Thursday, April 20, 2006

Automatic Structures: Part 1

Several weeks ago, my supervisor Leonid Libkin gave a presentation about automatic structures in our reading group meeting. I would like to talk about these nice animals in some details. This is the first part in the series of posts giving a flavor of automatic structures and sketching some important proofs. If there are interests, I will also talk about how one might apply automatic structures to program verification (this line of research is still under intense development).

Finite model theory primarily concerns finite structures, and has been successfully applied to database theory, and logical approaches to verification (i.e. model checking). On the other hand, finite structures are often too restrictive. For example, when modeling C programs, the use of infinite structures is often inevitable. For this reason, a lot of effort has been put into extending the framework of finite model theory to infinite structures. Several such approaches include metafinite model theory, embedded finite model theory, and automatic structures. In the sequel, we are primarily interested in automatic structures.

Roughly speaking, automatic structures are structures whose universe, and relations can each be represented by a finite automaton. A simple example of automatic structures is Presburger arithmetic (N,+), where N is the set of natural numbers. Here is how one might represent (N,+) using finite automata. Represent the universe N in binary in reverse order (e.g. 4 = 001, 2 = 01); call such a representation bin(N). So, bin(N) is a language over Σ = {0,1}. It is simple to devise a finite automaton that recognizes precisely bin(N). Now, how do we represent the 3-ary relation

+ = { (bin(i),bin(j),bin(k)) : i + j = k, i,j,k ∈ N }

with an automaton? First, it is possible to represent the relation + as a language L over the alphabet {0,1,#}3 defined in the following way. For i,j,k ∈ bin(N), concatenate i,j,k with the padding symbol # so that the resulting strings i',j',k' are of the same length. For example, if i = 001, j = 01, and k = 00001, then i' = 001##, j' = 01###, and k' = 00001. Now, we may treat the tuple (i',j',k') as a string over the alphabet {0,1,#}3, e.g., for i',j',k' in the above example, the resulting string is (0,0,0)(0,1,0)(1,#,0)(#,#,0)(#,#,1). Then, if i + j = k, put the string (i',j',k') in L. Having defined L, it is not hard to exhibit an automaton that recognizes precisely L. [This automaton resembles the commonplace algorithm for addition.]

What is so cool about automatic structures? First, it is somewhat immediate that automatic structures have decidable FO theories. Second, there exists a universal automatic structure Muniv, which is a structure in which all other automatic structures can be interpreted with FO translation (or reduction). Furthermore, the set of all relations definable in Muniv captures precisely all regular relations, which gives an easy way to prove properties about regular languages. By the way, automatic structures also give a nice way of proving that model checking some types of infinite transition systems be decidable.

Anyway, this post was merely intended to whet the reader's appetite. I hope this was enticing enough. In the next post, I will give a precise definition of automatic structures and prove the Buchi-Bruyer theorem that the set of relations definable in Muniv coincides with all regular relations.

marry said...

Blogs are so informative where we get lots of information on any topic. Nice job keep it up!!
_____________________________

International Finance Dissertation

Dissertation Help said...

Nicely explained. It's indeed an art to stop new visitors with your attractive writing style. Truly impressive and nice information. Thanks for sharing.
Dissertation Help | Custom Dissertation

Dissertation Writing service said...

This kind of information is very limited on internet. Nice to find the post related to my searching criteria. Your updated and informative post will be appreciated by blog loving people.

Dissertation Samples

Ciao Bella said...

This conclusion can be drived from secondary sources as well. Because writer point of view is not different from the claims of other indusry leaders. Author did not use primary sources.
Dissertation Help

sm@rt v@mpire said...

The data collected is based upon primary sources, which can be an essential step towards finding the actual
experience. I think that the writer can add secondary sources to satisfy some old users.
http://www.urgentcustomessays.com
http://www.urgentcustomessays.com

pamella said...

This is my Good luck that I found your post which is according to my search and topic, I think you are a great blogger, thanks for helping me outta my problem..
Literature Review

dallas area hotels said...

Excellent blog I like it so much. This is really very interesting and informative blog pots here. Good job! keep it up, hope to read your other updates.

DEWHURST TOULSON said...

precisely all regular relations, which gives an easy way to prove properties about regular languages.
dissertation-topics-examples.info

pak gendoet said...