The document discusses temporal planning and modeling of actions over time. It introduces the concept of representing planning problems using a time-oriented view with timelines rather than a state-oriented view. A timeline consists of temporal assertions about state variables over time intervals along with constraints. Actions are modeled as triples containing a name, a set of temporal assertions describing the effects over time, and constraints. This allows overlapping actions and reasoning about how state variable values change over time to be represented.
Solving connectivity problems via basic Linear Algebracseiitgn
Directed reachability and undirected connectivity are well studied problems in Complexity Theory. Reachability/Connectivity between distinct pairs of vertices through disjoint paths are well known but hard variations. We talk about recent algorithms to solve variants and restrictions of these problems in the static and dynamic settings by reductions to the determinant.
Efficient Scalar Multiplication for Ate Based Pairing over KSS Curve of Embed...Md. Al-Amin Khandaker Nipu
Efficiency of the next generation pairing based security pro- tocols rely not only on the faster pairing calculation but also on efficient scalar multiplication on higher degree rational points. In this paper we proposed a scalar multiplication technique in the context of Ate based pairing with Kachisa-Schaefer-Scott (KSS) pairing friendly curves with embedding degree k = 18 at the 192-bit security level. From the system- atically obtained characteristics p, order r and Frobenious trace t of KSS curve, which is given by certain integer z also known as mother parame- ter, we exploit the relation #E(Fp) = p+1−t mod r by applying Frobe- nius mapping with rational point to enhance the scalar multiplication. In addition we proposed z-adic representation of scalar s. In combination of Frobenious mapping with multi-scalar multiplication technique we ef- ficiently calculate scalar multiplication by s. Our proposed method can achieve 3 times or more than 3 times faster scalar multiplication com- pared to binary scalar multiplication, sliding-window and non-adjacent form method.
Solving connectivity problems via basic Linear Algebracseiitgn
Directed reachability and undirected connectivity are well studied problems in Complexity Theory. Reachability/Connectivity between distinct pairs of vertices through disjoint paths are well known but hard variations. We talk about recent algorithms to solve variants and restrictions of these problems in the static and dynamic settings by reductions to the determinant.
Efficient Scalar Multiplication for Ate Based Pairing over KSS Curve of Embed...Md. Al-Amin Khandaker Nipu
Efficiency of the next generation pairing based security pro- tocols rely not only on the faster pairing calculation but also on efficient scalar multiplication on higher degree rational points. In this paper we proposed a scalar multiplication technique in the context of Ate based pairing with Kachisa-Schaefer-Scott (KSS) pairing friendly curves with embedding degree k = 18 at the 192-bit security level. From the system- atically obtained characteristics p, order r and Frobenious trace t of KSS curve, which is given by certain integer z also known as mother parame- ter, we exploit the relation #E(Fp) = p+1−t mod r by applying Frobe- nius mapping with rational point to enhance the scalar multiplication. In addition we proposed z-adic representation of scalar s. In combination of Frobenious mapping with multi-scalar multiplication technique we ef- ficiently calculate scalar multiplication by s. Our proposed method can achieve 3 times or more than 3 times faster scalar multiplication com- pared to binary scalar multiplication, sliding-window and non-adjacent form method.
A Unifying Review of Gaussian Linear Models (Roweis 1999)Feynman Liang
Through a linear Gaussian process, we can unify a family of Gaussian linear models including Factor Analysis, PCA, Kalman Filters, Mixture of Gaussians, and Hidden Markov Models.
Control of Uncertain Nonlinear Systems Using Ellipsoidal Reachability CalculusLeo Asselborn
This paper proposes an approach to algorithmically synthesize control strategies for discrete-time nonlinear uncertain systems based on reachable set computations using the ellipsoidal calculus. For given ellipsoidal initial sets and bounded ellipsoidal disturbances, the proposed algorithm iterates over conservatively approximating and LMI-constrained optimization problems to compute stabilizing controllers. The method uses
first-order Taylor approximation of the nonlinear dynamics and a conservative approximation of the Lagrange remainder.
We present a proof of the Generalized Riemann hypothesis (GRH) based on asymptotic expansions and operations on series. The advantage of our method is that it only uses undergraduate maths which makes it accessible to a wider audience.
We present a proof of the Generalized Riemann hypothesis (GRH) based on asymptotic expansions and operations on series. The advantage of our method is that it only uses undergraduate maths which makes it accessible to a wider audience.
Ilya Shkredov – Subsets of Z/pZ with small Wiener norm and arithmetic progres...Yandex
It is proved that any subset of Z/pZ, p is a prime number, having small Wiener norm (l_1-norm of its Fourier transform) contains a subset which is close to be an arithmetic progression. We apply the obtained results to get some progress in so-called Littlewood conjecture in Z/pZ as well as in a quantitative version of Beurling-Helson theorem.
Multidimensional time series appear in many fields of application. Sometimes, it can be useful to use PCA to reach dimensionality reduction. However, formal inference procedures on PC rely on the independence of the variables. Therefore, several PC-like techniques, as Singular Spectrum Analysis, are used to attain this reduction by decomposing the original series into a sum of a small number of interpretable components. Here, SSA and its extension are described and applied to real datasets.
Playing Atari with Deep Reinforcement Learning郁凱 黃
Playing Atari with Deep Reinforcement Learning
- Author: Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Alex Graves, Ioannis Antonoglou, Daan Wierstra, Martin Riedmiller
- Origin: https://arxiv.org/abs/1312.5602
- Related: https://github.com/number9473/nn-algorithm/issues/250
Probabilistic Control of Switched Linear Systems with Chance ConstraintsLeo Asselborn
An approach to algorithmically synthesize control
strategies for set-to-set transitions of uncertain discrete-time
switched linear systems based on a combination of tree search
and reachable set computations in a stochastic setting is
proposed in this presentation. The initial state and disturbances
are assumed to be Gaussian distributed, and a time-variant
hybrid control law stabilizes the system towards a goal set.
The algorithmic solution computes sequences of discrete states
via tree search and the continuous controls are obtained
from solving embedded semi-definite programs (SDP). These
program taking polytopic input constraints as well as timevarying
probabilistic state constraints into account. An example
for demonstrating the principles of the solution procedure with
focus on handling the chance constraints is included.
This presentation discuss a sufficient and necessary condition for quadratic stability of a class of switched systems including two modes. This result has been published in proceeding of the ECC Conference in Zürich, 2013.
A Unifying Review of Gaussian Linear Models (Roweis 1999)Feynman Liang
Through a linear Gaussian process, we can unify a family of Gaussian linear models including Factor Analysis, PCA, Kalman Filters, Mixture of Gaussians, and Hidden Markov Models.
Control of Uncertain Nonlinear Systems Using Ellipsoidal Reachability CalculusLeo Asselborn
This paper proposes an approach to algorithmically synthesize control strategies for discrete-time nonlinear uncertain systems based on reachable set computations using the ellipsoidal calculus. For given ellipsoidal initial sets and bounded ellipsoidal disturbances, the proposed algorithm iterates over conservatively approximating and LMI-constrained optimization problems to compute stabilizing controllers. The method uses
first-order Taylor approximation of the nonlinear dynamics and a conservative approximation of the Lagrange remainder.
We present a proof of the Generalized Riemann hypothesis (GRH) based on asymptotic expansions and operations on series. The advantage of our method is that it only uses undergraduate maths which makes it accessible to a wider audience.
We present a proof of the Generalized Riemann hypothesis (GRH) based on asymptotic expansions and operations on series. The advantage of our method is that it only uses undergraduate maths which makes it accessible to a wider audience.
Ilya Shkredov – Subsets of Z/pZ with small Wiener norm and arithmetic progres...Yandex
It is proved that any subset of Z/pZ, p is a prime number, having small Wiener norm (l_1-norm of its Fourier transform) contains a subset which is close to be an arithmetic progression. We apply the obtained results to get some progress in so-called Littlewood conjecture in Z/pZ as well as in a quantitative version of Beurling-Helson theorem.
Multidimensional time series appear in many fields of application. Sometimes, it can be useful to use PCA to reach dimensionality reduction. However, formal inference procedures on PC rely on the independence of the variables. Therefore, several PC-like techniques, as Singular Spectrum Analysis, are used to attain this reduction by decomposing the original series into a sum of a small number of interpretable components. Here, SSA and its extension are described and applied to real datasets.
Playing Atari with Deep Reinforcement Learning郁凱 黃
Playing Atari with Deep Reinforcement Learning
- Author: Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Alex Graves, Ioannis Antonoglou, Daan Wierstra, Martin Riedmiller
- Origin: https://arxiv.org/abs/1312.5602
- Related: https://github.com/number9473/nn-algorithm/issues/250
Probabilistic Control of Switched Linear Systems with Chance ConstraintsLeo Asselborn
An approach to algorithmically synthesize control
strategies for set-to-set transitions of uncertain discrete-time
switched linear systems based on a combination of tree search
and reachable set computations in a stochastic setting is
proposed in this presentation. The initial state and disturbances
are assumed to be Gaussian distributed, and a time-variant
hybrid control law stabilizes the system towards a goal set.
The algorithmic solution computes sequences of discrete states
via tree search and the continuous controls are obtained
from solving embedded semi-definite programs (SDP). These
program taking polytopic input constraints as well as timevarying
probabilistic state constraints into account. An example
for demonstrating the principles of the solution procedure with
focus on handling the chance constraints is included.
This presentation discuss a sufficient and necessary condition for quadratic stability of a class of switched systems including two modes. This result has been published in proceeding of the ECC Conference in Zürich, 2013.
introduction to Knowledge - Types of Knowledge - Knowledge Management: goals and objectives of KM, Knowledge worker and its role importance of Knowledge worker and characteristics of Knowledge worker
Presentation of the paper:
Szymon Klarman and Thomas Meyer. Querying Temporal Databases via OWL 2 QL (with appendix). In Proceedings of the 8th International Conference on Web Reasoning and Rule Systems (RR-14), 2014.
Research Inventy : International Journal of Engineering and Scienceresearchinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
The derivative is a major tool for investigating the behavior of a function. Since functions are ubiquitous, so are their derivatives. Velocity, growth rates, marginal costs, and material strain are all examples of derivatives. We motivate and define the derivative and compute a few examples, then discuss how features of a function are manifested in its derivative.
Oscillation results for second order nonlinear neutral delay dynamic equation...inventionjournals
In this paper, we establish sufficient conditions for the oscillation of solutions of second order neutral delay dynamic equations [r (t )( x (t ) p (t ) x ( (t ))) ] q (t ) f ( x ( (t ))) = 0 on an arbitrary time scale 핋.
Text Book: An Introduction to Mechanics by Kleppner and Kolenkow
Chapter 1: Vectors and Kinematics
-Explain the concept of vectors.
-Explain the concepts of position, velocity and acceleration for different kinds of motion.
References:
Halliday, Resnick and Walker
Berkley Physics Volume-1
Tenser Product of Representation for the Group CnIJERA Editor
The main objective of this paper is to compute the tenser product of representation for the group Cn. Also
algorithms designed and implemented in the construction of the main program designated for the determination
of the tenser product of representation for the group Cn including a flow-diagram of the main program. Some
algorithms are followed by simple examples for illustration.
1. 1
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
This
work
is
licensed
under
a
CreaBve
Commons
AEribuBon-‐NonCommercial-‐ShareAlike
4.0
InternaBonal
License.
Chapter
4
Delibera.on
with
Temporal
Domain
Models
Dana S. Nau and Vikas Shivashankar
University of Maryland
2. 2
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
Temporal
Planning
time
statevariables
state
timeline
● Motivation: changes don’t occur instantaneously
Ø Actions take time, may overlap
Ø Need ways to reason about this
● Up to now, we’ve used a “state-oriented view”
• Sequence of states s0, s1, s2
• Actions that transform each state into the next one
Ø No way to reason about overlapping actions
● Switch to a “time-oriented view”
Ø Sequence of time points
• t = 1, 2, 3, …
Ø For each state variable x,
different values during
different time intervals
Ø At each time t, the state is
the set of atoms that are
true at time t
3. 3
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
time
loc(r1)
loc1
loc2
l
t1 t2 t3 t4
Change
Persistence
Temporal
Asser.ons
and
Constraints
● Temporal assertion:
Ø a state variable’s value
during a time interval
● Examples:
change: [t1,t2] loc(r1):(loc1, l) during [t1,t2], loc(r1)
changes from loc1
to l
persistence: [t2,t3] loc(r1)=l during [t2,t3], loc(r1)
= l
change: [t3,t4] loc(r1):(l, loc2) during [t3,t4], loc(r1)
changes from l to loc1
● These assertions entail the following temporal constraints
Ø t1 < t2 < t3 < t4
● They also entail the following object constraints
Ø l ≠ loc1, l ≠ loc2
● We may want to specify additional constraints that aren’t entailed
4. 4
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
Timelines
time
loc(r1)
loc1
loc2
l
t1 t2 t3 t4
Change
Persistence
● Timeline: a pair (T,C )
Ø T is a set of temporal
assertions for the
same state variable
Ø C is a set of constraints
● T = {[t1,t2] loc(r1):(loc1,l), [t2,t3] loc(r1)=l, [t3,t4] loc(r1):(l,loc2)}
● C = (t1 < t2 < t3 < t4, l ≠ loc1, l ≠ loc2)
Ø written with parentheses rather than set braces
● If a constraint is entailed by T then there’s no need to have it in C too
Ø In the above example, (T,C ) is equivalent to (T, ∅)
5. 5
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
Timelines
● [t1,t2] loc(r1):(loc1,l)
• sometime during [t1,t2], r1
leaves loc1
• sometime during [t1,t2], r1
arrives at l
Ø But the expression doesn’t say exactly when
● If we want to be more specific, we could add these to T
• [t1,t1+1] loc(r1):(loc1,route)
• [t2–1,t2] loc(r1):(route,l)
Ø Each change requires at least one unit of time
time
loc(r1)
loc1
loc2
l
t1 t2 t3 t4
Change
Persistence
6. 6
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
Separa.on
Constraints
● A set of assertions T is nonconflicting if it’s consistent for all variable assignments
Ø e.g., T1 = {[t1,t2] loc(r1)=loc1, [t2,t3] loc(r1):(loc1,
loc2)}
● Otherwise T is potentially conflicting
Ø e.g., T2 = {[t1,t2] loc(r1)=loc1, [t3,t4] loc(r1):(l,lʹ)}
• e.g., it would be inconsistent if t1 < t3 < t2 and l ≠ loc1
● Separation constraint: a set of constraints that prevents the conflict
Ø Some possible separation constraints for C2:
Ø (t2 < t3), (t4 < t1), (t2 ≤ t3, l=loc1), (t4 ≤ t1, l′=loc1)
● (T,C) is secure if T is nonconflicting, or C entails a separation constraint for T
• e.g., (T1, ∅) or (T2, (t2 < t3))
● (T,C) is viable if either T is nonconflicting, or there exists a separation constraint
Cʹ for T that is consistent with C
• (T1, ∅) nonconflicting
• (T2, ∅) Cʹ = (t2 < t3)
Ø If (T,C) is viable and Cʹ is as in the definition, then (T,C ∪ Cʹ) is secure
7. 7
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
Causal
Support
● In a timeline (T,C), let α be one of the following:
Ø a persistence assertion, e.g., [t,t′] x=v
Ø a change assertion, e.g., [t,t′] x:(v,v′)
● α is causally supported if T contains another temporal assertion that establishes
x=v at time t
Ø Causal support could have one of the following forms:
a persistence assertion [t′′,t] x=v
a change assertion [t′′,t] x:(v′′,v)
8. 8
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
Causal
Support
● Example
T = {[t1,t2] loc(r1):(loc1,l),
[t2,t3] loc(r1)=l,
[t3,t4] loc(r1):(l,loc2)}
C = (t4 < t5, l ≠ loc1, l ≠ loc2)
Ø [t2,t3] loc(r1)=l is causally supported by [t1,t2] loc(r1):(loc1,l)
Ø [t3,t4] loc(r1):(l,loc2) is causally supported by [t2,t3] loc(r1)=l
Ø [t1,t2] loc(r1):(loc1,l) isn’t causally supported
time
loc(r1)
loc1
loc2
l
t1 t2 t3 t4
Change
Persistence
9. 9
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
Temporal
Model
of
an
Ac.on
● Each action is represented by a triple (head,T,C), where
Ø head is the name and arguments
Ø T is a set of temporal assertions about one or more state variables
• i.e., T is the union of one or more timelines
Ø T is a set of constraints
● On the next two slides, an example
10. 10
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
Objects,
Rigid
Rela.ons,
State
Variables
● Robots moving among loading docks connected by a network of roads
● Objects and ranges of variables:
Ø r ∈ Robots, k ∈ Cranes, c ∈ Containers, p ∈ Piles, d ∈ Docks, w ∈ Roads
● Rigid relations:
Ø aEached ⊆ (Cranes ∪ Piles) × Docks
Ø adjacent ⊆ Docks × Roads
● State variables:
Ø loc(r) ∈ Docks ∪ Roads
Ø cargo(r) ∈ Containers ∪ {nil}
Ø pos(c) ∈ Containers ∪ Robots ∪ Cranes
Ø pile(c) ∈ Piles ∪ {nil}
Ø grip(k) ∈ Containers ∪ {nil}
Ø top(p) ∈ Containers
∪ {pallet}
Ø content(d) ∈ Robots ∪ {nil}
d1
r1
d2
r2
w1
11. 11
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
d1
Ac.ons
leave(r,d,w): robot r leaves dock d to an adjacent road w
enter(r,d,w): r enters d from an adjacent road w
navigate(w,w′): navigate from road w to w′
pickup(k,c,r): crane k picks up container c from robot r
putdown(k,c,r): crane k puts down container c on r
stack(k,c,p): crane k stacks container c on top of pile p
unstack(k,c,p): crane k takes a container c from top of p
● Examples:
leave(r,d,w)
[ts,t] loc(r): (d,w)
[t,te] content(d): (r,nil)
t ≤ ts + δ
adjacent(d,w)
● No separate “preconditions” and “effects”
Ø The precondition is to establish causal support
pickup(k,c,r)
[ts,t] grip(k): (nil, c)
[ts,t] pos(c): (r,k)
[ts,t] cargo(r): (c,nil)
[ts,te] loc(r) = d
t ≤ ts + δ, t ≤ te
aEached(k,d)
r1
d2
r2
w1
c1
What’s
wrong
with
this
drawing?
w2
12. 12
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
Chronicle
● Chronicle: a triple φ = (A,T,C)
Ø A is a set of temporally qualified actions and tasks
Ø T is a set of temporally qualified assertions for a set of state variables
Ø C is a consistent set of constraints on the variables in A and T
● The following definitions all generalize to chronicles
Ø Potentially conflicting assertions
Ø Separation constraints
Ø Secure and viable timelines
Ø Causal support
13. 13
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
Example
● φ = (A,T,C) , where
● A = {[t0,t1] leave(r1,dock1,w1),
[t1,t2] navigate(r1,w1,w2),
[t2,t3] enter(r1,dock2,w2),
[t′0,t′1] leave(r2,dock2,w2),
[t′1,t′2] navigate(r2,w2,w1),
[t′2, t′3] enter(r2,dock1,w1)}
● T = {temporal assertions for the
above actions}
● C = (t′1 < t2, t1 < t′2,
adjacent(dock1,w1),
adjacent(dock2,w2))
time
r1
leave
dock1
t1 t2 t3t0
navigate enter
dock2
r2
leave
dock2
t’1 t’2 t’3t’0
navigate
enter
dock1
dock1
r1
dock2
r2
w1
w2
14. 14
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
Another
Example
● φ = (∅, {[0,1] loc(r1)=dock1,
[0,1] loc(r2)=dock2,
[5,t] docked(ship1)=pier1,
[10,t′] docked(ship2)=pier2)},
(t ≥ 5 + δ, t′ ≥ 10 + δ′))
● No actions this time
Ø Just temporal assertions and constraints
● Represents a situation where various exogenous events are scheduled to occur
15. 15
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
move
t1
t2
t3
ts
uncover
pile(c)=p’
cargo(r)=nil
load
move
unload
t4 t6t5 t7 te
Temporal
Tasks
and
Refinement
Methods
● A task is an expression of the form [t,t′] taskname(arg1,arg2,…)
Ø the task starts at or after t, and ends at or before t′
● A method is a triple (head, task, body)
Ø Will normally write it as in the following example
● Method for bringing a container c to a pile p
● m-‐bring(c,p,p′,d,d′,r)
task: bring(c,p)
body:
[ts,t1] pile(c) = p′
[ts,t1] cargo(r) = nil
[ts,t2] move(r,d′)
[ts,t3] uncover(c)
[t1,t4] load(c,r)
[t5,t6] move(r,d)
[t7,te] unload(r,p)
aEached(p′,d′), aEached(p,d), d ≠ d′
t2 ≤ t1, t3 ≤ t1, t3 ≤ t5, t6 ≤ t7
16. 16
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
More
Refinement
Methods
● A second uncover method is needed, for the case where c = top(p)
17. 17
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
Temporal
Planning
Domains/Problems
● Planning domain:
Ø Objects, rigid relations, state variables, primitive actions, temporal
refinement methods
● Planning problem:
Ø A pair (Σ,φ0)
• Σ is a temporal planning domain
• φ0 is an initial chronicle
● φ0 = () includes
Ø initial state of the world
Ø future facts that are expected to occur independently of the activities to
be planned for
Ø the tasks to be performed
● All assertions in φ0 must be non-conflicting
18. 18
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
Solu.ons
to
Temporal
Planning
Problems
● Solution plan: a chronicle φ = (A,T,C) having the following properties
Ø (i) φ contains no tasks. It contains actions and the corresponding assertions
produced
• These refine all tasks in φ0, all tasks produced by refining those tasks, and
so forth, all the way down
▸ as in the definition of a solution to a refinement planning problem
Ø (ii) All assertions in φ except for those in φ0 must be causally supported
Ø (iii) φ is secure
● Planning is done by a combination of refinement and flaw repair
Ø three types of flaws
• φ has tasks: violates condition (i)
• φ has non-supported assertions: violates condition (ii)
• φ has potentially conflicting assertions: violates condition (iii)
19. 19
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
Planning
Algorithm
● Very similar to PSP in Chapter 2
Ø Both repeatedly select flaws and
choose resolvers
Ø One uses a loop, the other uses
recursion
• Just programming style,
can be rewritten either way
• Which do you think is clearer?
● In a deterministic implementation
Ø resolver selection is a
backtracking point
Ø flaw selection isn’t
20. 20
Dana
Nau
and
Vikas
Shivashankar:
Lecture
slides
for
Automated
Planning
and
Ac0ng
Updated
4/9/15
Resolvers
for
Flaws
● First kind of flaw: φ contains a task
Ø Resolver: any applicable instance m of a refinement method
• applicable if it matches the task and its constraints are consistent with φ’s
Ø Applying the resolver:
• Modify φ by replacing the task with m
● Second kind: φ contains a temporal assertion α that isn’t causally supported (like
an open goal in PSP)
Ø Resolvers:
• Add constraints to C
• Add a persistence assertion to T
• Add a new task or action to A that supports α
● Third kind: φ contains assertions that potentially conflict (like threats in PSP)
Ø These weren’t in φ0
• Must have been introduced by a resolver for another flaw
Ø Resolvers: separation constraints
21. 21
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Example
● leave(r,d,w)
[ts,t] loc(r): (d,w)
[t,te] occupant(d): (r,nil)
adjacent(d,w)
● enter(r,d,w)
[ts,t] occupant(d): (nil,r)
[t,te] loc(r): (w,d)
adjacent(d,w)
● m-‐move(r,d1,d2,w)
task: move(r,d2)
body: [ts,t1] loc(r) = d1
[t1,t2] leave(r,d,w)
[t3,te] enter(r,d,w)
adjacent(d1,w)
adjacent(d2,w)
d1≠d2
t2 < t3
d1
r1
d2
r2
w1
● φ0 = (∅,
{[t0,t1] loc(r1)=d1,
[t0,t1] occupant(d1)=r1,
[t1,t2] move(r1,d1,d2,w1),
[t′0,t′1] loc(r2)=d2,
[t′0,t′1] occupant(d2)=r2,
[t′1,t′2]
move(r2,d2,d1,w1)},
(adjacent(d1,w1), adjacent(d2,w1),
t1 < t′2, t′1 < t2))
● Like earlier example, but simplified by removing navigation
22. 22
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NOTE
● I told Malik about the problem with TemPlan never putting the action
specifications into φ. He agrees it’s a problem, and he’ll fix it.
23. 23
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Heuris.cs
for
guiding
TemPlan
● Flaw selection, resolver selection
heuristics similar to those in PSP
Ø Select the flaw with the smallest
number of resolvers
Ø Choose the resolver that rules out
the fewest resolvers for the other
flaws
● There is also a problem with
constraint management
Ø We ignored it when discussing
PSP
Ø We’ll discuss it now
24. 24
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Constraint
Management
● Each time TemPlan
applies a resolver, it modifies (T,C)
Ø Some resolvers will make (T,C) inconsistent
▸ No solution in this part of the search space
• Detect inconsistency => prune this part of the search space
• Don’t detect it => waste time looking for a solution
● Analogy: PSP checked simple cases of inconsistency
Ø E.g., can’t create a constraint a ≺ b if there’s already a constraint b ≺ a
Ø But PSP ignored more complicated cases
● Example:
Ø Containers = {c1, c2}
Ø Range(c1) = Range(c2) = Range(c3) = Containers
Ø Suppose that to resolve 3 threats,
PSP chooses these resolvers:
• c1 ≠ c2, c2 ≠ c3, c1 ≠ c3
Ø No solutions in this part of the search
space, but PSP searches it anyway …
… …
…
…
…
…
…
…
…
…
… …
…
… …
……… …… … … … … … … …
25. 25
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Constraint
Management
in
TemPlan
● At various points, check consistency of C
Ø If C is inconsistent, then (T,C) is inconsistent
• Can prune this part of the search space
Ø If C is consistent then (T,C) may or may not be consistent
● Example of a case where C is consistent but (T,C) isn’t:
• T = {[t1, t2] loc(r1)=loc1, [t3, t4] loc(r1)=loc2}
• C = (t1 < t3 < t4 < t2)
26. 26
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Consistency
of
C
● C contains two kinds of constraints
Ø Temporal constraints
• t1 < t3
Ø Object constraints
• loc(r) ≠ l2
● The way we’ve formulated them, they are decoupled
Ø no constraint involves both objects and time points
● Two separate subproblems
Ø check consistency of object constraints
Ø check consistency of temporal constraints
Ø C is consistent iff both are consistent
27. 27
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Object
Constraints
● Constraint-satisfaction problem (CSP)
Ø Checking consistency of CSPs is NP-hard
● Can write an algorithm that’s complete but runs in exponential time
• If there’s an inconsistency, always finds it
• Might prune large parts of the search space
• But may spend lots of time at each node visited
● Some well-known constraint-satisfaction techniques
that are incomplete but run in polynomial time
• arc consistency, path consistency
Ø Detect some inconsistencies but not others
Ø Might not prune as much of the search space
Ø But less time at each node
…
… …
…
…
…
…
…
…
…
…
… …
…
… …
……… …… … … … … … … …
28. 28
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Time
Constraints
● Can check of time constraints in time O(n3)
● Simple Temporal Network (STN): a pair (τ,φ), where
• τ = {a set of temporal variables t1, …, tn}
• φ is a set of expressions that represent constraints on the variables in τ
Ø Called a network because these correspond to nodes and edges in a graph
● Each expression in φ has the form rij = [aij,bij], where aij and bij are integers
Ø Denotes a binary constraint aij ≤ tj − ti ≤ bij
Ø Thus rij = [aij,bij] is equivalent to rji = [–b,–a]
● To represent unary constraints, include a dummy variable t0 = 0
Ø r0i = [a0i,b0i] represents a0i ≤ ti − t0 ≤ b0i
• i.e., a0i ≤ ti ≤ b0i
● Solution to an STN: an integer value for each ti
Ø An STN is consistent if it has a solution
Ø An STN is minimal if for every constraint rij,
every pair (ti, tj) that satisfies rij belongs to at least one solution
ti
tj
[aij,bij]
t0
[a0i,b0i]
29. 29
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Two
Examples
● STN (τ,φ), where
Ø τ = {t1, t2, t3}
Ø φ = {r12=[1,2], r23=[3,4], r13=[2,3]}
● Transitivity:
Ø 1 ≤ t2 – t1 ≤ 2
Ø 3 ≤ t3 – t2 ≤ 4
==> 4 ≤ t3 – t1 ≤ 6
● r′13 = [4,6]
● Can’t satisfy both r13 and r′13
Ø r13 ∩ r′13 = [2,3] ∩ [4,6] = ∅
● (τ,φ) is inconsistent
● STN (τ,φ), where
Ø τ = {t1, t2, t3}
Ø φ = {r12=[1,2], r23=[3,4], r13=[2,5]}
● As before, r′13 = [4,6]
● This time, (τ,φ) is consistent
Ø r13 ∩ r′13 = [4,5]
● Change r13 to [4,5]
Ø Minimal network:
t1
t2
t3
[1,2]
[3,4]
[2,3]
t1
t2
t3
[1,2]
[3,4]
[2,5]
t1
t2
t3
[1,2]
[3,4]
[4,5]
30. 30
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Opera.ons
on
STNs
● Suppose rik = [aik,bik], rkj = [akj,bkj], rij = [aij,bij]
Ø Composition:
• rik • rkj = [aik +akj, bik +bkj]
Ø Intersection:
• rij ∩ r′ij = [max(aij +a′ij), min(bij +b′ij)]
Ø Consistency checking:
• rik , rkj , rij are consistent if rij ∩ (rik • rkj) ≠ ∅
● PC (Path Consistency) algorithm:
Ø Do the above computations
on all triples of constraints
Ø In cases where the constraint
doesn’t exist, use [−∞, +∞]
Ø n constraints => n3 triples
=> running time O(n3)
PC(τ,φ)
for k = 1 to n do
for every i ∈ {1,…, n–1} such that i ≠ k do
for every j ∈ {i+1, …, n} such that j ≠ k do
rij ← rij ∩ [rik • rkj]
if rij = ∅ then return inconsistent
ti
tk
tj
[aik,bik]
[akj,bkj]
[aij,bij]
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Example
t2
[1, 2]
[1, 2]
t1 t4
t3
t5
[3, 4]
[6, 7]
[4, 5]
● For k = 2, PC adds a constraint r14 = [4,6]
Ø Additional iterations …
● What PC is supposed to accomplish
Ø Make the STN minimal
• Shrink each time interval rik to exclude
values that aren’t part of any solution
Ø Detect inconsistent networks
• If rik = [aik,bik] is empty (i.e., bik < aik), then inconsistent
● PC
is supposed to be complete, but I’m not convinced that it is
PC(τ,φ)
for k = 1 to n do
for every i ∈ {1,…, n–1} such that i ≠ k do
for every j ∈ {i+1, …, n} such that j ≠ k do
rij ← rij ∩ [rik • rkj]
if rij = ∅ then return inconsistent
ti
tk
tj
[aik,bik]
[akj,bkj]
[aij,bij]
32. 32
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Example
● Can modify PC to make it incremental
Ø Input: a consistent, minimal STN, and
a new constraint r′ij
Ø Incorporate r′ij in time O(n2)
PC(τ,φ)
for k = 1 to n do
for every i ∈ {1,…, n–1} such that i ≠ k do
for every j ∈ {i+1, …, n} such that j ≠ k do
rij ← rij ∩ [rik • rkj]
if rij = ∅ then return inconsistent
33. 33
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Controllability
● Suppose TemPlan gives you a chronicle and you want to execute it
Ø Constraints on time points
Ø Need to reason about these in order to decide when to start each action
● Section 4.4.3 is about this
Ø Uses STNs, but doesn’t explain how you would get them from chronicles
• I need to discuss this with Malik
Ø I’ll just assume we have the STNs, and proceed from there
34. 34
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Controllability
● Example:
Ø bring&move to be done by a robot, takes 30 to 50 time units
Ø uncover to be done by a crane, takes 5 to 10 time units
● Want at most 5 seconds between the two ending times
Ø Don’t want either the crane or robot to wait long
● Can we accomplish this?
● Run PC, get a minimized network
Ø Network is consistent
Ø There exists a set of time points that satisfies all the constraints
● That assumes we can control each action’s starting time and ending time
Ø But we can’t always do that
● Suppose the time durations are nondeterministic
● Then t1 is controllable and t2 is uncontrollable
Ø We can choose the value of t1
Ø We know t2 ∈ [t1+30, t1+50], but we can’t choose which value
t1
t3
t2
t4
[30, 50]
[5, 10]
[-5,5]
bring&move
uncover
t1
t3
t2
t4
[30, 50]
[5, 10]
[-5,5]
[15,50]
[0,15]
[25,55]
35. 35
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STNUs
● STNU (Simple Temporal Network with Uncertainty):
Ø A 4-tuple (τ,E,φ,C)
• τ and E are disjoint sets of time points
• φ and C are disjoint sets of binary constraints on time points
• τ and φ are controllable (e.g., actions’ starting times
• E and C are contingent (e.g., actions’ ending times)
● Consider an action that has a controllable starting time ts ∈ τ and a contingent
ending time te ∈ E
● Suppose C contains a constraint [l,u] on ts and te
Ø Then te – ts will be somewhere in [l, u], but we don’t know where
• Like a random variable from an unknown distribution
● Want to ensure that there exists a combination of values for the variables in τ
such that for every combination of values of the variables in E,
if C is satisfied then φ is satisfied
36. 36
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Dynamic
Execu.on
Strategies
● Dynamic execution strategy: procedure that assigns values to controllable time
points, in real time
Ø E.g., if these are the starting times of actions, the strategy selects actions
one at a time, and starts executing them
● Conditions under which the strategy operates (I think)
(1) Can’t ever assign a value that’s in the past
• Suppose the strategy selects a time point t ∈ τ and gives it a value v
• Then v ≥ the current time
(2) For each contingent time point t ∈ C
• the strategy won’t learn t’s value until t is the current time
● (to be continued)