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Reasoning about Memory Management in Resource-Bounded Agents

Stefania Costantini Valentina Pitoni DISIM, University of L’Aquila, Italy

Trieste, 20 June 2019

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Content

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

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Introduction

Memory

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

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Introduction

Starting Point

In recent work [1], Balbiani, Fernández-Duque and Lorini:

◮ propose a formalization

• f 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

• r through inference from

existing knowledge and beliefs. We extend DLEK with TIME!

Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

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Introduction

Previous work

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

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Characterization

Syntax:Static part

In this setting, a time interval I ⊆ (0, ∞) is an interval of natural

• numbers. The extension to the LEK language (LLEK) presented in [1]

to the T-LEK language (LT−LEK) is defined as follows:

Definition

ϕ, ψ := p(t1, t2) | ¬ϕ | I ϕ | B ϕ | K ϕ | ϕ ∧ ψ| ϕ → ψ

◮ ⊤, ⊥, →, ↔ are defined from ¬ and ∧ in the standard way; ◮ p(t1, t2) with t1 t2 and p, q, h are predicates, stands for “p is

true from the time instant t1 to t2" with t1, t2 ∈ N and (Temporal Representation of the external world); as a special case we can have p(t1, t1) which stands for “p is true in the time instant t1".;

◮ 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

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Characterization

Observation and Example

◮ 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

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Characterization

Syntax: Dynamic part

The extension to the DLEK language (LDLEK) presented in [1] to the T-DLEK language (LT−DLEK) is defined as follows:

Definition

ϕ, ψ := p(t1, t2) | ¬ϕ | I ϕ | B ϕ | K ϕ | ϕ ∧ ψ| ϕ → ψ| [α]ϕ 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

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Characterization

TIME Function

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(t1, t2)) = [t1, t2], which stands for “p is true in the time

interval [t1, t2]" where t1, t2 ∈ N; as a special case we have T(p(t1, t1)) = t1, which stands for “p is true in the time instant t1" where t1 ∈ N (time instant);

◮ T(¬p(t1, t2)) = T(p(t1, t2)), which stands for “p is not true in the

time interval [t1, t2]" where t1, t2 ∈ 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

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Characterization

TIME Function

◮ T(Bϕ) = T(ϕ); ◮ T(Kϕ) = T(ϕ); ◮ T(Iϕ) = 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

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Characterization

Semantic: T-LEK model

A T-LEK model is a tuple M = W; N; R; V; T where:

◮ V : W → 2Atm valuation function; ◮ T “time” function; ◮ W is the set of worlds; let t1 the minimum time instant of

T(p(t1, t1)) where p(t1, t1) ∈ V(w) and let t2 be the supremum time instant (we can have t2 = ∞) among the atoms in V(w). We denote w as wI where I = [t1, t2];

◮ R ⊆ W×W is the accessibility relation,

R(wI) = {vI ∈ W | wIR vI} called epistemic state of the agent in wI;

◮ N : W → 22W is a “neighbourhood” function, ∀wI ∈ W, N(w)

defines, in terms of sets of worlds, what the agent is allowed to explicitly believe in the world wI.

Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

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Characterization

Semantic: Truth conditions

Truth conditions for T-DLEK formulas are defined inductively as follows:

◮ M, wI |

= p(t1, t2) iff p(t1, t2) ∈ V(wI) and T(p(t1, t2)) ⊆ I;

◮ M, wI |

= ¬ϕ iff M, wI ϕ and T(¬ϕ) ⊆ I;

◮ M, wI |

= ϕ ∧ ψ iff M, wI | = ϕ and M, wI | = ψ with T(ϕ), T(ψ) ⊆ I;

◮ M, wI |

= ϕ ∨ ψ iff M, wI | = ϕ or M, wI | = ψ with T(ϕ), T(ψ) ⊆ I;

◮ M, wI |

= ϕ → ψ iff M, wI ϕ or M, wI | = ψ with T(ϕ), T(ψ) ⊆ I;

◮ M, wI |

= B ϕ iff ϕ M

wI∈ N(wI) and T(ϕ) ⊆ I; ◮ M, wI |

= Ki ϕ iff for all vI ∈ R(wI), it holds that M, vI | = ϕ and T(ϕ) ⊆ I;

◮ M, wI |

= Jϕ iff T(ϕ) ⊆ J ⊆ I and for all vI ∈ R(wI), it holds that M, vI | = ϕ;

Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

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Characterization

Semantic: Truth conditions

◮ M, wI |

= [α] ϕ iff Mα, wI | = ϕ and T(ϕ) ⊆ I where Mα = W; Nα(wI); R; V; T.

◮ N+ϕ(wI) = N(wI) ∪

• ϕ M

wI

• with T(ϕ) ⊆ I.

◮ N∩(ψ,χ)(wI) =

   N(wI) ∪

• ψ ∧ χ M

wI

• if M, wI |

= B(ψ) ∧ B(χ) and T(∩(ψ, χ)) ⊆ I N(i, wI)

• therwise

◮ N⊢(ψ,χ)(wI) =

   N(wI) ∪

• χ M

wI

• if M, wI |

= B(ψ) ∧ K(ψ → χ) and T(⊢ (ψ, χ)) ⊆ I N(wI)

• therwise

Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

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Characterization

Semantic: Truth conditions

◮ Given Q = q(j, k) s.t. T(q(j, k)) = T(q(t1, t2)) ∩ T(q(t3, t4)) with

j, k ∈ N and P =

• M, wI |

= B(p(t1, t2)) ∧ B(q(t3, t4)) ∧ K(p(t1, t2) → ¬q(t3, t4)) and T(⊣ (p(t1, t2), q(t3, t4))) ⊆ I and there is no interval J T(p(t1, t2)) s.t. B(q(t5, t6)) where T(q(t5, t6))=J

• :

N⊣(p(t1,t2),q(t3,t4))(wI) =

• N(wI) \
• Q M

wI

• if P

N(i, wI)

• therwise

Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

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Characterization

Example 1

• 1. B(raining(2, 2))
• 2. K(rain(t1, t2) → take(t1, t2, umbrella))
• 3. B(take(2, 2, umbrella))
• 4. K(rain(t1, t2)∧take(t1, t2, umbrella) →

go(t1 + 1, ∞, shops))

• 5. B(go(3, ∞, shops))

Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

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Characterization

Example 2

• 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

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Main Results

Theorems

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

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Conclusion and Future Work

In this work we extended an existing approach to the logical modeling

• f short-term and long-term memories in Intelligent

Resource-Bounded Agents by introducing the T function, which manages the interval when an atom is true. Through this function we are also able to assign a “timing" to the epistemic operators B and K. Future developments could be the extension to the multi-agent case also reconsidering the complexity.

Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

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References

[1] Philippe Balbiani, David Fernández-Duque and Emiliano Lorini A Logical Theory of Belief Dynamics for Resource-Bounded Agents. Proceedings of the 2016 International Conference on Autonomous Agents & Multiagent Systems, Singapore, May 9-13, 2016. [2] Stefania Costantini, Andrea Formisano and Valentina Pitoni Timed Memory in Resource-Bounded Agents Proceedings of AI*IA 2018, Trento, November 2018.

Stefania Costantini, Valentina Pitoni | Reasoning about Memory Management in Resource-Bounded Agents

Thank you! Any Questions?