Logic of resources and capabilities

Transcription

1 Logic of resources and capabilities Apostolos Tzimoulis joint work with Marta Bílková, Virginia Dignum, Giuseppe Greco and Alessandra Palmigiano Delft University of Technology Zürich - 10 February

2 Logic of Resources and Capabilities 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. Similarities with STIT logics: try to capture actions in terms of a hierarchy of more primitive notions, starting with agents capabilities. Main idea: reasoning about resources and their manipulation can provide a concrete handle on the notion of capabilities, which is elusive.

3 Logic of Resources and Capabilities A ::= p A A A A A A (α, A) (α, A), α ::= a 1 0 α α α + α. Axioms Diamond-like modality (α, A B) (α, A) (α, B) (α, ) Box-like modality (1, ) (0, A) (α + β, A) (α, A) (β, A) (α + β, A) (α, A) (β, A) (α β, A) (α, (β, A)) Box-Diamond interaction (α, A B) (α, A) (α, B) (α, ) (α, A) (α, A)

4 Resources + is commutative, associative and idempotent, is associative, α β α and α β β, α (β + γ) = (α β) + (α γ) and (β + γ) α = (β α) + (γ α). and closed under MP, US and the following: Rules A B (α, A) (α, B) α β (β, A) (α, A) A B (α, A) (α, B) α β (α, A) (β, A) α β (β, A) (α, A)

5 Resources + is commutative, associative and idempotent, is associative, α β α and α β β, α (β + γ) = (α β) + (α γ) and (β + γ) α = (β α) + (γ α). and closed under MP, US and the following: Rules A B (α, A) (α, B) α β (β, A) (α, A) A B (α, A) (α, B) α β (α, A) (β, A) α β (β, A) (α, A)

6 Resources + is commutative, associative and idempotent, is associative, α β α and α β β, α (β + γ) = (α β) + (α γ) and (β + γ) α = (β α) + (γ α). and closed under MP, US and the following: Rules A B (α, A) (α, B) 10 francs 5 francs (5, A) (10, A) A B (α, A) (α, B) α β (α, A) (β, A) α β (β, A) (α, A)

7 Resources + is commutative, associative and idempotent, is associative, α β α and α β β, α (β + γ) = (α β) + (α γ) and (β + γ) α = (β α) + (γ α). and closed under MP, US and the following: Rules A B (α, A) (α, B) α β (β, A) (α, A) A B (α, A) (α, B) 50 euros 55 francs (50, A) (55, A) α β (β, A) (α, A)

8 Completeness and Canonicity The logic is to be interpreted in LCR-albegraic frames, which are tuples of the form F = (A, Q,, ) where A is a distributive lattice Q is a finite quantale : Q A A : Q A A A standard Lindenbaum-Tarski argument guarantees (weak) completeness. Theorem The axioms of LRC are canonical. Hence, for every LRC-frame F, its canonical extension F δ is a perfect LRC-frame. Hence, the logic LRC is weakly complete w.r.t. the class of perfect LRC-algebraic frames.

9 Structural and logical connectives Structural connectives are interpreted contextually (like Gentzen s comma) : 1 1 a a

10 Introduction rules (1) Logical Rules Γ α (α, A) 1 (α, A) X (α, A) X A X (Γ, X) 1 X A 1 (α, X) (α, A) 1(α, A)]succ X Y [ X Y [ (α, A)] succ

11 Rules corresponding to axioms (2) Structural Rules 1 (1, I) (1, ) D3 I (1, ) 1 (α, X) Y 1 (β, X) Z D4 1 (α + β, X) Y ; Z Γ Γ a a (Y, W) (Y ; Z, W) a (Z, W) B6 X X (Γ, (, Y)) B7 (Γ, Y) 1 1 1

12 Display rules... but (3) Display Postulates 1 (Γ, X) Y X a (Γ, Y) 1 (Γ, X) Y Γ a (Y, X) X Γ a 1 a (Γ, Y) (X, Y) X (Γ, X) Y 1 (Γ, Y)

13 Cut rules (X Y)[A] succ A Z Γ α α (X Y)[Z/A] succ Γ Canonical Cut-elimination and subformula property Follow from a general meta-theorem.

14 Homework correction Sheet Sheet α β Carl c P P Dan d M M ( c (α, ), d (β, ) ) ( ) c (α, P α ) d (β, M β ) (P α d (α, M α )) (M β c (β, P β ))

15 The wisdom of the crow (σ, ) (σ, (ρ 1 ρ 2, (λ, (ϕ, ))))

16 The gift of the magi j ( ω, d (β, ) j (ω, ) ) ; d ( η, j (γ, ) d (η, ) ) ( d (β, ) j γ ) ( j (γ, ) d β )

17 References Bílková, Dignum, Greco, Palmigiano, T. Logic of resources and capabilities, In preparation. Frittella, Greco, Kurz, Palmigiano, Sikimić, A Proof Theoretic Semantic Analysis of Dynamic Epistemic Logic, JLC (2013). Frittella, Greco, Kurz, Palmigiano, Sikimić, Multi-Type Display Calculus for Dynamic Epistemic Logic, JLC (2014). Frittella, Greco, Kurz, Palmigiano, Multi-Type Display Calculus for Propositional Dynamic Logic, JLC (2014). Frittella, Greco, Kurz, Palmigiano, Multi-Type Sequent Calculi, Proc. Trends in Logics (2014). Greco, Kurz, Palmigiano, Dynamic Epistemic Logic Displayed, Proc. LORI (2013).