Stream Reasoning using Temporal Logic and Predictive Probabilistic - - PowerPoint PPT Presentation

Stream Reasoning using Temporal Logic and Predictive Probabilistic State Models

Mattias Tiger Fredrik Heintz

Artificial Intelligence and Integrated Computer Systems Department of Computer and Information Science Link¨

• ping University, Sweden

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary Motivation Stream Reasoning

Execution Monitoring in Robotics

Am I in a no-fly zone? - boolean Is it likely that I am in a no-fly zone? Is it likely that I am about to crash into the wall in the near future?

Mattias Tiger, Fredrik Heintz Link¨

• ping University

2/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary Motivation Stream Reasoning

Execution Monitoring in Robotics

Am I in a no-fly zone? - boolean Is it likely that I am in a no-fly zone? Is it likely that I am about to crash into the wall in the near future?

Mattias Tiger, Fredrik Heintz Link¨

• ping University

2/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary Motivation Stream Reasoning

Metric Temporal Logic (MTL) formulas are evaluated over the stream (infinite state sequence) using Progression. Incremental evaluation by formula re-writing to incorporate what has been observed so far.

Mattias Tiger, Fredrik Heintz Link¨

• ping University

3/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary Intuition P-MTL: Stochastic temporal term operator P-MTL: Syntax P-MTL: Grounding

What do we know (about terms) at tk?

Mattias Tiger, Fredrik Heintz Link¨

• ping University

4/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary Intuition P-MTL: Stochastic temporal term operator P-MTL: Syntax P-MTL: Grounding

What do we know (about terms) at tk?

Mattias Tiger, Fredrik Heintz Link¨

• ping University

5/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary Intuition P-MTL: Stochastic temporal term operator P-MTL: Syntax P-MTL: Grounding

What do we know (about terms) at tk?

Mattias Tiger, Fredrik Heintz Link¨

• ping University

5/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary Intuition P-MTL: Stochastic temporal term operator P-MTL: Syntax P-MTL: Grounding

What do we know (about terms) at tk?

Mattias Tiger, Fredrik Heintz Link¨

• ping University

5/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary Intuition P-MTL: Stochastic temporal term operator P-MTL: Syntax P-MTL: Grounding

What do we know (about terms) at tk?

Mattias Tiger, Fredrik Heintz Link¨

• ping University

5/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary Intuition P-MTL: Stochastic temporal term operator P-MTL: Syntax P-MTL: Grounding

Truth values of predicates Numerical values of terms Stochastic estimates of terms (Green, solid outline) Stochastic predictions of terms (Red, dashed outline)

Mattias Tiger, Fredrik Heintz Link¨

• ping University

6/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary Intuition P-MTL: Stochastic temporal term operator P-MTL: Syntax P-MTL: Grounding

An alternative view

Mattias Tiger, Fredrik Heintz Link¨

• ping University

7/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary Intuition P-MTL: Stochastic temporal term operator P-MTL: Syntax P-MTL: Grounding

P-MTL is MTL extended with a stochastic temporal term operator Estimated feature:

t|tAltitude[uav1] (t ≤ 0)

Predicted feature:

t′|tAltitude[uav1] (t ≤ 0, t′ = t)

• Altitude[uav1] − Altitude[roofA]) > 2
• Pr((

0|0Altitude[uav1] − 0|0Altitude[roofA]) > 2) ≥ 0.99

• Pr((

3|0Altitude[uav1] − 0|0Altitude[roofA]) > 2) ≥ 0.99

• Mattias Tiger, Fredrik Heintz

Link¨

• ping University

8/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary Intuition P-MTL: Stochastic temporal term operator P-MTL: Syntax P-MTL: Grounding

P-MTL is MTL extended with a stochastic temporal term operator Estimated feature:

t|tAltitude[uav1] (t ≤ 0)

Predicted feature:

t′|tAltitude[uav1] (t ≤ 0, t′ = t)

• Altitude[uav1] − Altitude[roofA]) > 2
• Pr((

0|0Altitude[uav1] − 0|0Altitude[roofA]) > 2) ≥ 0.99

• Pr((

3|0Altitude[uav1] − 0|0Altitude[roofA]) > 2) ≥ 0.99

• Mattias Tiger, Fredrik Heintz

Link¨

• ping University

8/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary Intuition P-MTL: Stochastic temporal term operator P-MTL: Syntax P-MTL: Grounding

P-MTL is MTL extended with a stochastic temporal term operator Estimated feature:

t|tAltitude[uav1] (t ≤ 0)

Predicted feature:

t′|tAltitude[uav1] (t ≤ 0, t′ = t)

• Altitude[uav1] − Altitude[roofA]) > 2
• Pr((

0|0Altitude[uav1] − 0|0Altitude[roofA]) > 2) ≥ 0.99

• Pr((

3|0Altitude[uav1] − 0|0Altitude[roofA]) > 2) ≥ 0.99

• Mattias Tiger, Fredrik Heintz

Link¨

• ping University

8/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary Intuition P-MTL: Stochastic temporal term operator P-MTL: Syntax P-MTL: Grounding

P-MTL is MTL extended with a stochastic temporal term operator Estimated feature:

t|tAltitude[uav1] (t ≤ 0)

Predicted feature:

t′|tAltitude[uav1] (t ≤ 0, t′ = t)

• Altitude[uav1] − Altitude[roofA]) > 2
• Pr((

0|0Altitude[uav1] − 0|0Altitude[roofA]) > 2) ≥ 0.99

• Pr((

3|0Altitude[uav1] − 0|0Altitude[roofA]) > 2) ≥ 0.99

• Mattias Tiger, Fredrik Heintz

Link¨

• ping University

8/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary Intuition P-MTL: Stochastic temporal term operator P-MTL: Syntax P-MTL: Grounding

P-MTL is MTL extended with a stochastic temporal term operator Estimated feature:

t|tAltitude[uav1] (t ≤ 0)

Predicted feature:

t′|tAltitude[uav1] (t ≤ 0, t′ = t)

• Altitude[uav1] − Altitude[roofA]) > 2
• Pr((

0|0Altitude[uav1] − 0|0Altitude[roofA]) > 2) ≥ 0.99

• Pr((

3|0Altitude[uav1] − 0|0Altitude[roofA]) > 2) ≥ 0.99

• Mattias Tiger, Fredrik Heintz

Link¨

• ping University

8/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary Intuition P-MTL: Stochastic temporal term operator P-MTL: Syntax P-MTL: Grounding

Predicates P(τ1, . . . , τn) | ¬α | α ∧ β | α ∨ β | α → β |

t1α | [t1,t2] α |

α |

[t1,t2] α |

α Terms ¯ f [const] |

t1 ¯

f [const] |

t1|t2 ¯

f [const] | const | f (τ1, . . . , τn) | Pr(g(τp, c1, . . . , cm))

Mattias Tiger, Fredrik Heintz Link¨

• ping University

9/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary Intuition P-MTL: Stochastic temporal term operator P-MTL: Syntax P-MTL: Grounding

Grounding of P-MTL terms in computational environment

Mattias Tiger, Fredrik Heintz Link¨

• ping University

10/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary Intuition P-MTL: Stochastic temporal term operator P-MTL: Syntax P-MTL: Grounding

Grounding of P-MTL terms in computational environment

E = T, O, F, ¯ F, X, D, T , P

Mattias Tiger, Fredrik Heintz Link¨

• ping University

10/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary

Example: Execution Monitoring A UAV may only move under the conditions that Its perception is precise

The estimate of its position to be within a 1m radius circle with 99% probability

Its near-time predictions are precise

The prediction of its position 3 seconds from now must be within a 1m radius circle with 95% probability

Its near-time prediction quality is high

The prediction must match with the then estimated position with at least 50% similarity.

Mattias Tiger, Fredrik Heintz Link¨

• ping University

11/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary

Example: Execution Monitoring A UAV may only move under the conditions that Its perception is precise

The estimate of its position to be within a 1m radius circle with 99% probability

Its near-time predictions are precise

The prediction of its position 3 seconds from now must be within a 1m radius circle with 95% probability

Its near-time prediction quality is high

The prediction must match with the then estimated position with at least 50% similarity.

Mattias Tiger, Fredrik Heintz Link¨

• ping University

11/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary

Example: Execution Monitoring A UAV may only move under the conditions that Its perception is precise

The estimate of its position to be within a 1m radius circle with 99% probability

Its near-time predictions are precise

The prediction of its position 3 seconds from now must be within a 1m radius circle with 95% probability

Its near-time prediction quality is high

The prediction must match with the then estimated position with at least 50% similarity.

Mattias Tiger, Fredrik Heintz Link¨

• ping University

11/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary

The estimate of its position to be within a 1m radius circle with 99% probability The prediction of its position 3 seconds from now must be within a 1m radius circle with 95% probability The prediction must match with the then estimated position with at least 50% similarity.

• Pr(insideRelative(

0|0Position[uav1], 1mCircle)) > 0.99

∧ Pr(insideRelative(

3|0Position[uav1], 1mCircle)) > 0.95

∧

3

• similarity(

0|0Position[uav1], 0|−3Position[uav1]) > 0.5

• Mattias Tiger, Fredrik Heintz

Link¨

• ping University

12/13

Introduction Stochastic and Predictive Stream Reasoning UAV example Summary

We introduce1 P-MTL as an extension to MTL Our contribution is a formal interface between existing logical reasoning and existing probabilistic reasoning methods: A formal framework with an explicit separation A selection of important temporal and probabilistic concepts from probability theory can be referred to at the logical level Both aspects retain strengths and computational complexities

• 1M. Tiger, F. Heintz, Stream Reasoning using Temporal Logic and

Predictive Probabilistic State Models, in Proc. TIME, 2016

Mattias Tiger, Fredrik Heintz Link¨

• ping University

13/13