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
Linköping 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
Linköping 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
Linköping 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
Linköping 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
Linköping 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
Linköping 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
Linköping 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
Linköping 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
Linköping 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
Linköping 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
Linköping 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|t Altitude[uav1]
Predicted feature:
t 0 |t Altitude[uav1]
(t ≤ 0)
(t ≤ 0, t 0 6= t)
Altitude[uav1] − Altitude[roofA]) > 2
Pr ((
0|0 Altitude[uav1]
−
0|0 Altitude[roofA])
> 2) ≥ 0.99
Pr ((
3|0 Altitude[uav1]
−
0|0 Altitude[roofA])
> 2) ≥ 0.99
Mattias Tiger, Fredrik Heintz
Linköping 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|t Altitude[uav1]
Predicted feature:
t 0 |t Altitude[uav1]
(t ≤ 0)
(t ≤ 0, t 0 6= t)
Altitude[uav1] − Altitude[roofA]) > 2
Pr ((
0|0 Altitude[uav1]
−
0|0 Altitude[roofA])
> 2) ≥ 0.99
Pr ((
3|0 Altitude[uav1]
−
0|0 Altitude[roofA])
> 2) ≥ 0.99
Mattias Tiger, Fredrik Heintz
Linköping 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|t Altitude[uav1]
Predicted feature:
t 0 |t Altitude[uav1]
(t ≤ 0)
(t ≤ 0, t 0 6= t)
Altitude[uav1] − Altitude[roofA]) > 2
Pr ((
0|0 Altitude[uav1]
−
0|0 Altitude[roofA])
> 2) ≥ 0.99
Pr ((
3|0 Altitude[uav1]
−
0|0 Altitude[roofA])
> 2) ≥ 0.99
Mattias Tiger, Fredrik Heintz
Linköping 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|t Altitude[uav1]
Predicted feature:
t 0 |t Altitude[uav1]
(t ≤ 0)
(t ≤ 0, t 0 6= t)
Altitude[uav1] − Altitude[roofA]) > 2
Pr ((
0|0 Altitude[uav1]
−
0|0 Altitude[roofA])
> 2) ≥ 0.99
Pr ((
3|0 Altitude[uav1]
−
0|0 Altitude[roofA])
> 2) ≥ 0.99
Mattias Tiger, Fredrik Heintz
Linköping 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|t Altitude[uav1]
Predicted feature:
t 0 |t Altitude[uav1]
(t ≤ 0)
(t ≤ 0, t 0 6= t)
Altitude[uav1] − Altitude[roofA]) > 2
Pr ((
0|0 Altitude[uav1]
−
0|0 Altitude[roofA])
> 2) ≥ 0.99
Pr ((
3|0 Altitude[uav1]
−
0|0 Altitude[roofA])
> 2) ≥ 0.99
Mattias Tiger, Fredrik Heintz
Linköping 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 |
t1 α
|
[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
Linköping 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
Linköping 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 = hT , O, F, F̄, X , D, T , Pi
Mattias Tiger, Fredrik Heintz
Linköping 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
Linköping 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
Linköping 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
Linköping 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|0 Position[uav1], 1mCircle))
> 0.99
3|0 Position[uav1], 1mCircle))
> 0.95
∧
Pr (insideRelative(
∧
3
similarity (
0|0 Position[uav1],
Mattias Tiger, Fredrik Heintz
0|−3 Position[uav1])
Linköping University
> 0.5
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
1
M. Tiger, F. Heintz, Stream Reasoning using Temporal Logic and
Predictive Probabilistic State Models, in Proc. TIME, 2016
Mattias Tiger, Fredrik Heintz
Linköping University
13/13