Formalising Algorithmic Correspondence for Modal Languages - PowerPoint PPT Presentation

Formalising Algorithmic Correspondence for Modal Languages Removing propositional variables with SQEMA in Coq by Merlin Göttlinger February 20, 2018

Fundamentals 2/52 Fundamentals

Modal Syntax Fundamentals 3/52 Let… All sets involved are assumed to be disjoint. τ = ( O 0 , ρ 0 ) be a modal similarity type consisting of a set O 0 of base modal terms together with an arity function ρ 0 : O 0 → N , A be the set of atoms, N be the set of nominals.

Modal Terms Fundamentals 4/52 Define now sets MT and MF and arity function ρ mutually: MT: α, β ::= ⊥ | ι 1 | ι 2 | ϕ | α ( β 1 , . . . , β ρ ( α ) ) | o | α − i where 0 < i ≤ ρ ( α ) , ϕ ∈ MF not mentioning any atoms or nominals, and o ∈ O 0 . Define arity function ρ as ρ ( ⊥ ) = 0 ρ ( ι 1 ) = 1 ρ ( ι 2 ) = 2 ρ ( ϕ ) = 0 ρ ( α ( β 1 , . . . , β n )) = ρ ( β 1 ) + · · · + ρ ( β n ) ρ ( o ) = ρ 0 ( o ) ρ ( α − i ) = ρ ( α )

Modal Formulas Fundamentals 5/52 Plus the following definitions: MF: ϕ, ψ ::= p | j | ¬ ϕ | ϕ ∨ ψ | ♦ α ( ϕ 1 , . . . , ϕ ρ ( α ) ) where α ∈ MT, p ∈ A and j ∈ N . ϕ → ψ := ¬ ϕ ∨ ψ ϕ ∧ ψ := ¬ ( ¬ ϕ ∨ ¬ ψ ) ϕ ↔ ψ := ( ϕ → ψ ) ∧ ( ψ → ϕ ) � α ( ϕ 1 , . . . , ϕ n ) := ¬ ♦ α ( ¬ ϕ 1 , . . . , ¬ ϕ n ) � − i α ( ϕ 1 , . . . , ϕ n ) := � α − i ( ϕ 1 , . . . , ϕ n )

Modal Term Semantics Fundamentals 6/52 Definition (Kripke Frame) each base modal term o . The relation for the nonrecursive elements in MT is defined as A (Kripke) τ -frame F = ( W , { R o } o ∈ O 0 ) consists of a set of worlds W and a ( ρ ( α ) + 1) -ary accessibility relation R o for R ⊥ := ∅ R ι 1 := { ( w , w ) | w ∈ W } R ι 2 := { ( w , w , w ) | w ∈ W } R ϕ := { w | ( F , w ) � ϕ }

Modal Term Semantics Fundamentals 7/52 The relations for the other modal terms are defined as follows: n R α − i := { ( w i , w 1 , . . . , w i − 1 , w , w i +1 , . . . , w n ) | ( w , w 1 , . . . , w n ) ∈ R α } R α ( β 1 ,...,β n ) := { ( w , w 11 , . . . , w 1 b 1 , . . . , w n 1 , . . . , w nb n ) | ∃ u 1 . . . u n . R α ( w , u 1 , . . . , u n ) ∧ � R β i ( u i , w i 1 , . . . , w ib i ) } i =1 where α has arity n and b 1 , . . . , b n are the arities of β 1 , . . . , β n .

Validity Fundamentals 8/52 Definition (Kripke Model) Definition (Pointedness) world. A Kripke τ -model based on a τ -frame F is a triple M = ( F , V , N ) where V : A → 2 W is the valuation and N : N → W the nominal valuation. A pointed τ -frame ( F , w ) is a pair of a frame F with a world w ∈ W . Similarly a pointed τ -model is a pair of a model with a

Validity Fundamentals 9/52 Validity of a modal formula in a pointed model is defined recursively: ( M , w ) � p ifg w ∈ V ( p ) ( M , w ) � j ifg w = N ( j ) ( M , w ) � ¬ ϕ ifg ( M , w ) � � ϕ ( M , w ) � ϕ ∨ ψ ifg ( M , w ) � ϕ or ( M , w ) � ψ ( M , w ) � ♦ α ( ϕ 1 , . . . , ϕ ρ ( α ) ) ifg there exist w 1 . . . w ρ ( α ) such that R α ( w , w 1 , . . . , w ρ ( α ) ) and ( M , w i ) � ϕ i for each 1 ≤ i ≤ ρ ( α )

Applied Modalities Fundamentals 10/52 What do those applied modalities mean? n so R α ( β 1 ,...,β n ) := { ( w , w 11 , . . . , w 1 b 1 , . . . , w n 1 , . . . , w nb n ) | ∃ u 1 . . . u n . R α ( wu 1 , . . . , u n ) ∧ � R β i ( u i , w i 1 , . . . , w ib i ) } i =1 ( M , w ) � ♦ α ( ϕ 1 , . . . , ϕ ρ ( α ) ) ifg there exist w 1 . . . w ρ ( α ) such that R α ( w , w 1 , . . . , w ρ ( α ) ) and ( M , w i ) � ϕ i for each 1 ≤ i ≤ ρ ( α ) ♦ α ( β 1 ,...,β n ) ( ϕ 11 , . . . , ϕ 1 b 1 , . . . ) ≡ ♦ α ( ♦ β 1 ( ϕ 11 , . . . , ϕ 1 b 1 ) , . . . )

Special Modalities Fundamentals 11/52 R ⊥ := ∅ R ϕ := { w | ( F , w ) � ϕ } R ι 1 := { ( w , w ) | w ∈ W } R ι 2 := { ( w , w , w ) | w ∈ W } ( M , w ) � ♦ α ( ϕ 1 , . . . , ϕ ρ ( α ) ) ifg there exist w 1 . . . w ρ ( α ) such that R α ( w , w 1 , . . . , w ρ ( α ) ) and ( M , w i ) � ϕ i for each 1 ≤ i ≤ ρ ( α ) ♦ ⊥ ≡ ⊥ � ι 1 ϕ ≡ ϕ ≡ ♦ ι 1 ϕ � ⊥ ≡ ⊤ ♦ ι 2 ( ϕ, ψ ) ≡ ϕ ∧ ψ ♦ ϕ ≡ ϕ ≡ � ϕ � ι 2 ( ϕ, ψ ) ≡ ϕ ∨ ψ

Standard Translation Fundamentals 12/52 Those semantics are expressible in FOL via the first-order/model standard translation ST given we have a unary predicate P i for each p i ∈ A a designated variable n j for each j ∈ N a ( ρ ( α ) + 1 )-ary predicate R α for each α ∈ MT ST ( p i , x ) := P i ( x ) ST ( j , x ) := x = n j ST ( ¬ ϕ, x ) := ¬ ST ( ϕ, x ) ST ( ϕ ∨ ψ, x ) := ST ( ϕ, x ) ∨ ST ( ψ, x ) ST ( ♦ α ( ϕ 1 , . . . , ϕ ρ ( α ) ) , x ) := ∃ z 1 . . . z ρ ( α ) . R α ( x , z 1 , . . . , z ρ ( α ) ) ∧ ρ ( α ) � ST ( ϕ i , z i ) i =1 ( M , w ) � ϕ ifg M | = ST ( ϕ, x )[ w / x ]

General Validity Fundamentals 13/52 … valid on a pointed frame… A modal formula ϕ can also be valid on a model… M � ϕ ifg ( M , w ) � ϕ for all w ∈ W ( F , w ) � ϕ ifg ( M , w ) � ϕ for all M over F … valid on a frame… F � ϕ ifg ( F , w ) � ϕ for all w ∈ W … or valid � ϕ ifg F � ϕ for all F

Standard Translation every assignment for the nominals. n , are vectors of unary predicate symbols Fundamentals quantifying over the atom/nominal variables in the result of the first-order translation: every valuation for the atoms, We have to capture the notions of What about standard translation for pointed frame validity? 14/52 We obtain the second-order/frame standard translation by = ∀ ¯ P ∀ ¯ ( F , w ) � ϕ ifg F | n . ST ( ϕ, x )[ w / x ] where ¯ P and ¯ representing atoms, and variables resembling nominals in ϕ .

Correspondents Fundamentals 15/52 Definition (Frame-Correspondents) A modal formula ϕ and a first-order formula θ ( x ) are local frame-correspondents if for every pointed frame ( F , w ) ( F , w ) � ϕ ifg F | = θ ( x )[ w / x ] They are called global frame-correspondents if for every frame F F � ϕ ifg F | = θ

Elementary Formulas Fundamentals 16/52 If we can have a first-order formula expressing the same property as a second-order one it is beneficial to use that. Computationally easier to reason about More algorithms handling those Easier to read as a human Definition (Elementary Formula) A modal formula that has a first-order correspondent is called elementary. elementary class of frames nor is implied by it. Note that this does not imply completeness w.r.t some

Automatic Computation Fundamentals 17/52 Can we compute those correspondents in an automatic way? Theorem (Chagrova’s Theorem) It is undecidable whether an arbitrary basic modal formula has a global first-order correspondent. We can however have algorithms working on subclasses of the class of elementary formulas.

Algorithms Fundamentals 18/52 SCAN - Gabbay, D. M., & Ohlbach, H. J. (1992) Eliminates second-order existential quantifiers in Uses skolemization, resolution and unskolemization. DLS - Doherty, P., Łukaszewicz, W., & Szałas, A. (1997) Eliminates second-order quantifiers from arbitrary second-order formulas; Uses skolemization, the Ackermann lemma and unskolemization. formulas of the shape ∃ P 1 . . . P n .ϕ ;

Algorithms Fundamentals 19/52 SQEMA - Conradie, W., Goranko, V., & Vakarelov, D. (2006) Eliminates atoms from modal formulas using a modal variation of the Ackermann lemma; Avoids skolemization and unskolemization; Computes local frame-correspondents. ALBA - Conradie, W., & Palmigiano, A. (2012) Counterpart to SQEMA for distributive modal logics; Using a variation of the Ackermann lemma.

Polarity Fundamentals 20/52 Definition (Polarity) A formula is… positive (negative) in p if any occurrence of atom p in it is under an even (odd) number of negations. pure in p if it doesn’t contain p . variable-positive (-negative) if it is positive (negative) in all its atoms. pure if it contains no atoms.

Monotonicity Fundamentals 21/52 Definition (Monotonicity) implies If a formula is positive(negative) in p it is also upwards(downwards) monotone in p . A formula ϕ is upwards monotone in an atom p if V ( p ) ⊆ V ′ ( p ) � ϕ � V ⊆ � ϕ � V ′ or downwards monotone if it implies � ϕ � V ′ ⊆ � ϕ � V

Modal Ackermann Lemma Fundamentals 22/52 Lemma (Modal Ackermann Lemma) valuation of p such that Let β and α be modal formulas such that α pure in p and β negative in p. Then for any model M M � β [ α / p ] ifg there exists a model M ′ difgering from M at most in the M ′ � ( α → p ) ∧ β

Inductive Formulas Fundamentals 23/52 Definition (Box Formula) where p is a single propositional variable (called head) and box formula : either a headed or a headless box. An occurrence of an atom p in a box formula is called… essential if it occurs as head, inessential otherwise. headed box : � α ( N 1 , . . . , p , . . . , N n − 1 ) , headless box : � α ( N 1 , . . . , N n ) , N 1 , . . . , N n are variable-negative formulas.