SLIDE 1
Logic of resources and capabilities
Apostolos Tzimoulis
joint work with Marta Bílková, Giuseppe Greco, Alessandra Palmigiano and Nachoem Wijnberg
TACL 2017 - Prague
SLIDE 2 Motivation
◮ 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
SLIDE 3
Main 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.
SLIDE 4
Language
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’
SLIDE 5
The logic of resources and capabilities
Axiom schemas for and
D1.
(A ∨ B) ↔ A ∨ B
D3.
(α ⊔ β) ↔ α ∨ β
D2.
⊥ ↔ ⊥
D4.
0 ↔ ⊥
Axiom schemas for ⊲ and ⊲
B1.
(α ⊔ β) ⊲ A ↔ α ⊲ A ∧ β ⊲ A
B4.
(α ⊔ β) ⊲γ ↔ α ⊲γ ∧ β ⊲γ
B2.
0 ⊲ A
B5.
0 ⊲α
B3.
α ⊲ β ⊲ A → α · β ⊲ A
B6.
α ⊲(β ⊓ γ) ↔ α ⊲β ∧ α ⊲γ
B7.
α ⊲1
Interaction axiom schemas
BD1.
α ∧ α ⊲ A → A
BD2.
α ⊲β → α ⊲ β
SLIDE 6
The 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 · γ ⊢ β · γ A ⊢ B α ⊲ A ⊢ α ⊲ B A ⊢ B A ⊢ B α ⊢ β γ ⊲α ⊢ γ ⊲β α ⊢ β γ · α ⊢ γ · β α ⊢ β β ⊲ A ⊢ α ⊲ A α ⊢ β α ⊢ β α ⊢ β β ⊲γ ⊢ α ⊲γ
SLIDE 7
Completeness, 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
Theorem
The axioms of LRC are canonical. Hence, for every heterogenerous LRC-algebra F, its canonical extension Fδ is a perfect LRC-algebra. Hence, the logic LRC is complete w.r.t. the class of perfect LRC-algebras.
SLIDE 8 Display-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.
() (◮) () () (◮)
SLIDE 9
Introduction rules
Γ ⊢ α A ⊢ X ⊲ L α ⊲ A ⊢ Γ ✶ X X ⊢ α ✶ A ⊲ R X ⊢ α ⊲ A Γ ⊢ α β ⊢ ∆ ⊲ L α ⊲α ⊢ Γ ✶ ∆ Γ ⊢ α ✶ α ⊲R Γ ⊢ α ⊲α
SLIDE 10 Rules corresponding to axioms
Γ ⊢ (Y ❛ ∆) , (Z ❛ ∆)
B4
Γ ⊢ (Y ; Z) ❛ ∆ Γ ⊢ (Y ❛ W) , (Z ❛ W)
B1
Γ ⊢ (Y ; Z) ❛ W (Γ ❛X) , (Γ ❛Y) ⊢ ∆
B6
Γ ❛(X ; Y) ⊢ ∆ X ⊢ Γ ✶ (∆ ✶ Y)
B3
X ⊢ Γ ⊙ ∆ ✶ Y X ⊢ Γ ✶
X ⊢ ◦Γ > Y X ⊢ Γ ✶
X ⊢ Γ ✶ Y
SLIDE 11 Display rules ... but
X ⊢ •Y
Γ ⊢ •X X ⊢ Γ ✶ Y Γ ⊢ X ❛ Y X ⊢ Γ ✶ ∆ Γ ❛X ⊢ ∆ X ⊢ Γ ✶ ∆ Γ ⊢ X ❛ ∆
Notice the argument of the second coordinate of ⊲ cannot be displayed
SLIDE 12
Cut rules
(X ⊢ Y)[A]succ A ⊢ Z (X ⊢ Y)[Z/A]succ Γ ⊢ α α ⊢ ∆ Γ ⊢ ∆
Canonical Cut-elimination and subformula property
Follow from a general meta-theorem.
SLIDE 13
Homework correction
Capabilities initial state planning
α ⊲cPα β ⊲cPβ cα Mβ → cβ α ⊲dMα β ⊲dMβ dβ Pα → dα X ⊢ Y
Exi ◦iX ⊢ Y
SLIDE 14
The wisdom of the crow
Σ ⊙ Σ ⊢ Ω Σ ⊢ Ω (Γ ❛X) ⊙ (Π ❛Y) ⊢ ∆ (Γ ⊙ Π) ❛(X ; Y) ⊢ ∆ (Γ ❛X) ❛Y ⊢ ∆ Γ ❛(X ; Y) ⊢ ∆
SLIDE 15 The gift of the magi
1σ ∧ 2ξ ∧ [σ, ξ] ⊲χ → 2χ,
which is equivalent on perfect LRC-algebras to the following analytic rule:
- ◦2[Σ, Ξ] ❛X ⊢ Y RR
- 1Σ ; ◦2Ξ ; X ⊢ Y
