Binary decision diagrams for random Boolean functions

نویسنده

  • Clemens Gröpl
چکیده

Binary Decision Diagrams (BDDs) are a data structure for Boolean functions which are also known as branching programs. In ordered binary decision diagrams (OBDDs), the tests have to obey a fixed variable ordering. In free binary decision diagrams (FBDDs), each variable can be tested at most once. The efficiency of new variants of the BDD concept is usually demonstrated with spectacular (worst-case) examples. We pursue another approach and compare the representation sizes of almost all Boolean functions. Whereas I. Wegener proved that for ‘most’ values of the expected OBDD size of a random Boolean function of variables is equal to the worst-case size up to terms of lower order, we show that this is not the case for within intervals of constant length around the values . Furthermore, ranges of exist for which minimal FBDDs are almost always at least a constant factor smaller than minimal OBDDs. Our main theorems have doubly exponentially small probability bounds (in ). We also investigate the evolution of random OBDDs and their worst-case size, revealing an oscillating behaviour that explains why certain results cannot be improved in general. Zusammenfassung Binary Decision Diagrams (BDDs) sind eine Datenstruktur für Boolesche Funktionen, die auch unter dem Namen branching program bekannt ist. In ordered binary decision diagrams (OBDDs) müssen die Tests einer festen Variablenordnung genügen. In free binary decision diagrams (FBDDs) darf jede Variable höchstens einmal getestet werden. Die Effizienz neuer Varianten des BDD-Konzepts wird gewöhnlich anhand spektakulärer (worst-case) Beispiele aufgezeigt. Wir verfolgen einen anderen Ansatz und vergleichen die Darstellungsgrößen für fast alle Booleschen Funktionen. Während I. Wegener bewiesen hat, daß für die ‘meisten’ die erwartete OBDD-Größe einer zufälligen Booleschen Funktion von Variablen gleich der worst-case Größe bis auf Terme kleinerer Ordnung ist, zeigen wir daß dies nicht der Fall ist für innerhalb von Intervallen konstanter Länge um die Werte . Ferner gibt es Bereiche von , in denen minimale FBDDs fast immer um mindestens einen konstanten Faktor kleiner sind als minimale OBDDs. Unsere Hauptsätze haben doppelt exponentielle Wahrscheinlichkeitsschranken (in ). Außerdem untersuchen wir die Entwicklung zufälliger OBDDs und ihrer worst-case Größe und decken dabei ein oszillierendes Verhalten auf, das erklärt, warum gewisse Aussagen im allgemeinen nicht verstärkt werden können. Acknowledgements I am grateful to everybody who supported and contributed to this work: Hans Jürgen Prömel (my supervisor), Anand Srivastav, Mathias Block, Harry Preuß, Martin Skutella – as my coauthors; also to all the many other people with whom I discussed these things: Stefan Hougardy, Bernd Kreuter, Christoph Meinel, Paul Molitor, Till Nierhoff, Ralf Oelschlägel, Martin Sauerhoff, Detlef Sieling, Anusch Taraz, Thorsten Theobald, ; and my wife Antje . The graduate program ‘Algorithmische Diskrete Mathematik’ provided financial allowance and a high quality scientific framework. The graduate school ‘Algorithmische Diskrete Mathematik’ is supported by the Deutsche Forschungsgemeinschaft, grant GRK 219/2-97.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Function-driven Linearly Independent Expansions of Boolean Functions

The paper presents a family of new expansions of Boolean functions called Function-driven Linearly Independent (fLI) expansions. On the basis of this expansion a new kind of a canonical representation of Boolean functions is constructed: Function-driven Linearly Independent Binary Decision Diagrams (fLIBDDs). They generalize both Function-driven Shannon Binary Decision Diagrams (fShBDDs) and Li...

متن کامل

Probabilistic manipulation of Boolean functions using free Boolean diagrams

We propose a data structure for Boolean functions termed the Free Boolean Diagram. A Free Boolean Diagram allows decision vertices as in the conventional Binary Decision Diagram, but also allows function vertices corresponding to the and and xor functions. It has been shown previously that the equivalence of two Free Boolean Diagrams can be decided probabilistically in polynomial time. Based on...

متن کامل

Discrete Function Representations Utilizing Decision Diagrams and Spectral Techniques

All discrete function representations become exponential in size in the worst case. Binary decision diagrams have become a common method of representing discrete functions in computer-aided design applications. For many functions, binary decision diagrams do provide compact representations. This work presents a way to represent large decision diagrams as multiple smaller partial binary decision...

متن کامل

Ordered Binary Decision Diagrams and Minimal Trellises John La erty

Ordered binary decision diagrams (OBDDs) are graph-based data structures for representing Boolean functions. They have found widespread use in computer-aided design and in formal veri cation of digital circuits. Minimal trellises are graphical representations of error-correcting codes that play a prominent role in coding theory. This paper establishes a close connection between these two graphi...

متن کامل

Combinational Logic-Level Veri cation using Boolean Expression Diagrams

Boolean Expression Diagrams (BEDs) is a new data structure for representing and manipulating Boolean functions. BEDs are a generalization of Binary Decision Diagrams (BDDs) that are capable of representing any Boolean circuit in linear space and still maintain many of the desirable properties of BDDs. This paper demonstrates that BEDs are well suited for solving the combinational logic-level ve...

متن کامل

Binary Decision Diagrams

Decision diagrams are a natural representation of finite functions. The obvious complexity measures are length and size which correspond to time and space of computations. Decision diagrams are the right model for considering space lower bounds and time-space trade-offs. Due to the lack of powerful lower bound techniques, various types of restricted decision diagrams are investigated. They lead...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 1999