Algebraic Dynamic Programming
Unit 3.b: Products of Evaluation Algebras
Robert Giegerich1 (Lecture)
Benedikt Löwes (Exercises)
Faculty of Technology
Bielefeld University
http://www.techfak.uni-bielefeld.de/ags/pi/lehre/ADP
Summer 2014
1
robert@techfak.uni-bielefeld.de
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
1 / 23
Product Algebras
Motivation for a bag-of-surprises
You have worked with the ADP examples
Relaxed Palindromes
NW alignments
Nussinow RNA folding
using different algebras:
minimizing cost, distance
maximizing base pairs, similarity
counting numbers of candidates
enumerating candidate trees or pretty printing
So far, we have always used one algebra at a time.
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
2 / 23
Product Algebras
Algebras in combination
In working with applications, often we want several types of information
about the candidates in our search space.
Optimal solutions under a primary and a secondary objectives (the
cheapest of the largest pizzas, or vice versa)
the optimal score and the prettyprint of the candidate(s) that
achieve(s) it
the optial score and the number of co-optial candidates
enumeration of candidates in tree and prettyprint form (for
debugging)
... or combinations of the above
Whenever the desired information can be derived via various evaluation
algebras, this amounts to using several algebras simultaneously and in
cooperation.
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
3 / 23
Product Algebras
Ad-hoc combinations of evaluation algebras
Given two Σ-algebras A and B, holding (say) functions fA , fB and hA , hB ,
we can define algebra C operating on value pairs from A and B:
fC ((a, b)) =
some function using fA (a) and fB (b)
hC ([(a, b)]) = some function on lists of pairs,
implementing the desired combination of hA and hB
This means coding a new algebra (using the available fA , fB ), a new proof
of Bellman’s Principle for fC versus hC , and new debugging.
There are “standard” cases where we can do better . . . !
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
4 / 23
Product Algebras
Algebra products
We define several product operations *, ×, ... to combine two algebras in
specific ways.
Instead of hand-coding algebra C and calling
G(C , x)
we directly call
G(A ∗ B, x)
with no coding and debugging effort on our side.
The ADP compiler knows the definitin of A ∗ B and constructs and uses a
correct implementation of C .
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
5 / 23
Product Algebras
General scheme of product operations
Product algebras A ∗ b
– compute answer value pairs (a, b)
– using functions
fA ∗ fB :
hA ∗ hB :
component wise
dependent on our purpose
In fact, there are at least four useful variants of hA ∗ hB .
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
6 / 23
Product Algebras
The Cartesian product (×)
Definition
Let A and B be evaluation algebras over Σ. Their Cartesian product
algebra C = A × B is an evaluation algebra over Σ and with
fC (x1 , . . . , xk ) = (fA (a1 , . . . , ak ), fB (b1 , . . . , bk ))
if xi = (xiA , xiB ), then ai = xiA , bi = xiB ,
if xi ∈ A,
then ai = xi = bi
for each f in Σ, and with the objective function
hC [(a1 , b1 ), ..., (ak , bk )] = [(l, r )|
l ∈ hA [a1 , . . . , ak ],
r ∈ hB [b1 , ..., bk ] ]
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
7 / 23
Product Algebras
Meaning of the Cartesian Product
The Cartesian product algebra A × B
optimizes independently according to A and B
returns the Cartesian product of the result sets from A and B
a result value (a, b) may or may not be derived from the same
candidate
This appears too simple to be useful – but be patient.
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
8 / 23
Product Algebras
Definition of lexicographic product *
Definition
Let A and B be evaluation algebras over Σ.
Their lexicographic product algebra A ∗ B is an evaluation algebra C over
Σ with
fC defined component-wise (as with ×) for each f in Σ,
and with the objective function
hA∗B ([(a1 , b1 ), ..., (ak , bk )]) = [(l, r )]|
l ← set(hA [a1 , . . . , ak ]),
r ← hB ([r |(l 0 , r ) ← [(a1 , b1 ), ..., (ak , bk )], l 0 = l])
where [] denotes a multi-set and the function set returns a multi-set in
which all duplicate elements are removed.
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
9 / 23
Product Algebras
The meaning of G(A ∗ B, x)
is actually lready explained in ADP definitions, but still ...
Intuitive explanation:
Phase 1 evaluates all candidates via functions fA ∗ fB into a gigantic list
[..., (ai , bi ), ...]
Phase 2 chooses pairs with hA based on the ai component, and
Phase 3 chooses from those the final pairs, based on te bi component
Thus, if hA maximizes under ordering <A and hB maximizes under
ordering <B , the product maximizes under the lexicographic combination
of orderinge, (<A , <B ).
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
10 / 23
Product Algebras
How about Bellman’s Principle?
In general, there is a proof obligation with *
A ∗ B must satisfy Bellman’s principle
. . . because the three phases are not executed separately. Instead, hA∗B is
applied at intermediate steps.
Bellman’s Principle guarantees that we loose no final result along the way.
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
11 / 23
Product Algebras
Advantages over hand-coded algebra combinations
Benefits of ∗ products:
the ADP compiler knows the definition of * and produces
automatically code for A ∗ B.
the resulting functions fA∗B etc. maybe a little less efficient than in a
handcoded product, BUT:
products formed with ∗ require no coding and testing
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
12 / 23
Product Algebras
Versatility of the lexicographic product
A ∗ B has a surprising number of uses
optimization under a lexicographic ordering
class-wise optimization
reporting candidates
The latter two cases use algebras A or B that do not do optimization.
More later ...
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
13 / 23
Product Algebras
Useful properties of the lexicographic product (1)
Theorem
(1) Selectivity:
Let hA (X ) ⊆ X and hB (X ) ⊆ X . For each value (a, b) computed by
hA ∗ hB , there is a candidate t such that
(a, b) = (A(t), B(t)).
I.e., each returned value pair is backed up by a candidate that evaluates to
these two values.
Selectivity does not hold e.g. for the counting algebra.
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
14 / 23
Product Algebras
Useful properties of the lexicographic product (2)
Theorem
(2) Identity:
For any algebras A and B, and multiset X ,
(idA ∗ idB )(X ) = X .
Combining two identity functions gives the identity function on pairs.
(On lists, (idA ∗ idB )(X ) is a permutation of X .)
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
15 / 23
Product Algebras
Useful elementary properties (3)
Since A ∗ B is just another algebra, we can make bigger products ...
Theorem
(3) Associativity:
G(A ∗ (B ∗ C ), x) = G((A ∗ B) ∗ C , x)
modulo re-grouping of tuples of result values, e.g.
(a, (b, c)) = ((a, b), c) = (a, b, c)
Hence we can simply write G(A ∗ B ∗ C , x).
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
16 / 23
Product Algebras
Useful elementary properties (4)
Theorem
(4) Lexicographic optimization:
Let A and B optimize over <A and <B .
Then A ∗ B satisfies Bellman’s Principle and optimizes over the
lexicographic ordering (<A , <B ).
So, for optimizing over two orderings in lexicographic fashion, there is a
general proof that Bellman’s Principle is always satisfied.
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
17 / 23
Interleaved product
The interleaved product: Motivation
Let B(k) be an algebra that computes the best k scores under some
optimization scheme. Let A be a an algebra that assigns a class attribute
to each candidate. With G(A ∗ B(k), x), we obtain the best k for each
class.
How can we get the best 3 B-values from different A-classes?
We first compute B(1) for each A-class,
and then choose the classes with the best k B-values.
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
18 / 23
Interleaved product
The interleaved product: Definition
Let A be a Σ- algebra and B(k) a generic Σ-algebra, and note that B(1)
is unitary.
Definition
The interleaved product (A/B)(k) is a generic Σ-algebra and has the
functions
fA/B = fA×B
(1)
for each f ∈ Σ, and the objective function
h(A/B)(k) [(a1 , b1 ), . . . , (am , bm )] =
[ (l, r ) | (l, r ) ← U, p ← V , p = r ]
where
U = hA∗B(1) [(a1 , b1 ), . . . , (am , bm )]
V = set(hB(k) [ v | ( , v ) ← U])
Robert Giegerich, Benedikt Löwes
ADP Lecture
(2)
Summer 2014
19 / 23
Interleaved product
Interleaved product (2)
In Equation 2, U stores the best results in each A-class, V stores the
global rank of all solutions from U, and the main set comprehension
filters U to contain only the results from V .
This product is useful when B(k) alone produces the k-best-scoring
candidates, and A maps candidates to different classes (not making
any choices). Then, A/B yields the 1-best-scoring candidates from k
different classes.
This situation required sophisticated hand-coding in RNA shape
analysis in 2004 – it is now simply written as (Shape/Mfe)(k).
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
20 / 23
Pareto product
Pareto-Optimization
Pareto-optimization is a general way of optimizing with two (or more)
idependent criteria.
The solution set one computes is the Pareto front of the search space
X ⊆ A × B.
Taking hA and hB as maximization, the Pareto front operator pf is defined
as
pf(X ) = {(a, b) ∈ X | 6 ∃(a0 , b 0 ) ∈ X \ {(a, b)} with a ≤ a0 , b ≤ b 0 }.
In words: No element of the Pareto front is dominated in both dimensions
by another element of X .
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
21 / 23
Pareto product
The Pareto-Product
Let A and B be two Σ-algebras which maximize with respect to <A and
<B .
Definition
The Pareto-product AB is a Σ-algebra C with
fC defined component-wise from fA and fB , and
hC (X ) = pf(X ) with pf as defined above for <A and <B .
The Pareto product is a recent development and not yet implemented in
Bellman’s GAP.
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
22 / 23
Pareto product
Conclusion on algebra products
A number of algebra product operators have been defined – there may be
more.
it is faster and safer to use products where suitable than hand-coding
a combined evalation algebra.
since products yield new algebras, these can be used with further
products, as in G((A/B) ∗ (C × D), x), which makes them a powerful
programming device on a very high level of abstraction.
Understanding the definitions of products is one thing.
Using products effectively takes quite some practice!!
That’s the topic of our next set of slides.
Robert Giegerich, Benedikt Löwes
ADP Lecture
Summer 2014
23 / 23