Signatures and terms
Trees and tree grammars
Evaluation algebras
Algebraic Dynamic Programming
Session 3
ADP Theory I: Basic Definitions
Mathias Möhl, Rolf Backofen (Lecture)
Daniel Maticzka, Sita Lange (Exercises)
Department of Computer Science
Albert-Ludwigs-University Freiburg
Summer 2010
http://www.bioinf.uni-freiburg.de/Lehre/Courses/2010 SS/V ADP
mmohl@informatik.uni-freiburg.de.de
Mathias Möhl, Rolf Backofen (Lecture) Daniel Maticzka, Sita Lange (Exercises)
ADP Lecture Summer 2010
Albert-Ludwigs-University Freiburg
Signatures and terms
Trees and tree grammars
Evaluation algebras
Alphabets
An alphabet A is a finite set of symbols (also called characters).
Symbols can be compared for equality, and often, there is a total
ordering defined on them (“alphabetical order”).
Examples:
the ASCII alphabet
the single-letter or the 3-letter IUPAC code for amino acid
sequences
{A, C , G , T } for DNA
a finite subset of N × N , denoting e.g. matrix dimensions
Sequences of symbols over some alphabet are called texts, words,
sequences, strings, genes, proteins, . . . – depending on the
application domain
Mathias Möhl, Rolf Backofen (Lecture) Daniel Maticzka, Sita Lange (Exercises)
ADP Lecture Summer 2010
Albert-Ludwigs-University Freiburg
Signatures and terms
Trees and tree grammars
Evaluation algebras
A signature Σ over A is a family of function declarations. There is
a name for a base set (say S) – merely a placeholder for a yet
unspecified data domain
a family of function names, together with their argument and
result types
argument types are either S or A
result type is always S
Mathias Möhl, Rolf Backofen (Lecture) Daniel Maticzka, Sita Lange (Exercises)
ADP Lecture Summer 2010
Albert-Ludwigs-University Freiburg
Signatures and terms
Trees and tree grammars
Evaluation algebras
Term language
A signature describes a language of terms.
Terms are all well-typed formulas that can be formed from the
symbols in A and the function names of the signature.
The term language defined by Σ over A is denoted TΣ .
Subsets of TΣ are also term languages.
If we allow variables, taken from a set V , in the terms, we speak of
TΣ (V ), a term language with variables.
Mathias Möhl, Rolf Backofen (Lecture) Daniel Maticzka, Sita Lange (Exercises)
ADP Lecture Summer 2010
Albert-Ludwigs-University Freiburg
Signatures and terms
Trees and tree grammars
Evaluation algebras
Signature and term language for the edit distance problem
Alphabet: A
Base set name: Ali
Simple edit distance model:
r : (A, Ali, A) → Ali
d : (A, Ali)
→ Ali
i : (Ali, A)
→ Ali
e:A
→ Ali
For the affine gap cost model we add:
dx : (A, Ali) → Ali
ix : (Ali, A) → Ali
Mathias Möhl, Rolf Backofen (Lecture) Daniel Maticzka, Sita Lange (Exercises)
ADP Lecture Summer 2010
Albert-Ludwigs-University Freiburg
Signatures and terms
Trees and tree grammars
Evaluation algebras
Some terms in the term language
e($)
d(x3 , r (x4 , e($), y2 )
r (x1 , d(x2 , d(x3 , r (x4 , e($), y2 ))), y1 )
r (x1 , d(x2 , r (x3 , d(x4 , e($)), y2 )), y1 )
r (x1 , d(x2 , dx(x3 , r (x4 , e($), y2 ))), y1 )
r (x1 , dx(x2 , dx(x3 , r (x4 , e($), y2 ))), y1 )
Although the function names have (yet) no meaning, we can think
of the last four terms as representing alternative alignments of two
strings x1 x2 x3 x4 and y1 y2 .
More general: terms can describe candidates in a DP problem.
Mathias Möhl, Rolf Backofen (Lecture) Daniel Maticzka, Sita Lange (Exercises)
ADP Lecture Summer 2010
Albert-Ludwigs-University Freiburg
Signatures and terms
Trees and tree grammars
Evaluation algebras
From terms to trees
Terms (or any formula) can be seen as trees –
the outmost function (or “operator”) is the root of the tree,
its arguments are the subtrees.
Examples:
e($)
Mathias Möhl, Rolf Backofen (Lecture) Daniel Maticzka, Sita Lange (Exercises)
ADP Lecture Summer 2010
d(x3 , r (x4 , e($), y2 ))
Albert-Ludwigs-University Freiburg
Signatures and terms
Trees and tree grammars
Evaluation algebras
Term languages TΣ are not specific enough
Reconsider the term
r (x1 , dx(x2 , dx(x3 , r (x4 , e($), y2 ))), y1 )
This term should denote an alignment (no candidate), but it holds
a two-letter deletion that is NOT charged a gap opening cost – as
there is no use of function d. This violates the affine gap model.
We need a way to describe specific, well-formed subsets of term
languages. Only terms built in a special way represent the objects
of interest that constitute our search space.
→ use tree grammars
Mathias Möhl, Rolf Backofen (Lecture) Daniel Maticzka, Sita Lange (Exercises)
ADP Lecture Summer 2010
Albert-Ludwigs-University Freiburg
Signatures and terms
Trees and tree grammars
Evaluation algebras
Term languages via tree grammars
Term languages can be described by (different types of) tree
grammars.
Recall:
A formal language (for example context-free language or
regular language) is a subset of A∗ .
A formal tree language is a subset of TΣ .
Different types of grammars describe languages of different
complexity (for example regular or context-free).
Mathias Möhl, Rolf Backofen (Lecture) Daniel Maticzka, Sita Lange (Exercises)
ADP Lecture Summer 2010
Albert-Ludwigs-University Freiburg
Signatures and terms
Trees and tree grammars
Evaluation algebras
Regular tree grammars
A regular tree grammar over Σ has
a set V of nonterminal symbols
a designated symbol Z ∈ V , called the axiom
a set of productions of the form v → t with v ∈ V and
t ∈ TΣ (V ).
The language described by a tree grammar is
L(G) = {t ∈ TΣ |Z ⇒∗ t}.
Derivation with tree grammars is just as it is with context free grammars –
substituting righthand-side trees for nonterminal symbols generates a tree.
Notions of terminal trees, leftmost derivation, derivation tree, and syntactic
ambiguity carry over. More on this after the next example.
Mathias Möhl, Rolf Backofen (Lecture) Daniel Maticzka, Sita Lange (Exercises)
ADP Lecture Summer 2010
Albert-Ludwigs-University Freiburg