Automata for the mu-calculus and Related Results

The propositional mu-calculus as introduced by Kozen in [4] isconsidered. The notion of disjunctive formula is defined and it is shownthat every formula is semantically equivalent to a disjunctive formula.For these formulas many difficulties encountered in the general case maybe avoided. For instance, satisfiability checking is linear for disjunctiveformulas. This kind of formula gives rise to a new notion of finite automatonwhich characterizes the expressive power of the mu-calculus overall transition systems

Constructing Fixed-Point Combinators Using Application Survival

The theory of application survival was developed in our Ph.D. thesisas an approach for reasoning about application in general and self-applicationin particular. In this paper, we show how application survivalprovides a uniform framework not only for for reasoning aboutfixed-points, fixed-point combinators, but also for deriving and comparingknown and new fixed-point combinators

Succinct Indexable Dictionaries with Applications to Encoding kk-ary Trees, Prefix Sums and Multisets

We consider the {\it indexable dictionary} problem, which consists of storing a set $S \subseteq \{0,...,m-1\}$ for some integer $m$, while supporting the operations of \Rank(x), which returns the number of elements in $S$ that are less than $x$ if $x \in S$, and -1 otherwise; and \Select(i) which returns the $i$-th smallest element in $S$. We give a data structure that supports both operations in O(1) time on the RAM model and requires ${\cal B}(n,m) + o(n) + O(\lg \lg m)$ bits to store a set of size $n$, where {\cal B}(n,m) = \ceil{\lg {m \choose n}} is the minimum number of bits required to store any $n$-element subset from a universe of size $m$. Previous dictionaries taking this space only supported (yes/no) membership queries in O(1) time. In the cell probe model we can remove the $O(\lg \lg m)$ additive term in the space bound, answering a question raised by Fich and Miltersen, and Pagh. We present extensions and applications of our indexable dictionary data structure, including: An information-theoretically optimal representation of a $k$-ary cardinal tree that supports standard operations in constant time, A representation of a multiset of size $n$ from $\{0,...,m-1\}$ in ${\cal B}(n,m+n) + o(n)$ bits that supports (appropriate generalizations of) \Rank and \Select operations in constant time, and A representation of a sequence of $n$ non-negative integers summing up to $m$ in ${\cal B}(n,m+n) + o(n)$ bits that supports prefix sum queries in constant time.Comment: Final version of SODA 2002 paper; supersedes Leicester Tech report 2002/1

From Branching to Linear Metric Domains (and back)

A branching and a linear metric domain - both turned into a category - are related by means of a reflection and a coreflection

An n log n Algorithm for Online BDD Refinement

Binary Decision Diagrams are in widespread use in verification systemsfor the canonical representation of Boolean functions. A BDD representinga function phi : B^nu -> N can easily be reduced to its canonical form inlinear time.In this paper, we consider a natural online BDD refinement problemand show that it can be solved in O(n log n) if n bounds the size of theBDD and the total size of update operations.We argue that BDDs in an algebraic framework should be understoodas minimal fixed points superimposed on maximal fixed points. We proposea technique of controlled growth of equivalence classes to make theminimal fixed point calculations be carried out efficiently. Our algorithmis based on a new understanding of the interplay between the splittingand growing of classes of nodes.We apply our algorithm to show that automata with exponentiallylarge, but implicitly represented alphabets, can be minimized in timeO(n log n), where n is the total number of BDD nodes representing theautomaton

Tables Should Be Sorted (on Random Access Machines)

We consider the problem of storing an n element subset S of a universe of size m, so that membership queries (is x 2 S?) can be answered efficiently. The model of computation is a random access machine with the standard instruction set (direct and indirect adressing, conditional branching, addition, subtraction, and multiplication). We show that if s memory registers are used to store S, where n s m=n , then query time \Omega\Gammame/ n) is necessary in the worst case. That is, under these conditions, the solution consisting of storing S as a sorted table and doing binary search is optimal. The condition s m=n is essentially optimal; we show that if n + m=n o(1) registers may be used, query time o(log n) is possible

Nearly Optimal Static Las Vegas Succinct Dictionary

Given a set $S$ of $n$ (distinct) keys from key space $[U]$, each associated with a value from $\Sigma$, the \emph{static dictionary} problem asks to preprocess these (key, value) pairs into a data structure, supporting value-retrieval queries: for any given $x\in [U]$, $\mathtt{valRet}(x)$ must return the value associated with $x$ if $x\in S$, or return $\bot$ if $x\notin S$. The special case where $|\Sigma|=1$ is called the \emph{membership} problem. The "textbook" solution is to use a hash table, which occupies linear space and answers each query in constant time. On the other hand, the minimum possible space to encode all (key, value) pairs is only $\mathtt{OPT}:= \lceil\lg_2\binom{U}{n}+n\lg_2|\Sigma|\rceil$ bits, which could be much less. In this paper, we design a randomized dictionary data structure using $\mathtt{OPT}+\mathrm{poly}\lg n+O(\lg\lg\lg\lg\lg U)$ bits of space, and it has \emph{expected constant} query time, assuming the query algorithm can access an external lookup table of size $n^{0.001}$. The lookup table depends only on $U$, $n$ and $|\Sigma|$, and not the input. Previously, even for membership queries and $U\leq n^{O(1)}$, the best known data structure with constant query time requires $\mathtt{OPT}+n/\mathrm{poly}\lg n$ bits of space (Pagh [Pag01] and P\v{a}tra\c{s}cu [Pat08]); the best-known using $\mathtt{OPT}+n^{0.999}$ space has query time $O(\lg n)$; the only known non-trivial data structure with $\mathtt{OPT}+n^{0.001}$ space has $O(\lg n)$ query time and requires a lookup table of size $\geq n^{2.99}$ (!). Our new data structure answers open questions by P\v{a}tra\c{s}cu and Thorup [Pat08,Tho13]. We also present a scheme that compresses a sequence $X\in\Sigma^n$ to its zeroth order (empirical) entropy up to $|\Sigma|\cdot\mathrm{poly}\lg n$ extra bits, supporting decoding each $X_i$ in $O(\lg |\Sigma|)$ expected time.Comment: preliminary version appeared in STOC'2