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Probabilistic Event Calculus based on Markov Logic Networks Anastasios Skarlatidis 1 , 2 , Georgios Paliouras 1 , George Vouros 2 and Alexander Artikis 1 1 Institute of Informatics and Telecommunications NCSR Demokritos, Athens 15310, Greece

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Probabilistic Event Calculus based on Markov Logic Networks

Anastasios Skarlatidis1,2, Georgios Paliouras1, George Vouros2 and Alexander Artikis1

1 Institute of Informatics and Telecommunications

NCSR “Demokritos”, Athens 15310, Greece {anskarl,paliourg,a.artikis}@iit.demokritos.gr

2 Department of Information and Communication Systems Engineering,

University of the Aegean, Samos, Greece georgev@aegean.gr

4 Nov 2011

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Outline

Introduction Event Calculus in Markov Logic Networks Representing Event Calculus Axioms in MLN Behaviour of Event Calculus in MLN Experiments: Activity Recognition

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Event Recognition

◮ Variety of application domains, e.g. health care monitoring,

public transport management, activity recognition etc

◮ Input:

◮ Low-level events (LLE) — time-stamped symbols ◮ LLE come from different sources/sensors ◮ e.g. happens(walking(id1), 10), happens(active(id2), 10), ...

◮ Output:

◮ Recognised high-level events (HLE) ◮ e.g. holdsAt(meeting(id1,id2), 11), holdsAt(meeting(id1,id2), 12), ...

◮ HLE: Relational structure over other sub-events (HLE or LLE)

Methods:

◮ Logic-based — very expressive ◮ Probabilistic-based — handle uncertainty

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Event Recognition

Requirements:

◮ Formal representation language ◮ Handle uncertainty

The method combines:

◮ Event Calculus — representation ◮ Markov Logic Networks — probabilistic inference

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Markov Logic Networks (MLN) — in a nutshell

◮ First-order logic → set of hard constraints ◮ Syntactically: weighted first-order logic formulas (Fi, wi) ◮ Semantically: (Fi, wi) represents a probability distribution

  • ver possible worlds (or Herbrand interpretations)

P( X = x ) = 1 Z exp

  • i

wi ni(x)

  • ◮ possible world

◮ partition function ◮ number of satisfied ground formulas

A world violating formulas becomes less probable, but not impossible!

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Markov Logic Networks (MLN) — in a nutshell

LLE HLE Knowledge base Grounding Markov-logic Markov Network Sensors

◮ Formula → clausal form ◮ Clauses are grounded according to the domain of their distinct variables ◮ Existentially quantified variables are replaced by the disjunction of their groundings: ∀X ∃Y p(Y ) ∧ q(X) ≡ (p(1) ∧ q(X)) ∨ (p(2) ∧ q(X)) ∨ ... ◮ Open-world assumption for non-evidence predicates

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Event Calculus

◮ Reasoning about events and their effects ◮ A variety of different dialects ◮ Ontology

◮ Timepoints ◮ Events → low-level events ◮ Fluents → high-level events

◮ Core domain-independent axioms

◮ Define whether a fluent holds or not at a specific timepoint ◮ Inertia: fluents persist over time, unless affected by some

event

◮ Domain-dependent definitions → HLE definitions

◮ Initiation of a fluent ◮ Termination of a fluent 7 / 20

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Representing Event Calculus in MLN

Some axioms from Shanahan’s Full Event Calculus: holdsAt(F, T) ←happens(E, T0) ∧ initiates(E, F, T0) ∧ T0 < T ∧ ¬clipped(F, T0, T)          F × E × T × T0 clipped(F, T0, T1) ↔ ∃ E, T happens(E, T) ∧ T0 ≤ T < T1 ∧ terminates(E, F, T)            F × E × T × T0 × T1

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Representing Event Calculus in MLN

Some axioms from Shanahan’s Full Event Calculus: holdsAt(F, T) ←happens(E, T0) ∧ initiates(E, F, T0) ∧ T0 < T ∧ ¬clipped(F, T0, T)          F × E × T × T0 clipped(F, T0, T1) ↔ ∃ E, T happens(E, T) ∧ T0 ≤ T < T1 ∧ terminates(E, F, T)            F × E × T × T0 × T1

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Representing Event Calculus in MLN

Some axioms from Shanahan’s Full Event Calculus: holdsAt(F, T) ←happens(E, T0) ∧ initiates(E, F, T0) ∧ T0 < T ∧ ¬clipped(F, T0, T)          F × E × T × T0 clipped(F, T0, T1) ↔ ∃ E, T happens(E, T) ∧ T0 ≤ T < T1 ∧ terminates(E, F, T)            F × E × T × T0 × T1

◮ Huge number of groundings

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Representing Event Calculus in MLN

Some axioms from Shanahan’s Full Event Calculus: holdsAt(F, T) ←happens(E, T0) ∧ initiates(E, F, T0) ∧ T0 < T ∧ ¬clipped(F, T0, T)          F × E × T × T0 clipped(F, T0, T1) ↔ ∃ E, T happens(E, T) ∧ T0 ≤ T < T1 ∧ terminates(E, F, T)            F × E × T × T0 × T1

◮ Huge number of groundings ◮ Combinatorial explosion

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Discrete Event Calculus

◮ Logically equivalent with EC, when the domain of timepoints

is limited to integers

◮ Axioms are defined over successive timepoints

holdsAt(F, T + 1) ← happens(E, T) ∧ initiates(E, F, T)    F × E × T holdsAt(F, T + 1) ← holdsAt(F, T) ∧ ¬∃ E happens(E, T) ∧ teminates(E, F, T)        F × E × T

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Discrete Event Calculus

◮ Logically equivalent with EC, when the domain of timepoints

is limited to integers

◮ Axioms are defined over successive timepoints

holdsAt(F, T + 1) ← happens(E, T) ∧ initiates(E, F, T)    F × E × T holdsAt(F, T + 1) ← holdsAt(F, T) ∧ ¬∃ E happens(E, T) ∧ teminates(E, F, T)        F × E × T Creates 2|E| clauses

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Simplifying Discrete Event Calculus

General form of domain-dependent HLE definition: initiatedAt(fluentk, T) ← happens(eventn, T) ∧ ... ∧ Conditions[T] terminatedAt(fluentk, T) ← happens(eventn, T) ∧ ... ∧ Conditions[T]

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Simplifying Discrete Event Calculus

holdsAt(F, T + 1) ← happens(E, T) ∧ initiates(E, F, T) holdsAt(F, T + 1) ← initiatedAt(F, T)

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Simplifying Discrete Event Calculus

holdsAt(F, T + 1) ← happens(E, T) ∧ initiates(E, F, T) holdsAt(F, T + 1) ← initiatedAt(F, T) holdsAt(F, T + 1) ← holdsAt(F, T) ∧ ¬∃ E happens(E, T) ∧ teminates(E, F, T) holdsAt(F, T + 1) ← holdsAt(F, T) ∧ ¬teminatedAt(F, T)

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Simplified Discrete Event Calculus

When a fluent holds: holdsAt(F, T + 1) ← initiatedAt(F, T)

  • F × T

holdsAt(F, T + 1) ← holdsAt(F, T) ∧ ¬teminatedAt(F, T)    F × T When a fluent does not hold: ¬holdsAt(F, T + 1) ← terminatedAt(F, T)

  • F × T

¬holdsAt(F, T + 1) ← ¬holdsAt(F, T) ∧ ¬initiatedAt(F, T)    F × T

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Example: HLE definition

When the fluent ’meeting’ is initiated:

initiatedAt(meeting, T) ← happens(event1, T) ∧ ¬happens(event2, T) ∧ distance(close, T) initiatedAt(meeting, T) ← happens(event3, T) ∧ ¬happens(event1, T) ∧ ¬happens(event2, T) ∧ distance(close, T)

When the fluent ’meeting’ is terminated:

terminatedAt(meeting, T) ← happens(event4, T) ...

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Open-world semantics in MLN

Domain-dependent definitions:

◮ Conditions under which HLE are initiated or terminated ◮ Open-world assumption for non-evidence predicates:

initiatedAt, terminatedAt and holdsAt When something is happening that it is not defined in the domain-dependent definitions:

◮ Cannot determine whether a fluent holds or not ◮ Loss of the inertia ◮ This is also known as the frame problem ◮ Solution: predicate completion

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Predicate completion

HLE definitions =                           

initiatedAt(meeting, T) ← happens(event1, T) ∧ ¬happens(event2, T) ∧ distance(close, T) initiatedAt(meeting, T) ← happens(event3, T) ∧ ¬happens(event1, T) ∧ ¬happens(event2, T) ∧ distance(close, T) ...

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Predicate completion

HLE definitions =                           

initiatedAt(meeting, T) ← happens(event1, T) ∧ ¬happens(event2, T) ∧ distance(close, T) initiatedAt(meeting, T) ← happens(event3, T) ∧ ¬happens(event1, T) ∧ ¬happens(event2, T) ∧ distance(close, T) ...

Completion constraints = (automatically generated)                       

initiatedAt(meeting, T) → [happens(event1, T) ∧ ¬happens(event2, T) ∧ distance(close, T) ] [happens(event3, T) ∧ ¬happens(event1, T) ∧ ¬happens(event2, T) ∧ distance(close, T) ] ...

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Predicate completion

HLE definitions =                           

1.5 initiatedAt(meeting, T) ← happens(event1, T) ∧ ¬happens(event2, T) ∧ distance(close, T) 0.25 initiatedAt(meeting, T) ← happens(event3, T) ∧ ¬happens(event1, T) ∧ ¬happens(event2, T) ∧ distance(close, T) ...

Completion constraints = (automatically generated)                       

4.0 initiatedAt(meeting, T) → [happens(event1, T) ∧ ¬happens(event2, T) ∧ distance(close, T) ] [happens(event3, T) ∧ ¬happens(event1, T) ∧ ¬happens(event2, T) ∧ distance(close, T) ] ...

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Inertia in MLN

EC domain-independent axioms are hard-constrained and:

  • 1. Only HLE definitions are soft-constrained

3 10 20 Time 0.0 0.5 1.0

Initiated Initiated Terminated

P(world) ∝ exp

  • (weights of formulas it satisfies)
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Inertia in MLN

EC domain-independent axioms are hard-constrained and:

  • 2. Only HLE definitions are soft-constrained and the termination

rules in the completion constraints 3 10 20 Time 0.0 0.5 1.0

Initiated Initiated Terminated

P(world) ∝ exp

  • (weights of formulas it satisfies)
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Inertia in MLN

EC domain-independent axioms are hard-constrained and:

  • 2. Only HLE definitions are soft-constrained and the termination

rules in the completion constraints 3 10 20 Time 0.0 0.5 1.0

Initiated Initiated Terminated

P(world) ∝ exp

  • (weights of formulas it satisfies)
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Experiments — CAVIAR dataset

◮ 28 surveillance videos ◮ LLE: active, inactive, walking, running, enter and exit ◮ HLE: meeting, moving, fighting and leaving an object Input: ... happens(active(id1),10) happens(walking(id2),10) happens(enter(id3),10) ... happens(running(id3),90) ... close(id1,id2,25,10) close(id1,id2,25,11) ...

MLN

Output: ... 0.7 holdsAt(meet(id1,id2), 10) 0.6 holdsAt(meet(id1,id2), 11) ... 0.2 holdsAt(meet(id1,id2), 90) ... Knowledge base: Event Calculus axioms HLE definitions Additional constraints

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Experiments — results

◮ EC-LP:

◮ Logic-programming based EC ◮ Knowledge base of HLE for CAVIAR dataset

◮ DEC-MLN:

◮ Markov Logic Networks based EC ◮ Manualy adjusted weight values for the HLE meeting: ◮ weak values — low confidence ◮ strong values — high confidence ◮ DEC-MLNa: soft-constrained HLE definitions ◮ DEC-MLNb: soft-constrained HLE definitions and termination

rules in the completion constraints

Method TP FP FN Precision Recall EC-LP 3099 2258 525 0.578 0.855 DEC-MLNa 3048 1762 576 0.633 0.841 DEC-MLNb 3048 1154 576 0.725 0.841

Source KB and dataset files can be found in http://www.iit.demokritos.gr/∼anskarl

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Conclusions

◮ Probabilistic extension of Event Calculus ◮ Formal and declarative semantics ◮ Probabilistic inference ◮ Emphasis on simplifying the axioms of the EC — reduce the

size and the complexity of the network

◮ Deterministic or probabilistic control of the inertia

Future directions:

◮ Machine learning — estimate weight values ◮ Experiments in other real-world data sets Source KB and dataset files can be found in http://www.iit.demokritos.gr/∼anskarl

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Thank you for your attention! Any questions?

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