Nested Antichains for WS1S s Fiedor 1 , 2 k 2 Tom a Luk a s Hol - PowerPoint PPT Presentation

Nested Antichains for WS1S s Fiedor 1 , 2 ık 2 Tom´ aˇ Luk´ aˇ s Hol´ 1 Red Hat, Czech Republic al 2 s Vojnar 2 Ondˇ rej Leng´ Tom´ aˇ 2 Brno University of Technology, Czech Republic AVM’15

WS1S weak monadic second-order logic of one successor ◮ second-order ⇒ quantification over relations; ◮ monadic ⇒ relations are unary (i.e. sets); ◮ weak ⇒ sets are finite; ◮ of one successor ⇒ reasoning about linear structures. corresponds to finite automata [B¨ uchi’60] decidable T. Fiedor Nested Antichains for WS1S AVM’15 2 / 17

WS1S weak monadic second-order logic of one successor ◮ second-order ⇒ quantification over relations; ◮ monadic ⇒ relations are unary (i.e. sets); ◮ weak ⇒ sets are finite; ◮ of one successor ⇒ reasoning about linear structures. corresponds to finite automata [B¨ uchi’60] decidable — but NONELEMENTARY ◮ constructive proof via translation to finite automata T. Fiedor Nested Antichains for WS1S AVM’15 2 / 17

Application of WS1S allows one to define rich invariants famous decision procedure: the MONA tool ◮ often efficient (in practice) used in tools for checking structural invariants ◮ Pointer Assertion Logic Engine ( PALE ) ◮ STRucture ANd Data ( STRAND ) many other applications ◮ program and protocol verifications, linguistics, theorem provers . . . T. Fiedor Nested Antichains for WS1S AVM’15 3 / 17

Application of WS1S allows one to define rich invariants famous decision procedure: the MONA tool ◮ often efficient (in practice) used in tools for checking structural invariants ◮ Pointer Assertion Logic Engine ( PALE ) ◮ STRucture ANd Data ( STRAND ) many other applications ◮ program and protocol verifications, linguistics, theorem provers . . . but sometimes the complexity strikes back ◮ unavoidable in general ◮ however, we try to push the usability border further • using the recent advancements in non-deterministic automata T. Fiedor Nested Antichains for WS1S AVM’15 3 / 17

WS1S Syntax: ◮ term ψ ::= X ⊆ Y | Sing ( X ) | X = { 0 } | X = σ ( Y ) ◮ formula ϕ ::= ψ | ϕ ∧ ϕ | ϕ ∨ ϕ | ¬ ϕ | ∃ X .ϕ Interpretation: over finite subsets of N ◮ models of formulae = assignments of sets to variables sets can be encoded as binary strings: Index: 012345 0123456 01234567 ◮ { 1 , 4 , 5 } → , or . . . Membership: x � xx �� x � xx �� x x � xx �� xx Encoding: 010011 0100110 01001100 for each variable we have one track in the alphabet � 0 � ◮ e.g. is symbol 0 = ϕ def � 0 �� 0 �� 0 �� 0 �� 0 � ⇔ X 1 : Example: { X 1 �→ ∅ , X 2 �→ { 4 , 2 }} | ∈ L ( A ϕ ) X 2 : 0 0 1 0 1 T. Fiedor Nested Antichains for WS1S AVM’15 4 / 17

Deciding WS1S using deterministic automata example of base automaton for X = σ ( Y ) � 0 � � 0 � X: X: Y: 0 Y: 0 X: � 0 � X: � 1 � Y: 1 Y: 0 0 1 2 Example: ¬ ( X ⊆ Y ) ∧ ∃ Z . Sing ( Z ) ∨ ∃ W . W = σ ( Z ) T. Fiedor Nested Antichains for WS1S AVM’15 5 / 17

Deciding WS1S using deterministic automata example of base automaton for X = σ ( Y ) � 0 � � 0 � X: X: Y: 0 Y: 0 X: � 0 � X: � 1 � Y: 1 Y: 0 0 1 2 Example: ¬ ( X ⊆ Y ) ∧ ∃ Z . Sing ( Z ) ∨ ∃ W . W = σ ( Z ) A 3 A 1 A 2 T. Fiedor Nested Antichains for WS1S AVM’15 5 / 17

Deciding WS1S using deterministic automata example of base automaton for X = σ ( Y ) � 0 � � 0 � X: X: Y: 0 Y: 0 X: � 0 � X: � 1 � Y: 1 Y: 0 0 1 2 Example: ¬ ( X ⊆ Y ) ∧ ∃ Z . Sing ( Z ) ∨ ∃ W . W = σ ( Z ) A 3 A 1 A 2 A 4 project W T. Fiedor Nested Antichains for WS1S AVM’15 5 / 17

Deciding WS1S using deterministic automata example of base automaton for X = σ ( Y ) � 0 � � 0 � X: X: Y: 0 Y: 0 X: � 0 � X: � 1 � Y: 1 Y: 0 0 1 2 Example: ¬ ( X ⊆ Y ) ∧ ∃ Z . Sing ( Z ) ∨ ∃ W . W = σ ( Z ) A 3 A 1 A 2 A 4 project W A 2 ∪ A 4 T. Fiedor Nested Antichains for WS1S AVM’15 5 / 17

Deciding WS1S using deterministic automata example of base automaton for X = σ ( Y ) � 0 � � 0 � X: X: Y: 0 Y: 0 X: � 0 � X: � 1 � Y: 1 Y: 0 0 1 2 Example: ¬ ( X ⊆ Y ) ∧ ∃ Z . Sing ( Z ) ∨ ∃ W . W = σ ( Z ) A 3 A 1 A 2 A 4 project W A 2 ∪ A 4 project Z A 7 T. Fiedor Nested Antichains for WS1S AVM’15 5 / 17

Deciding WS1S using deterministic automata example of base automaton for X = σ ( Y ) � 0 � � 0 � X: X: Y: 0 Y: 0 X: � 0 � X: � 1 � Y: 1 Y: 0 0 1 2 Example: ¬ ( X ⊆ Y ) ∧ ∃ Z . Sing ( Z ) ∨ ∃ W . W = σ ( Z ) A 3 A 1 A 2 A 4 project W A 6 complement A 2 ∪ A 4 project Z A 7 T. Fiedor Nested Antichains for WS1S AVM’15 5 / 17

Deciding WS1S using deterministic automata example of base automaton for X = σ ( Y ) � 0 � � 0 � X: X: Y: 0 Y: 0 X: � 0 � X: � 1 � Y: 1 Y: 0 0 1 2 Example: ¬ ( X ⊆ Y ) ∧ ∃ Z . Sing ( Z ) ∨ ∃ W . W = σ ( Z ) A 3 A 1 A 2 A 4 project W A 6 complement A 2 ∪ A 4 project Z A 7 A 6 ∩ A 7 T. Fiedor Nested Antichains for WS1S AVM’15 5 / 17

How to handle quantification issue with projection (existential quantification) ◮ after removing of the tracks not all models would be accepted ◮ so we need to adjust the final states � 0 � � 0 � X: X: Y: 0 Y: 0 � 0 � � 1 � X: X: Y: 1 Y: 0 1 2 3 A X = σ ( Y ) T. Fiedor Nested Antichains for WS1S AVM’15 6 / 17

How to handle quantification issue with projection (existential quantification) ◮ after removing of the tracks not all models would be accepted ◮ so we need to adjust the final states � 0 � � 0 � � 0 � � 0 � X: X: X: X: Y: 0 Y: 0 Y: 0 Y: 0 � 0 � � 1 � � 0 � � 1 � X: X: X: X: Y: 1 Y: 0 Y: 1 Y: 0 1 2 3 1 2 3 A X = σ ( Y ) → Projection T. Fiedor Nested Antichains for WS1S AVM’15 6 / 17

How to handle quantification issue with projection (existential quantification) ◮ after removing of the tracks not all models would be accepted ◮ so we need to adjust the final states � 0 � � 0 � � 0 � � 0 � X: X: X: X: Y: 0 Y: 0 Y: 0 Y: 0 � 0 � � 1 � � 0 � � 1 � X: X: X: X: Y: 1 Y: 0 Y: 1 Y: 0 1 2 3 1 2 3 A X = σ ( Y ) → Projection T. Fiedor Nested Antichains for WS1S AVM’15 6 / 17

How to handle quantification issue with projection (existential quantification) ◮ after removing of the tracks not all models would be accepted ◮ so we need to adjust the final states � 0 � � 0 � � 0 � � 0 � X: X: X: X: Y: [0] Y: [0] Y: 0 Y: 0 Y: 0 Y: 0 � 0 � � 1 � � 0 � � 1 � X: X: X: X: Y: [1] Y: [0] Y: 1 Y: 0 Y: 1 Y: 0 1 2 3 1 2 3 1 2 3 → Adjust states A X = σ ( Y ) → Projection to accept models: 1, 01, 001, . . . T. Fiedor Nested Antichains for WS1S AVM’15 6 / 17

How to handle quantification issue with projection (existential quantification) ◮ after removing of the tracks not all models would be accepted ◮ so we need to adjust the final states � 0 � � 0 � � 0 � � 0 � X: X: X: X: Y: [0] Y: 0 Y: 0 Y: [0] Y: 0 Y: 0 � 0 � � 1 � � 0 � � 1 � X: X: X: X: Y: [1] Y: [0] Y: 1 Y: 0 Y: 1 Y: 0 1 2 3 1 2 3 1 2 3 → Adjust states A X = σ ( Y ) → Projection to accept models: 1, 01, 001, . . . T. Fiedor Nested Antichains for WS1S AVM’15 6 / 17

Deciding WS1S using non-deterministic automata we consider only formulae in Prenex Normal Form ( ∃ PNF) ◮ we focus on dealing with prefix and alternations of quantifications based on number of alternations m ϕ = ¬ ∃X m ¬ . . . ¬ ∃X 2 ¬ ∃X 1 : ϕ 0 ( X ) (1) � �� � ϕ 1 ... � �� � ϕ m T. Fiedor Nested Antichains for WS1S AVM’15 7 / 17

Deciding WS1S using non-deterministic automata we consider only formulae in Prenex Normal Form ( ∃ PNF) ◮ we focus on dealing with prefix and alternations of quantifications based on number of alternations m ϕ = ¬ ∃X m ¬ . . . ¬ ∃X 2 ¬ ∃X 1 : ϕ 0 ( X ) (1) � �� � ϕ 1 ... � �� � ϕ m → hierarchical family of automata defined as follows: ◮ A ϕ 0 = by composition of atomic automata (previously described) ◮ A ϕ m = ( 2 2 ··· 2 Q 0 , ∆ m , I m , F m ) � �� � m T. Fiedor Nested Antichains for WS1S AVM’15 7 / 17

The intuition behind the procedure Key observation for ground formulae ϕ | = iff I m ∩ F m � = ∅

The intuition behind the procedure Key observation for ground formulae ϕ | = iff I m ∩ F m � = ∅ Why? ◮ eventually the symbols degenerate to empty ones . . . Y: [0] Y: [0] Y: [0] Y: [0] Y: [1] Y: [0] Y: [1] Y: [0] 3 1 2 1 2 3 A ∃ X . X = σ ( Y ) → Projection T. Fiedor Nested Antichains for WS1S AVM’15 8 / 17

The intuition behind the procedure Key observation for ground formulae ϕ | = iff I m ∩ F m � = ∅ Why? ◮ eventually the symbols degenerate to empty ones . . . Y: [0] Y: [0] Y: [0] Y: [0] Y: [1] Y: [0] Y: [1] Y: [0] 3 1 2 1 2 3 A ∃ X . X = σ ( Y ) → Projection T. Fiedor Nested Antichains for WS1S AVM’15 8 / 17