Binary Decision Diagrams: from Tree Compaction to Sampling

Any Boolean function corresponds with a complete full binary decision tree. This tree can in turn be represented in a maximally compact form as a direct acyclic graph where common subtrees are factored and shared, keeping only one copy of each unique subtree. This yields the celebrated and widely used structure called reduced ordered binary decision diagram (ROBDD). We propose to revisit the classical compaction process to give a new way of enumerating ROBDDs of a given size without considering fully expanded trees and the compaction step. Our method also provides an unranking procedure for the set of ROBDDs. As a by-product we get a random uniform and exhaustive sampler for ROBDDs for a given number of variables and size

On Uniquely Closable and Uniquely Typable Skeletons of Lambda Terms

Uniquely closable skeletons of lambda terms are Motzkin-trees that predetermine the unique closed lambda term that can be obtained by labeling their leaves with de Bruijn indices. Likewise, uniquely typable skeletons of closed lambda terms predetermine the unique simply-typed lambda term that can be obtained by labeling their leaves with de Bruijn indices. We derive, through a sequence of logic program transformations, efficient code for their combinatorial generation and study their statistical properties. As a result, we obtain context-free grammars describing closable and uniquely closable skeletons of lambda terms, opening the door for their in-depth study with tools from analytic combinatorics. Our empirical study of the more difficult case of (uniquely) typable terms reveals some interesting open problems about their density and asymptotic behavior. As a connection between the two classes of terms, we also show that uniquely typable closed lambda term skeletons of size $3n+1$ are in a bijection with binary trees of size $n$.Comment: Pre-proceedings paper presented at the 27th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2017), Namur, Belgium, 10-12 October 2017 (arXiv:1708.07854

Laws relating runs, long runs, and steps in gambler's ruin, with persistence in two strata

Define a certain gambler's ruin process \mathbf{X}_{j}, \mbox{ \ }j\ge 0, such that the increments $\varepsilon_{j}:=\mathbf{X}_{j}-\mathbf{X}_{j-1}$ take values $\pm1$ and satisfy $P(\varepsilon_{j+1}=1|\varepsilon_{j}=1, |\mathbf{X}_{j}|=k)=P(\varepsilon_{j+1}=-1|\varepsilon_{j}=-1,|\mathbf{X}_{j}|=k)=a_k$, all $j\ge 1$, where $a_k=a$ if $0\le k\le f-1$, and $a_k=b$ if $f\le k<N$. Here $0<a, b <1$ denote persistence parameters and $f ,N\in \mathbb{N}$ with $f<N$. The process starts at $\mathbf{X}_0=m\in (-N,N)$ and terminates when $|\mathbf{X}_j|=N$. Denote by ${\cal R}'_N$, ${\cal U}'_N$, and ${\cal L}'_N$, respectively, the numbers of runs, long runs, and steps in the meander portion of the gambler's ruin process. Define $X_N:=\left ({\cal L}'_N-\frac{1-a-b}{(1-a)(1-b)}{\cal R}'_N-\frac{1}{(1-a)(1-b)}{\cal U}'_N\right )/N$ and let $f\sim\eta N$ for some $0<\eta <1$. We show $\lim_{N\to\infty} E\{e^{itX_N}\}=\hat{\varphi}(t)$ exists in an explicit form. We obtain a companion theorem for the last visit portion of the gambler's ruin.Comment: Presented at 8th International Conference on Lattice Path Combinatorics, Cal Poly Pomona, Aug., 2015. The 2nd version has been streamlined, with references added, including reference to a companion document with details of calculations via Mathematica. The 3rd version has 2 new figures and improved presentatio

Dynamic programming for graphs on surfaces

We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2O(k·log k). Our approach combines tools from topological graph theory and analytic combinatorics.Postprint (updated version

Dynamic Programming for Graphs on Surfaces

We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2^{O(k log k)} n steps. Our approach combines tools from topological graph theory and analytic combinatorics. In particular, we introduce a new type of branch decomposition called "surface cut decomposition", generalizing sphere cut decompositions of planar graphs introduced by Seymour and Thomas, which has nice combinatorial properties. Namely, the number of partial solutions that can be arranged on a surface cut decomposition can be upper-bounded by the number of non-crossing partitions on surfaces with boundary. It follows that partial solutions can be represented by a single-exponential (in the branchwidth k) number of configurations. This proves that, when applied on surface cut decompositions, dynamic programming runs in 2^{O(k)} n steps. That way, we considerably extend the class of problems that can be solved in running times with a single-exponential dependence on branchwidth and unify/improve most previous results in this direction.Comment: 28 pages, 3 figure

Multiplicative anomaly and zeta factorization

Some aspects of the multiplicative anomaly of zeta determinants are investigated. A rather simple approach is adopted and, in particular, the question of zeta function factorization, together with its possible relation with the multiplicative anomaly issue is discussed. We look primordially into the zeta functions instead of the determinants themselves, as was done in previous work. That provides a supplementary view, regarding the appearance of the multiplicative anomaly. Finally, we briefly discuss determinants of zeta functions that are not in the pseudodifferential operator framework.Comment: 20 pages, AIP styl

Fully Analyzing an Algebraic Polya Urn Model

This paper introduces and analyzes a particular class of Polya urns: balls are of two colors, can only be added (the urns are said to be additive) and at every step the same constant number of balls is added, thus only the color compositions varies (the urns are said to be balanced). These properties make this class of urns ideally suited for analysis from an "analytic combinatorics" point-of-view, following in the footsteps of Flajolet-Dumas-Puyhaubert, 2006. Through an algebraic generating function to which we apply a multiple coalescing saddle-point method, we are able to give precise asymptotic results for the probability distribution of the composition of the urn, as well as local limit law and large deviation bounds.Comment: LATIN 2012, Arequipa : Peru (2012

Uniform generation in trace monoids

We consider the problem of random uniform generation of traces (the elements of a free partially commutative monoid) in light of the uniform measure on the boundary at infinity of the associated monoid. We obtain a product decomposition of the uniform measure at infinity if the trace monoid has several irreducible components-a case where other notions such as Parry measures, are not defined. Random generation algorithms are then examined.Comment: Full version of the paper in MFCS 2015 with the same titl

Laguerre-type derivatives: Dobinski relations and combinatorial identities

We consider properties of the operators D(r,M)=a^r(a^\dag a)^M (which we call generalized Laguerre-type derivatives), with r=1,2,..., M=0,1,..., where a and a^\dag are boson annihilation and creation operators respectively, satisfying [a,a^\dag]=1. We obtain explicit formulas for the normally ordered form of arbitrary Taylor-expandable functions of D(r,M) with the help of an operator relation which generalizes the Dobinski formula. Coherent state expectation values of certain operator functions of D(r,M) turn out to be generating functions of combinatorial numbers. In many cases the corresponding combinatorial structures can be explicitly identified.Comment: 14 pages, 1 figur