Algebra and Proof Theory for a logic of propositions, actions, and - PowerPoint PPT Presentation

Algebra and Proof Theory for a logic of propositions, actions, and adjoint modalities Joint work with Roy and Julian Truffault Mehrnoosh Sadrzadeh EPSRC Career Acceleration Research Fellow Oxford University, Department of CS

The muddy children puzzle There are n children playing in the mud and k of them have muddy foreheads. Their father announces to them that ‘at least one of you has a dirty forehead’. Then asks ‘do you know it is you who has a muddy forehead?’. The children look around and think and all of them say ‘no!’, then again they look around and think and say ‘no!’ again, and so on. The question: will the dirty children ever know they are dirty? If so, after how many rounds of no answers? How about the clean ones? Modern twists: what if the father is a liar? what if the children are liars or imperfect reasoners? Can one prove the answers in a logic? If so which logic?

An analysis of the puzzle We need to model: Propositions: being dirty or not Actions: announcing no answers Modalities: knowing that one is dirty Learning: how do the actions update the knowledge Modern twists: knowledge about the actions

Existing work Epistemic Logics: Semantic proof: Fagin and Halpern Syntactic proof: Huth and Ryan, Natural Deduction Defect of Both: actions are not part of the logic, so the solution only formalizes half of the puzzle.

Bad attempt at taking actions into account Take Full Linear Logic This has both actions and propositions Add epistemic modalities to it (A. Martin, U of Ottawa) Use it to prove muddy children Automate the proof in COQ

Correct Approach Linear Logic of Actions Logic of Propositions Adjoint Modalities on both Make the two interact to model update and learning

Why adjoint modalities? Kripke Frame W = ( W, R A ) A where R A ⊆ W × W Satisfaction Relation | = ⊆ W × L ML Defined by induction on the structure of the formulae W, w | = ✷ A φ iff ∀ z, ( w, z ) ∈ R A implies z | = φ Read ✷ A φ as “A believes that φ .” := ✷ A φ ∧ φ . Define Knowledge as K A φ Knowledge becomes secondary to belief.

Adjoint Modalities Consider the converse of the relation ( W, R A , R c A ) A Modalities for the converse relation too! ∃ z, ( w, z ) ∈ R c W, w | = � A φ iff and z | = φ Verify W, w | = φ → ✷ A � A φ W, w | = � A ✷ A φ → φ Balck diamond and box are adjoint. � A ⊣ ✷ A

Algebraic Modal Logic Start with the Kripke frame ( W, R A ) Lift it to its powerset: Boolean Algebra ( P ( W ) , f A ) where operator f A : P ( W ) → P ( W ) is canonically defined by � f A ( X ) = R A [ X ] x ∈ X = { y ∈ W | ∃ x ∈ X, ( x, y ) ∈ R A } { y ∈ W | ∃ x ∈ X, ( y, x ) ∈ R c = A } = � A X

✷ A as adjoint to � A Since � A preserves all the unions, it has a right adjoint. Canonically defined by � g A ( X ) = { Y | f A ( Y ) ⊆ X } � = { Y | R A [ Y ] ⊆ X } = { y ∈ W | ∀ z, ( y, z ) ∈ R A z ∈ X } implies = ✷ A X In fact: these hold for any complete lattice.

Epistemic Interpretation � A X All possibilities, choices, options that A has wrt to X . All propositions that A might consider true if X is true. All actions A might consider happening if X is happening. A ’s uncertainty about X . Appearance to A of X . Knowledge is secondary to belief. Belief is secondary to uncertainty.

Developing the logic

Logic of Actions The set Q of actions q of the logic is generated over a set A of agents A and a set B of basic actions σ by the following grammar: q ::= ⊥ | ⊤ | 1 | σ | q ∧ q | q ∨ q | q • q | ✷ A q | � A q

Algebra of Actions with Adjoint Modalities Definition. Let A be a set of agents . A lattice monoid with adjoint modalities (an LMAM ) over A is both (1) a bounded lattice ( Q, ∨ , ∧ , ⊤ , ⊥ ) and (2) a unital monoid ( Q, 1 , • , ≤ ) , and we have q • ( q ′ ∨ q ′′ ) = ( q • q ′ ) ∨ ( q • q ′′ ) ( q ′ ∨ q ′′ ) • q = ( q ′ • q ) ∨ ( q ′′ • q ) and (1) q • 1 = q and 1 • q = q (2) q ≤ q ′ � A q ≤ � A q ′ implies (3) q ≤ q ′ ✷ A q ≤ ✷ A q ′ implies (4) � A q ≤ q ′ q ≤ ✷ A q ′ iff (5) An LMAM Q over A is multiplicative whenever we have � A ( q • q ′ ) ≤ � A q • � A q ′ (6) (7) � A 1 ≤ 1

Proposition. In any LMAM Q over A , the following hold: � A ( q ∨ q ′ ) � A q ∨ � A q ′ = (8) ✷ A ( q ∧ q ′ ) = ✷ A q ∧ ✷ A q (9) � A ( q ∧ q ′ ) � A q ∧ � A q ′ ≤ (10) ✷ A q ∨ ✷ A q ′ ✷ A ( q ∨ q ′ ) ≤ (11) � A ⊥ = ⊥ (12) ✷ A ⊤ = ⊤ (13) q • ( q ′ ∧ q ′′ ) ( q • q ′ ) ∧ ( q • q ′′ ) ≤ (14) ( q ′ ∧ q ′′ ) • q ( q ′ • q ) ∧ ( q ′′ • q ) ≤ (15) ≤ (16) � A ✷ A q q ≤ (17) q ✷ A � A q ✷ A q • ✷ A q ′ ✷ A ( q • q ′ ) ≤ (18) 1 ≤ ✷ A 1 (19)

Sequent Calculus for Actions We have action items Q and action contexts Θ generated by the following syntax: Q ::= q | Θ A Θ ::= Q list where Θ A will be interpreted as � A ( � Θ) , for � Θ the composition of the interpretations of elements in Θ . If one of the items inside a context is replaced by a “hole” [ ] , we have a context-with-a-hole . More precisely, we have the notions of context-with-a-hole Σ and item-with-a-hole R , defined using mutual recursion as follows: Σ ::= Θ , R, Θ ′ R ::= [ ] | Σ A Initial Sequents Σ[ ⊥ ] ⊢ q ⊥ L ⊢ 1 1 R Θ ⊢ ⊤ ⊤ R σ ⊢ σ Id

Rules for the lattice operations, composition and modalities: Σ[ ] ⊢ q Σ[1] ⊢ q 1 L Σ[ q i ] ⊢ q Θ ⊢ q 1 Θ ⊢ q 2 Σ[ q 1 ∧ q 2 ] ⊢ q ∧ L i ∧ R Θ ⊢ q 1 ∧ q 2 Σ[ q 1 ] ⊢ q Σ[ q 2 ] ⊢ q Θ ⊢ q 1 Θ ⊢ q 2 Θ ⊢ q 1 ∨ q 2 ∨ R 1 Θ ⊢ q 1 ∨ q 2 ∨ R 2 ∨ L Σ[ q 1 ∨ q 2 ] ⊢ q Σ[ q 1 , q 2 ] ⊢ q Θ 1 ⊢ q 1 Θ 2 ⊢ q 2 Σ[ q 1 • q 2 ] ⊢ q • L Θ 1 , Θ 2 ⊢ q 1 • q 2 • R Σ[ q A ] ⊢ q ′ Θ ⊢ q Θ A ⊢ � A q � A R Σ[ � A q ] ⊢ q ′ � A L Σ[ q ] ⊢ q ′ Θ A ⊢ q Σ[( ✷ A q ) A ] ⊢ q ′ ✷ A L Θ ⊢ ✷ A q ✷ A R

And structural Rules, encoding the multiplicative axioms: Σ[Θ A , Θ ′ A ] ⊢ q Σ[ � � ] ⊢ q Σ[(Θ , Θ ′ ) A ] ⊢ q Dist Σ[ � � A ] ⊢ q Unit

Example of Derivation q ′ ⊢ q ′ Id q ⊢ q Id q B ⊢ � B q � B R q ′ B ⊢ � B q ′ � B R ( q B ) A ⊢ � A � B q � A R ( q ′ B ) A ⊢ � A � B q ′ � A R • R ( q B ) A , ( q ′ B ) A ⊢ � A � B q • � A � B q ′ Dist q ′′ ⊢ q ′′ Id ( q B , q ′ B ) A ⊢ � A � B q • � A � B q ′ q ′′ B ⊢ � B q ′′ � B R (( q, q ′ ) B ) A ⊢ � A � B q • � A � B q ′ Dist (( q • q ′ ) B ) A ⊢ � A � B q • � A � B q ′ • L ( q ′′ B ) A ⊢ � A � B q ′′ � A R ((( q • q ′ ) ∧ q ′′ ) B ) A ⊢ � A � B q • � A � B q ′ ∧ L ((( q • q ′ ) ∧ q ′′ ) B ) A ⊢ � A � B q ′′ ∧ L ∧ R ((( q • q ′ ) ∧ q ′′ ) B ) A ⊢ ( � A � B q • � A � B q ′ ) ∧ � A � B q ′′ � B (( q • q ′ ) ∧ q ′′ ) A ⊢ ( � A � B q • � A � B q ′ ) ∧ � A � B q ′′ � B L � A � B (( q • q ′ ) ∧ q ′′ ) ⊢ ( � A � B q • � A � B q ′ ) ∧ � A � B q ′′ � A L

Admissibility of Cut Rule Theorem. The following Cut rule is admissible Σ ′ [ q ] ⊢ q ′ Θ ⊢ q Cut Σ ′ [Θ] ⊢ q ′ Proof. Strong induction on the rank of the cut, where the rank is given by the pair (size of cut formula q , sum of heights of derivations of premisses). This involved checking 17 × 17 cases.

Example of a cases The cut-formula is of the form � A q ′′ : Θ ⊢ q ′′ Σ ′ [ q ′′ A ] ⊢ q ′ Θ A ⊢ � A q ′′ � A R Σ ′ [ � A q ′′ ] ⊢ q ′ � A L Cut Σ ′ [Θ A ] ⊢ q ′ transforms to Θ ⊢ q ′′ Σ ′ [ q ′′ A ] ⊢ q ′ Cut Σ ′ [Θ A ] ⊢ q ′

Logic of Propositions Given sets A of agents A , At a set of (propositional) atoms p ; the set M of propositions m is generated by the following grammar: m ::= ⊥ | ⊤ | p | m ∧ m | m ∨ m | ✷ A m | � A m | m · q | [ q ] m The last two binary connectives are mixed action-proposition con- nectives: the operator [ q ] is the dynamic modality operator and · q is (as we shall see) its left adjoint, called update , just as � A is the left adjoint of ✷ A .

Algebra of Propositions with Adjoint Modalities Definition. Let A be a set, with elements called agents . A DLAM over A is a bounded distributive lattice ( L, ∧ , ∨ , ⊤ , ⊥ ) with two A - indexed families such that m ≤ m ′ � A m ≤ � A m ′ implies (20) m ≤ m ′ ✷ A m ≤ ✷ A m ′ implies (21) � A m ≤ m ′ m ≤ ✷ A m ′ iff (22)

Sequent Calculus for Propositions As in the action logic, we have propositional contexts Γ and propo- sitional items I (abbreviated to p–contexts and p-items ), generated by the following grammar: I ::= m | Γ A | Γ Θ Γ ::= I multiset where Γ A will be interpreted as � A ( � Γ) , for � Γ the conjunction of the interpretations of elements in Γ , and Γ Θ as ( � Γ) · � Θ , for � Θ the composition of the interpretations of elements in Θ . Contexts with holes are defined as before, with more cases. Initial Sequents ∆[ ⊥ ] ⊢ m ⊥ L Γ ⊢ ⊤ ⊤ R Γ , p ⊢ p Id