Academic Tutorials



English | French | Portugese | German | Italian
Home Advertise Payments Recommended Websites Interview Questions FAQs
News Source Codes E-Books Downloads Jobs Web Hosting
Chats

Lisp Programming
Introduction to Lisp
Expressions
Editing, Loading and Compiling LISP Programs
Control Stuctures
Multiple Recursions
Lists
Structural Recursion
Symbols
Examples
Using Lists as Sets

HTML Tutorials
HTML Tutorial
XHTML Tutorial
CSS Tutorial
TCP/IP Tutorial
CSS 1.0
CSS 2.0
HLML
XML Tutorials
XML Tutorial
XSL Tutorial
XSLT Tutorial
DTD Tutorial
Schema Tutorial
XForms Tutorial
XSL-FO Tutorial
XML DOM Tutorial
XLink Tutorial
XQuery Tutorial
XPath Tutorial
XPointer Tutorial
RDF Tutorial
SOAP Tutorial
WSDL Tutorial
RSS Tutorial
WAP Tutorial
Web Services Tutorial
Browser Scripting
JavaScript Tutorial
VBScript Tutorial
DHTML Tutorial
HTML DOM Tutorial
WMLScript Tutorial
E4X Tutorial
Server Scripting
ASP Tutorial
PERL Tutorial
SQL Tutorial
ADO Tutorial
CVS
Python
Apple Script
PL/SQL Tutorial
SQL Server
PHP
.NET (dotnet)
Microsoft.Net
ASP.Net
.Net Mobile
C# : C Sharp
ADO.NET
VB.NET
VC++
Multimedia
SVG Tutorial
Flash Tutorial
Media Tutorial
SMIL Tutorial
Photoshop Tutorial
Gimp Tutorial
Matlab
Gnuplot Programming
GIF Animation Tutorial
Scientific Visualization Tutorial
Graphics
Web Building
Web Browsers
Web Hosting
W3C Tutorial
Web Building
Web Quality
Web Semantic
Web Careers
Weblogic Tutorial
SEO
Web Site Hosting
Domain Name
Java Tutorials
Java Tutorial
JSP Tutorial
Servlets Tutorial
Struts Tutorial
EJB Tutorial
JMS Tutorial
JMX Tutorial
Eclipse
J2ME
JBOSS
Programming Langauges
C Tutorial
C++ Tutorial
Visual Basic Tutorial
Data Structures Using C
Cobol
Assembly Language
Mainframe
Forth Programming
Lisp Programming
Pascal
Delphi
Fortran
OOPs
Data Warehousing
CGI Programming
Emacs Tutorial
Gnome
ILU
Soft Skills
Communication Skills
Time Management
Project Management
Team Work
Leadership Skills
Corporate Communication
Negotiation Skills
Database Tutorials
Oracle
MySQL
Operating System
BSD
Symbian
Unix
Internet
IP-Masquerading
IPC
MIDI
Software Testing
Testing
Firewalls
SAP Module
ERP
ABAP
Business Warehousing
SAP Basis
Material Management
Sales & Distribution
Human Resource
Netweaver
Customer Relationship Management
Production and Planning
Networking Programming
Corba Tutorial
Networking Tutorial
Microsoft Office
Microsoft Word
Microsoft Outlook
Microsoft PowerPoint
Microsoft Publisher
Microsoft Excel
Microsoft Front Page
Microsoft InfoPath
Microsoft Access
Accounting
Financial Accounting
Managerial Accounting
Network Sites


Structural Recursion with Lists


Previoushome Next






Structural induction is a proof method that is used in mathematical logic (e.g., the proof of Los' theorem), computer science, graph theory, and some other mathematical fields.

A D V E R T I S E M E N T
It is a generalization of mathematical induction. Structural recursion is a recursion method bearing the same relationship to structural induction as ordinary recursion bears to ordinary mathematical induction.

In general, the idea is that one wishes to prove some proposition P(x), where x is any instance of some sort of recursively-defined structure such as lists or trees. A well-founded partial order is defined on the structures. The structural induction proof is a proof that the proposition holds for all the minimal structures, and that if it holds for the immediate substructures of a certain structure S, then it must hold for S also. (Formally speaking, this then satisfies the premises of an axiom of well-founded induction, which asserts that these two conditions are sufficient for the proposition to hold for all x.)

A structurally recursive function uses the same idea to define a recursive function: "base cases" handle each minimal structure and a rule for recursion. Structural recursion is usually proved correct by structural induction; in particularly easy cases, the inductive step is often left out. The length and ++ functions in the example below are structurally recursive. For example, if the structures are lists, one usually introduces the partial order in which L < M whenever list L is the tail of list M. Under this ordering, the empty list [] is the unique minimal element. A structural induction proof of some proposition P(l) then consists of two parts: A proof that P([]) is true, and a proof that if P(L) is true for some list L, and if L is the tail of list M, then P(M) must also be true.

USER(33): (list-length '(2 3 5 7 11 13 17 19))
8
Let us try to see how such a function can be implemented recursively. A given list L is created by either one of the two constructors, namely nil or a cons:
  • Case 1: L is nil.
    The length of an empty list is zero.
  • Case 2: L is constructed by cons.
    Then L is composed of two parts, namely, (first L) and (rest L). In such case, the length of L can be obtained inductively by adding 1 to the length of (rest L).

Formally, we could implement our own version of list-length as follows:

(defun recursive-list-length (L)
  "A recursive implementation of list-length."
  (if (null L)
      0
    (1+ (recursive-list-length (rest L)))))
Here, we use the recognizer null to differentiate how L is constructed. In case L is nil, we return 0 as its length. Otherwise, L is a cons, and we return 1 plus the length of (rest L). Recall that (1+ n) is simply a shorthand for (+ n 1).

Again, it is instructive to use the trace facilities to examine the unfolding of recursive invocations:

Structural Recursion with Lists

USER(40): (trace recursive-list-length)
(RECURSIVE-LIST-LENGTH)
USER(41): (recursive-list-length '(2 3 5 7 11 13 17 19))
0: (RECURSIVE-LIST-LENGTH (2 3 5 7 11 13 17 19))
v 1: (RECURSIVE-LIST-LENGTH (3 5 7 11 13 17 19))
2: (RECURSIVE-LIST-LENGTH (5 7 11 13 17 19))
3: (RECURSIVE-LIST-LENGTH (7 11 13 17 19))
4: (RECURSIVE-LIST-LENGTH (11 13 17 19))
5: (RECURSIVE-LIST-LENGTH (13 17 19))
6: (RECURSIVE-LIST-LENGTH (17 19))
7: (RECURSIVE-LIST-LENGTH (19))
8: (RECURSIVE-LIST-LENGTH NIL)
8: returned 0
7: returned 1
6: returned 2
5: returned 3
4: returned 4
3: returned 5
2: returned 6
1: returned 7
0: returned 8
8

The kind of recursion we see here is called structural recursion. Its standard pattern is as follows. To process an instance X of a recursive data type:

  1. Use the recognizers to determine how X is created (i.e. which constructor creates it). In our example, we use null to decide if a list is created by nil or cons.
  2. For instances that are atomic (i.e. those created by constructors with no components), return a trivial value. For example, in the case when a list is nil, we return zero as its length.
  3. If the instance is composite, then use the selectors to extract its components. In our example, we use first and rest to extract the two components of a nonempty list.
  4. Following that, we apply recursion on one or more components of X. For instance, we recusively invoked recursive-list-length on (rest L).
  5. Finally, we use either the constructors or some other functions to combine the result of the recursive calls, yielding the value of the function. In the case of recursive-list-length, we return one plus the result of the recursive call.

 

Sometimes, long traces like the one for list-length may be difficult to read on a terminal screen. Common LISP allows you to capture screen I/O into a file so that you can, for example, produce a hard copy for more comfortable reading. To capture the trace of executing (recursive-list-length '(2 3 5 7 11 13 17 19)), we use the dribble command:

USER(42): (dribble "output.txt")
dribbling to file "output.txt"
NIL
USER(43): (recursive-list-length '(2 3 5 7 11 13 17 19))
0: (RECURSIVE-LIST-LENGTH (2 3 5 7 11 13 17 19))
1: (RECURSIVE-LIST-LENGTH (3 5 7 11 13 17 19))
2: (RECURSIVE-LIST-LENGTH (5 7 11 13 17 19))
3: (RECURSIVE-LIST-LENGTH (7 11 13 17 19))
4: (RECURSIVE-LIST-LENGTH (11 13 17 19))
5: (RECURSIVE-LIST-LENGTH (13 17 19))
6: (RECURSIVE-LIST-LENGTH (17 19))
7: (RECURSIVE-LIST-LENGTH (19))
8: (RECURSIVE-LIST-LENGTH NIL)
8: returned 0
7: returned 1
6: returned 2
5: returned 3
4: returned 4
3: returned 5
2: returned 6
1: returned 7
0: returned 8
8
USER(44): (dribble)
The form (dribble "output.txt") instructs Common LISP to begin capturing all terminal I/O into a file called output.txt. The trailing (dribble) form instructs Common LISP to stop I/O capturing, and closes the file output.txt. If we examine output.txt, we will see the following:
dribbling to file "output.txt"
NIL
USER(43): (recursive-list-length '(2 3 5 7 11 13 17 19))
0: (RECURSIVE-LIST-LENGTH (2 3 5 7 11 13 17 19))
   1: (RECURSIVE-LIST-LENGTH (3 5 7 11 13 17 19))
     2: (RECURSIVE-LIST-LENGTH (5 7 11 13 17 19))
       3: (RECURSIVE-LIST-LENGTH (7 11 13 17 19))
         4: (RECURSIVE-LIST-LENGTH (11 13 17 19))
           5: (RECURSIVE-LIST-LENGTH (13 17 19))
             6: (RECURSIVE-LIST-LENGTH (17 19))
               7: (RECURSIVE-LIST-LENGTH (19))
                 8: (RECURSIVE-LIST-LENGTH NIL)
                 8: returned 0
               7: returned 1
             6: returned 2
           5: returned 3
         4: returned 4
       3: returned 5
     2: returned 6
   1: returned 7
0: returned 8
8
USER(44): (dribble)


Be the first one to comment on this page.




  Lisp Programming eBooks

No eBooks on Lisp Programming could be found as of now.

 
 Lisp Programming FAQs
More Links » »
 
 Lisp Programming Interview Questions
More Links » »
 
 Lisp Programming Articles

No Lisp Programming Articles could be found as of now.

 
 Lisp Programming News

No News on Lisp Programming could be found as of now.

 
 Lisp Programming Jobs

No Lisp Programming Articles could be found as of now.


Share And Enjoy:These icons link to social bookmarking sites where readers can share and discover new web pages.
  • blinkbits
  • BlinkList
  • blogmarks
  • co.mments
  • connotea
  • del.icio.us
  • De.lirio.us
  • digg
  • Fark
  • feedmelinks
  • Furl
  • LinkaGoGo
  • Ma.gnolia
  • NewsVine
  • Netvouz
  • RawSugar
  • Reddit
  • scuttle
  • Shadows
  • Simpy
  • Smarking
  • Spurl
  • TailRank
  • Wists
  • YahooMyWeb

Previoushome Next

Keywords: Lisp string,lisp command,lisp comment,lisp manual,lisp programme,lisp function list,lisp object

HTML Quizzes
HTML Quiz
XHTML Quiz
CSS Quiz
TCP/IP Quiz
CSS 1.0 Quiz
CSS 2.0 Quiz
HLML Quiz
XML Quizzes
XML Quiz
XSL Quiz
XSLT Quiz
DTD Quiz
Schema Quiz
XForms Quiz
XSL-FO Quiz
XML DOM Quiz
XLink Quiz
XQuery Quiz
XPath Quiz
XPointer Quiz
RDF Quiz
SOAP Quiz
WSDL Quiz
RSS Quiz
WAP Quiz
Web Services Quiz
Browser Scripting Quizzes
JavaScript Quiz
VBScript Quiz
DHTML Quiz
HTML DOM Quiz
WMLScript Quiz
E4X Quiz
Server Scripting Quizzes
ASP Quiz
PERL Quiz
SQL Quiz
ADO Quiz
CVS Quiz
Python Quiz
Apple Script Quiz
PL/SQL Quiz
SQL Server Quiz
PHP Quiz
.NET (dotnet) Quizzes
Microsoft.Net Quiz
ASP.Net Quiz
.Net Mobile Quiz
C# : C Sharp Quiz
ADO.NET Quiz
VB.NET Quiz
VC++ Quiz
Multimedia Quizzes
SVG Quiz
Flash Quiz
Media Quiz
SMIL Quiz
Photoshop Quiz
Gimp Quiz
Matlab Quiz
Gnuplot Programming Quiz
GIF Animation Quiz
Scientific Visualization Quiz
Graphics Quiz
Web Building Quizzes
Web Browsers Quiz
Web Hosting Quiz
W3C Quiz
Web Building Quiz
Web Quality Quiz
Web Semantic Quiz
Web Careers Quiz
Weblogic Quiz
SEO Quiz
Web Site Hosting Quiz
Domain Name Quiz
Java Quizzes
Java Quiz
JSP Quiz
Servlets Quiz
Struts Quiz
EJB Quiz
JMS Quiz
JMX Quiz
Eclipse Quiz
J2ME Quiz
JBOSS Quiz
Programming Langauges Quizzes
C Quiz
C++ Quiz
Visual Basic Quiz
Data Structures Using C Quiz
Cobol Quiz
Assembly Language Quiz
Mainframe Quiz
Forth Programming Quiz
Lisp Programming Quiz
Pascal Quiz
Delphi Quiz
Fortran Quiz
OOPs Quiz
Data Warehousing Quiz
CGI Programming Quiz
Emacs Quiz
Gnome Quiz
ILU Quiz
Soft Skills Quizzes
Communication Skills Quiz
Time Management Quiz
Project Management Quiz
Team Work Quiz
Leadership Skills Quiz
Corporate Communication Quiz
Negotiation Skills Quiz
Database Quizzes
Oracle Quiz
MySQL Quiz
Operating System Quizzes
BSD Quiz
Symbian Quiz
Unix Quiz
Internet Quiz
IP-Masquerading Quiz
IPC Quiz
MIDI Quiz
Software Testing Quizzes
Testing Quiz
Firewalls Quiz
SAP Module Quizzes
ERP Quiz
ABAP Quiz
Business Warehousing Quiz
SAP Basis Quiz
Material Management Quiz
Sales & Distribution Quiz
Human Resource Quiz
Netweaver Quiz
Customer Relationship Management Quiz
Production and Planning Quiz
Networking Programming Quizzes
Corba Quiz
Networking Quiz
Microsoft Office Quizzes
Microsoft Word Quiz
Microsoft Outlook Quiz
Microsoft PowerPoint Quiz
Microsoft Publisher Quiz
Microsoft Excel Quiz
Microsoft Front Page Quiz
Microsoft InfoPath Quiz
Microsoft Access Quiz
Accounting Quizzes
Financial Accounting Quiz
Managerial Accounting Quiz
Testimonials | Contact Us | Link to Us | Site Map
Copyright 2008. Academic Tutorials.com. All rights reserved Privacy Policies | About Us
Our Portals : Academic Tutorials | Best eBooksworld | Beyond Stats | City Details | Interview Questions | Discussions World | Excellent Mobiles | Free Bangalore | Give Me The Code | Gog Logo | Indian Free Ads | Jobs Assist | New Interview Questions | One Stop FAQs | One Stop GATE | One Stop GRE | One Stop IAS | One Stop MBA | One Stop SAP | One Stop Testing | Webhosting in India | Dedicated Server in India | Sirf Dosti | Source Codes World | Tasty Food | Tech Archive | Testing Interview Questions | Tests World | The Galz | Top Masala | Vyom | Vyom eBooks | Vyom International | Vyom Links | Vyoms | Vyom World
Copyright 2003-2019 Vyom Technosoft Pvt. Ltd., All Rights Reserved.