Reasoning about Memory Management in Resource-Bounded Agents - PowerPoint PPT Presentation

Reasoning about Memory Management in Resource-Bounded Agents Stefania Costantini Valentina Pitoni DISIM, University of L’Aquila, Italy Trieste, 20 June 2019

Content 1 Introduction Memory Starting Point Previous work Characterization Syntax TIME Function Semantic Example Main Results Conclusion References Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

Introduction Memory 2 Memory is a basic component of every reasoning process, and vice versa interaction between the agent and the environment can play an important role in creating memory and can affect future behaviour. Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

Introduction Starting Point 3 In recent work [1], Balbiani, Fernández-Duque and Lorini: ◮ propose a formalization of SOAR architecture in a particular modal logic (LEK/DLEK); ◮ DLEK logic helps in clarifying how a non-omniscient agent can form new beliefs either through perception We extend DLEK with or through inference from TIME! existing knowledge and beliefs. Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

Introduction Previous work 4 This work is an extended version of [2], where: ◮ we extend DLEK using Metric Temporal Logic; ◮ we introduce explicit time instants and time intervals in formulas. But to avoid problems with the formalization, in this paper we introduce a: TIME Function Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

Characterization Syntax:Static part 5 In this setting, a time interval I ⊆ ( 0 , ∞ ) is an interval of natural numbers. The extension to the LEK language ( L LEK ) presented in [1] to the T-LEK language ( L T − LEK ) is defined as follows: Definition ϕ, ψ := p ( t 1 , t 2 ) | ¬ ϕ | � I ϕ | B ϕ | K ϕ | ϕ ∧ ψ | ϕ → ψ ◮ ⊤ , ⊥ , → , ↔ are defined from ¬ and ∧ in the standard way; ◮ p ( t 1 , t 2 ) with t 1 � t 2 and p , q , h are predicates, stands for “p is true from the time instant t 1 to t 2 " with t 1 , t 2 ∈ N and ( Temporal Representation of the external world); as a special case we can have p ( t 1 , t 1 ) which stands for “p is true in the time instant t 1 " .; ◮ the operator B denotes belief and the operator K denotes knowledge; ◮ � I ϕ is always” operator applies to a formula. Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

Characterization Observation and Example 6 ◮ we can have predicates with more terms than only two but in that case we fix that the first two must be those that identify the time duration of the belief (i.e. open(1,3,door) which means “the agent knows that the door is open from time 1 to time 3”); ◮ If at time t = 2 it is starting raining, in the agent’s working memory there will be the following belief: B (raining(2,2)) . Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

Characterization Syntax: Dynamic part 7 The extension to the DLEK language ( L DLEK ) presented in [1] to the T-DLEK language ( L T − DLEK ) is defined as follows: Definition := p ( t 1 , t 2 ) | ¬ ϕ | � I ϕ | B ϕ | K ϕ | ϕ ∧ ψ | ϕ → ψ | [ α ] ϕ ϕ, ψ α denotes a mental operation ; we consider: ◮ + ϕ : learning perceived belief; ◮ ∩ ( ϕ, ψ ) : beliefs conjunction; ◮ ⊢ ( ϕ, ψ ) : belief inference; ◮ ⊣ ( ϕ, ψ ) : belief revision. Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

Characterization TIME Function 8 We define the “time” function T that associates to each formula the time interval in which this formula is true and operates as follows: ◮ T ( p ( t 1 , t 2 )) = [ t 1 , t 2 ] , which stands for “p is true in the time interval [ t 1 , t 2 ] " where t 1 , t 2 ∈ N ; as a special case we have T ( p ( t 1 , t 1 )) = t 1 , which stands for “p is true in the time instant t 1 " where t 1 ∈ N (time instant); ◮ T ( ¬ p ( t 1 , t 2 )) = T ( p ( t 1 , t 2 )) , which stands for “p is not true in the time interval [ t 1 , t 2 ] " where t 1 , t 2 ∈ N ; ◮ T ( ϕ op ψ ) = T ( ϕ ) � T ( ψ ) with op ∈ {∨ , ∧ , →} , which means the unique smallest interval including both T ( ϕ ) and T ( ψ ) ; Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

Characterization TIME Function 9 ◮ T ( B ϕ ) = T ( ϕ ) ; ◮ T ( K ϕ ) = T ( ϕ ) ; ◮ T ( � I ϕ ) = I where I is a time interval in N ; ◮ T ([ α ] ϕ ) there are different cases: ◮ T (+ ϕ ) = T ( ϕ ) ; ◮ T ( ∩ ( ϕ, ψ )) = T ( ϕ ) � T ( ψ ) ; ◮ T ( ⊢ ( ϕ, ψ )) = T ( ψ ) ; ◮ T ( ⊣ ( ϕ, ψ )) returns the restored interval where ψ is true. Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

Characterization Semantic: T-LEK model 10 A T-LEK model is a tuple M = � W ; N ; R ; V ; T � where: ◮ V : W → 2 Atm valuation function; ◮ T “time” function; ◮ W is the set of worlds; let t 1 the minimum time instant of T ( p ( t 1 , t 1 )) where p ( t 1 , t 1 ) ∈ V ( w ) and let t 2 be the supremum time instant (we can have t 2 = ∞ ) among the atoms in V ( w ) . We denote w as w I where I = [ t 1 , t 2 ] ; ◮ R ⊆ W × W is the accessibility relation, R ( w I ) = { v I ∈ W | w I R v I } called epistemic state of the agent in w I ; ◮ N : W → 2 2 W is a “neighbourhood” function, ∀ w I ∈ W , N ( w ) defines, in terms of sets of worlds, what the agent is allowed to explicitly believe in the world w I . Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

Characterization Semantic: Truth conditions 11 Truth conditions for T-DLEK formulas are defined inductively as follows: ◮ M , w I | = p ( t 1 , t 2 ) iff p ( t 1 , t 2 ) ∈ V ( w I ) and T ( p ( t 1 , t 2 )) ⊆ I ; ◮ M , w I | = ¬ ϕ iff M , w I � ϕ and T ( ¬ ϕ ) ⊆ I ; ◮ M , w I | = ϕ ∧ ψ iff M , w I | = ϕ and M , w I | = ψ with T ( ϕ ) , T ( ψ ) ⊆ I ; ◮ M , w I | = ϕ ∨ ψ iff M , w I | = ϕ or M , w I | = ψ with T ( ϕ ) , T ( ψ ) ⊆ I ; ◮ M , w I | = ϕ → ψ iff M , w I � ϕ or M , w I | = ψ with T ( ϕ ) , T ( ψ ) ⊆ I ; = B ϕ iff � ϕ � M ◮ M , w I | w I ∈ N ( w I ) and T ( ϕ ) ⊆ I ; ◮ M , w I | = K i ϕ iff for all v I ∈ R ( w I ) , it holds that M , v I | = ϕ and T ( ϕ ) ⊆ I ; ◮ M , w I | = � J ϕ iff T ( ϕ ) ⊆ J ⊆ I and for all v I ∈ R ( w I ) , it holds that M , v I | = ϕ ; Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

Characterization Semantic: Truth conditions 12 ◮ M , w I | = [ α ] ϕ iff M α , w I | = ϕ and T ( ϕ ) ⊆ I where M α = � W ; N α ( w I ); R ; V ; T � . ◮ N + ϕ ( w I ) = N ( w I ) ∪ � � ϕ � M � with T ( ϕ ) ⊆ I . w I ◮ N ∩ ( ψ,χ ) ( w I ) =  � ψ ∧ χ � M � � N ( w I ) ∪ if M , w I | = B ( ψ ) ∧ B ( χ ) w I  and T ( ∩ ( ψ, χ )) ⊆ I N ( i , w I ) otherwise  ◮ N ⊢ ( ψ,χ ) ( w I ) = � χ � M  � � N ( w I ) ∪ if M , w I | = B ( ψ ) ∧ K ( ψ → χ ) w I  and T ( ⊢ ( ψ, χ )) ⊆ I N ( w I ) otherwise  Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

Characterization Semantic: Truth conditions 13 ◮ Given Q = q ( j , k ) s.t. T ( q ( j , k )) = T ( q ( t 1 , t 2 )) ∩ T ( q ( t 3 , t 4 )) with j , k ∈ N and � P = M , w I | = B ( p ( t 1 , t 2 )) ∧ B ( q ( t 3 , t 4 )) ∧ K ( p ( t 1 , t 2 ) → ¬ q ( t 3 , t 4 )) and T ( ⊣ ( p ( t 1 , t 2 ) , q ( t 3 , t 4 ))) ⊆ I and there is no interval � J � T ( p ( t 1 , t 2 )) s.t. B ( q ( t 5 , t 6 )) where T ( q ( t 5 , t 6 ))= J : � Q � M � � � N ( w I ) \ if P N ⊣ ( p ( t 1 , t 2 ) , q ( t 3 , t 4 )) ( w I ) = w I N ( i , w I ) otherwise Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

Characterization Example 1 14 1. B ( raining ( 2 , 2 )) 2. K ( rain ( t 1 , t 2 ) → take ( t 1 , t 2 , umbrella )) 3. B ( take ( 2 , 2 , umbrella )) 4. K ( rain ( t 1 , t 2 ) ∧ take ( t 1 , t 2 , umbrella ) → go ( t 1 + 1 , ∞ , shops )) 5. B ( go ( 3 , ∞ , shops )) Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

Characterization Example 2 15 1. K ( marry ( T , T ) A → married ( T + 1 , ∞ )) K ( divorce ( T , T ) A → divorced ( T + 1 , ∞ )) 2. B ( married ( 6 , ∞ )) 3. K ( married ( T , ∞ ) → ¬ divorced ( T , ∞ )) K ( divorced ( T , ∞ ) → ¬ married ( T , ∞ )) 4. B ( married ( 6 , 8 )) plus B ( divorced ( 9 , ∞ )) Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

Main Results Theorems 16 Theorem T-LEK is strongly complete for the class of T-LEK models. Theorem T-DLEK is strongly complete for the class of T-LEK models. Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents