L ogic of resources and capabilities Apostolos Tzimoulis joint work - PowerPoint PPT Presentation

L ogic of resources and capabilities Apostolos Tzimoulis joint work with Marta Bílková, Giuseppe Greco, Alessandra Palmigiano and Nachoem Wijnberg TACL 2017 - Prague

M otivation ◮ Organizations are social units of agents structured and managed to meet a need, or pursue collective goals. ◮ Competitive advantage lends itself to be explained terms of agency , knowledge , goals , capabilities and inter-agent coordination . ◮ Resource-based view: Central role in determining the success of an organization is played by the acquisition, management, and transformation of resources within that organization.

M ain features ◮ STIT-logic approach of capabilities captured via modalities and use of resources and their manipulation to provide a concrete handle on the notion of capabilities. ◮ Constructive character guarantees that each theorem translates into an effective procedure.

L anguage α :: = a ∈ AtRes | 1 | 0 | α · α | α ⊔ α | α ⊓ α, A :: = p ∈ AtProp | ⊤ | ⊥ | A ∨ A | A ∧ A | A → A | α ⊲ A | � A | � α | α ⊲ α. ◮ � A : ‘the agent is able to bring about state of affairs A ’ ◮ � α : ‘the agent is in possession of resource α ’ ◮ α ⊲ A : ‘whenever resource α is in possession of the agent, using α the agent is capable to bring about A ’ ◮ α ⊲ β : ‘the agent is capable of getting β from α , whenever in possession of α ’

T he logic of resources and capabilities Axiom schemas for � and � � ( A ∨ B ) ↔ � A ∨ � B � ( α ⊔ β ) ↔ � α ∨ � β D1. D3. � ⊥ ↔ ⊥ � 0 ↔ ⊥ D2. D4. Axiom schemas for ⊲ and ⊲ ( α ⊔ β ) ⊲ A ↔ α ⊲ A ∧ β ⊲ A ( α ⊔ β ) ⊲ γ ↔ α ⊲ γ ∧ β ⊲ γ B1. B4. 0 ⊲ A 0 ⊲ α B2. B5. α ⊲ β ⊲ A → α · β ⊲ A α ⊲ ( β ⊓ γ ) ↔ α ⊲ β ∧ α ⊲ γ B3. B6. α ⊲ 1 B7. Interaction axiom schemas � α ∧ α ⊲ A → � A BD1. α ⊲ β → α ⊲ � β BD2.

T he logic of resources and capabilities Pure-resource entailments schemas R1. ⊔ and ⊓ are commutative, associative, idempotent, and distribute over each other; R2. · is associative with unit 1 ; R3. α ⊢ 1 and 0 ⊢ α R4. α · ( β ⊔ γ ) ⊢ ( α · β ) ⊔ ( α · γ ) and ( β ⊔ γ ) · α ⊢ ( β · α ) ⊔ ( γ · α ) . and closed under modus ponens, uniform substitution and the following rules: α ⊢ β α ⊢ β A ⊢ B A ⊢ B α ⊲ A ⊢ α ⊲ B � A ⊢ � B α · γ ⊢ β · γ γ ⊲ α ⊢ γ ⊲ β α ⊢ β α ⊢ β α ⊢ β α ⊢ β γ · α ⊢ γ · β β ⊲ A ⊢ α ⊲ A � α ⊢ � β β ⊲ γ ⊢ α ⊲ γ

C ompleteness , canonicity and disjunction property Heterogeneous LRC- algebras are tuples of the form F = ( A , Q , ⊲ , � , ⊲ , � ) where ◮ A is a Heyting algebra, ◮ Q = ( Q , ⊔ , ⊓ , · , 0 , 1) is a bounded distributive lattice with binary join-preserving operator · with unit 1 . ◮ ⊲ : Q × A → A , � : A → A , ⊲ : Q × Q → A , � : Q → A . ◮ Lindenbaum-Tarski argument guarantees completeness ◮ Standard argument guarantees disjunction property T heorem The axioms of LRC are canonical. Hence, for every heterogenerous LRC-algebra F , its canonical extension F δ is a perfect LRC-algebra. Hence, the logic LRC is complete w.r.t. the class of perfect LRC-algebras.

D isplay - style calculus Structural and operational symbols for pure Res-connectives: ⊙ , ⋗ ⋖ ( ⊏ ) Str. ⊐ · ⊓ ⊔ ( · \ ) ( / · ) ( ⊔ \ ) ( ⊓ \ ) ( / ⊔ ) ( / ⊓ ) Op. Structural and operational symbols for the modal operators: ✶ ✶ ◦ ◦ Str. � � Op. ⊲ ⊲ Structural and operational symbols for the adjoints and residuals of the modal operators: ❛ ❛ ❛ • • Str. ( � ) ( ◮ ) ( � ) ( � ) ( ◮ ) Op.

I ntroduction rules ✶ Γ ⊢ α A ⊢ X X ⊢ α A ⊲ L ⊲ R ✶ α ⊲ A ⊢ Γ X ⊢ α ⊲ A X Γ ⊢ α β ⊢ ∆ ✶ Γ ⊢ α α ⊲ L ⊲ R ✶ α ⊲ α ⊢ Γ ∆ Γ ⊢ α ⊲ α

R ules corresponding to axioms ❛ ❛ ❛ ❛ Γ ⊢ ( Y ∆ ) , ( Z ∆ ) Γ ⊢ ( Y W ) , ( Z W ) B4 B1 ❛ ❛ Γ ⊢ ( Y ; Z ) ∆ Γ ⊢ ( Y ; Z ) W ( Γ ❛ X ) , ( Γ ❛ Y ) ⊢ ∆ ✶ ✶ X ⊢ Γ ( ∆ Y ) B6 B3 Γ ❛ ( X ; Y ) ⊢ ∆ ✶ X ⊢ Γ ⊙ ∆ Y ✶ ✶ X ⊢ Γ • Y BD2 X ⊢ Γ • Y BD1 ✶ X ⊢ ◦ Γ > Y X ⊢ Γ Y

D isplay rules ... but ✶ ✶ ✶ X ⊢ Γ Y X ⊢ Γ ∆ X ⊢ Γ ∆ ◦ X ⊢ Y ◦ Γ ⊢ X X ⊢ • Y Γ ⊢ • X Γ ❛ X ⊢ ∆ ❛ ❛ Γ ⊢ X Y Γ ⊢ X ∆ Notice the argument of the second coordinate of ⊲ cannot be displayed

C ut rules ( X ⊢ Y )[ A ] succ A ⊢ Z Γ ⊢ α α ⊢ ∆ ( X ⊢ Y )[ Z / A ] succ Γ ⊢ ∆ C anonical C ut - elimination and subformula property Follow from a general meta-theorem.

H omework correction Capabilities initial state planning α ⊲ c P α β ⊲ c P β � c α M β → � c β α ⊲ d M α β ⊲ d M β � d β P α → � d α X ⊢ Y Ex i ◦ i X ⊢ Y

T he wisdom of the crow ( Γ ❛ X ) ⊙ ( Π ❛ Y ) ⊢ ∆ ( Γ ❛ X ) ❛ Y ⊢ ∆ Σ ⊙ Σ ⊢ Ω Σ ⊢ Ω ( Γ ⊙ Π ) ❛ ( X ; Y ) ⊢ ∆ Γ ❛ ( X ; Y ) ⊢ ∆

T he gift of the magi � 1 σ ∧ � 2 ξ ∧ [ σ, ξ ] ⊲ χ → �� 2 χ, which is equivalent on perfect LRC-algebras to the following analytic rule: ◦◦ 2 [ Σ , Ξ ] ❛ X ⊢ Y RR ◦ 1 Σ ; ◦ 2 Ξ ; X ⊢ Y