Non-polynomial Worst-Case Analysis of Recursive Programs

We study the problem of developing efficient approaches for proving worst-case bounds of non-deterministic recursive programs. Ranking functions are sound and complete for proving termination and worst-case bounds of nonrecursive programs. First, we apply ranking functions to recursion, resulting in measure functions. We show that measure functions provide a sound and complete approach to prove worst-case bounds of non-deterministic recursive programs. Our second contribution is the synthesis of measure functions in nonpolynomial forms. We show that non-polynomial measure functions with logarithm and exponentiation can be synthesized through abstraction of logarithmic or exponentiation terms, Farkas' Lemma, and Handelman's Theorem using linear programming. While previous methods obtain worst-case polynomial bounds, our approach can synthesize bounds of the form $\mathcal{O}(n\log n)$ as well as $\mathcal{O}(n^r)$ where $r$ is not an integer. We present experimental results to demonstrate that our approach can obtain efficiently worst-case bounds of classical recursive algorithms such as (i) Merge-Sort, the divide-and-conquer algorithm for the Closest-Pair problem, where we obtain $\mathcal{O}(n \log n)$ worst-case bound, and (ii) Karatsuba's algorithm for polynomial multiplication and Strassen's algorithm for matrix multiplication, where we obtain $\mathcal{O}(n^r)$ bound such that $r$ is not an integer and close to the best-known bounds for the respective algorithms.Comment: 54 Pages, Full Version to CAV 201

A Tutorial on the Expectation-Maximization Algorithm Including Maximum-Likelihood Estimation and EM Training of Probabilistic Context-Free Grammars

The paper gives a brief review of the expectation-maximization algorithm (Dempster 1977) in the comprehensible framework of discrete mathematics. In Section 2, two prominent estimation methods, the relative-frequency estimation and the maximum-likelihood estimation are presented. Section 3 is dedicated to the expectation-maximization algorithm and a simpler variant, the generalized expectation-maximization algorithm. In Section 4, two loaded dice are rolled. A more interesting example is presented in Section 5: The estimation of probabilistic context-free grammars.Comment: Presented at the 15th European Summer School in Logic, Language and Information (ESSLLI 2003). Example 5 extended (and partially corrected

Factoring Predicate Argument and Scope Semantics : underspecified Semantics with LTAG

In this paper we propose a compositional semantics for lexicalized tree-adjoining grammar (LTAG). Tree-local multicomponent derivations allow separation of the semantic contribution of a lexical item into one component contributing to the predicate argument structure and a second component contributing to scope semantics. Based on this idea a syntax-semantics interface is presented where the compositional semantics depends only on the derivation structure. It is shown that the derivation structure (and indirectly the locality of derivations) allows an appropriate amount of underspecification. This is illustrated by investigating underspecified representations for quantifier scope ambiguities and related phenomena such as adjunct scope and island constraints