about SAT and FO 2 on arbitrary structures � We are interested in finite satisfiability problems � Models = purely relational structures, no constants, no functions � Some classical results: � FO undecidable (Church, Turing; 1930s) 4 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Facts about SAT and FO 2 on arbitrary structures � We are interested in finite satisfiability problems � Models = purely relational structures, no constants, no functions � Some classical results: � FO undecidable (Church, Turing; 1930s) � FO 3 undecidable (Kahr, Moore, Wang; 1959) 4 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Facts about SAT and FO 2 on arbitrary structures � We are interested in finite satisfiability problems � Models = purely relational structures, no constants, no functions � Some classical results: � FO undecidable (Church, Turing; 1930s) � FO 3 undecidable (Kahr, Moore, Wang; 1959) � FO 2 decidable (Mortimer; 1975) � FO 2 enjoys exponential model property (Gradel, Kolaitis, Vardi; 1997) - NE XP T IME -completeness 4 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Facts about SAT and FO 2 on arbitrary structures � We are interested in finite satisfiability problems � Models = purely relational structures, no constants, no functions � Some classical results: � FO undecidable (Church, Turing; 1930s) � FO 3 undecidable (Kahr, Moore, Wang; 1959) � FO 2 decidable (Mortimer; 1975) � FO 2 enjoys exponential model property (Gradel, Kolaitis, Vardi; 1997) - NE XP T IME -completeness � Connection between FO 2 and modal, temporal, description logics; � many applications in verification and databases 4 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Facts about SAT and FO 2 on arbitrary structures � We are interested in finite satisfiability problems � Models = purely relational structures, no constants, no functions � Some classical results: � FO undecidable (Church, Turing; 1930s) � FO 3 undecidable (Kahr, Moore, Wang; 1959) � FO 2 decidable (Mortimer; 1975) � FO 2 enjoys exponential model property (Gradel, Kolaitis, Vardi; 1997) - NE XP T IME -completeness � Connection between FO 2 and modal, temporal, description logics; � many applications in verification and databases Example formula: from each element there exists a path of length 3 ∀ x ∃ y ( E ( x , y ) ∧ ∃ x ( E ( y , x ) ∧ ∃ y E ( x , y ))) 4 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Facts about SAT and FO 2 on arbitrary structures � We are interested in finite satisfiability problems � Models = purely relational structures, no constants, no functions � Some classical results: � FO undecidable (Church, Turing; 1930s) � FO 3 undecidable (Kahr, Moore, Wang; 1959) � FO 2 decidable (Mortimer; 1975) � FO 2 enjoys exponential model property (Gradel, Kolaitis, Vardi; 1997) - NE XP T IME -completeness � Connection between FO 2 and modal, temporal, description logics; � many applications in verification and databases Example formula: from each element there exists a path of length 3 ∀ x ∃ y ( E ( x , y ) ∧ ∃ x ( E ( y , x ) ∧ ∃ y E ( x , y ))) Conclusion: FO 2 decidable, but limited in terms of expressivity. 4 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Logics on trees 5 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Possible variations There are several scenarios which may influence decidability/complexity. E.g., we may consider: 6 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Possible variations There are several scenarios which may influence decidability/complexity. E.g., we may consider: � Ordered vs Unordered trees 6 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Possible variations There are several scenarios which may influence decidability/complexity. E.g., we may consider: � Ordered vs Unordered trees � Ranked vs Unranked trees 6 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Possible variations There are several scenarios which may influence decidability/complexity. E.g., we may consider: � Ordered vs Unordered trees � Ranked vs Unranked trees � Finite vs Infinite trees 6 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Possible variations There are several scenarios which may influence decidability/complexity. E.g., we may consider: � Ordered vs Unordered trees � Ranked vs Unranked trees � Finite vs Infinite trees � With unary alphabet restriction (UAR) or without UAR � precisely one unary predicate holds at each node � . . . 6 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Possible variations There are several scenarios which may influence decidability/complexity. E.g., we may consider: � Ordered vs Unordered trees � Ranked vs Unranked trees � Finite vs Infinite trees � With unary alphabet restriction (UAR) or without UAR � precisely one unary predicate holds at each node � . . . We will focus on Finite, Ordered, Unranked Trees, where multiple predicates can hold at one node (without UAR). 6 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Tree notions 7 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Signature τ = τ 0 ∪ τ nav � τ 0 – unary symbols (usually P , Q , etc.) � τ nav – navigational binary symbols with fixed interpretation � words: ≤ (order over positions), + 1 (it’s induced successor) � unordered trees: ↓ (child), ↓ + (descendant, TC of ↓ ) � ordered trees: ↓ , ↓ + , → (next sibling), → + (TC of → ) A word: P , Q Q P , Q P P P a c e g b d f An unordered tree: P An ordered tree: P P Q P Q P , Q P , Q P , Q P , Q 8 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Complexity results � FO is T OWER -complete, even for FO 3 (Stockmeyer; 1974). 9 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Complexity results � FO is T OWER -complete, even for FO 3 (Stockmeyer; 1974). � FO 2 [ ≤ , + 1 ] on finite words � FO 2 is NE XP T IME -complete (Etessami et al, LICS 1997) � Equally expressive to Unary Temporal Logic � FO 2 + ∃ ≤ k + ∃ ≥ k still in NE XP T IME (Charatonik et al, CSL 2015) 9 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Complexity results � FO is T OWER -complete, even for FO 3 (Stockmeyer; 1974). � FO 2 [ ≤ , + 1 ] on finite words � FO 2 is NE XP T IME -complete (Etessami et al, LICS 1997) � Equally expressive to Unary Temporal Logic � FO 2 + ∃ ≤ k + ∃ ≥ k still in NE XP T IME (Charatonik et al, CSL 2015) � FO 2 [ ↓ , ↓ + , → , → + ] on finite trees � FO 2 on trees is E XP S PACE -complete (Benaim et al, ICALP 2013). � Equally expressive to Navigational XPath (Marx et al, 2004). � FO 2 + ∃ ≤ k + ∃ ≥ k still in E XP S PACE (Bednarczyk et al, CSL 2017) 9 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions 2-E XP T IME E XP S PACE FO 2 [ ↓ + ] , FO 2 [ ↓ , ↓ + , → , → + ] NE XP T IME FO 2 [ ≤ , + 1 ] FO 2 [] 10 / 33
Introduction Tree structures Our contribution Why modulo? Lower bound Upper bound Conclusions Our results 11 / 33
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