The Complexity of Constraints: Dichotomies and Beyond

ABSTRACT. This talk is devoted to the recent results on the Dichotomy Conjecture for the Constraint Satisfaction Problem (CSP) posed by Feder and Vardi in 1993. We will briefly review the research leading up to the dichotomy theorem for the CSP, outline the solution algorithm, and discuss remaining open problems and future research directions.

ABSTRACT. Additive number theory is the study of the additive properties of sets of natural numbers. One of the most famous results in this field is Lagrange's theorem from 1770: every natural number is the sum of four squares.

In this talk I will show how logic and finite automata can be used to mechanically construct proofs of theorems in additive number theory of genuine interest to number theorists. For example, we can prove that every natural number is the sum of four binary palindromes (numbers whose base-2 representation reads the same forwards and backwards).

I will give examples of what can be proven use this idea, mention some open problems, and talk about future applications and limitations. This represents joint work with Jason Bell, Dirk Nowotka, Tim Smith, Aayush Rajasekaran, Finn Lidbetter, P. Madhusudan, Kathryn Hare, Daniel Kane, and Carlo Sanna.

Continuous Models of Computation: Computability, Complexity, Universality

ABSTRACT. In 1941, Claude Shannon introduced a continuous-time analog model of computation, namely the General Purpose Analog Computer (GPAC). The GPAC is a physically feasible model in the sense that it can be implemented in practice through the use of analog electronics or mechanical devices. It can be proved that the functions computed by a GPAC are precisely the solutions of a special class of differential equations where the right-hand side is a polynomial. Analog computers
have since been replaced by digital counterpart. Nevertheless, one can wonder how the GPAC could be compared to Turing machines.

A few years ago, it was shown that Turing-based paradigms and the GPAC have the same computational power. However, this result did not shed any light on what happens at a computational complexity level. In other words, analog computers do not make a difference about what can be computed; but maybe they could compute faster than a digital computer. A fundamental difficulty of continuous-time model is to define a proper notion of complexity. Indeed, a troubling problem is that many models exhibit the so-called Zeno's phenomenon, also known as space-time contraction.

In this talk, I will present results from my thesis that give several fundamental contributions to these questions. We show that the GPAC has the same computational power as the Turing machine, at the complexity level. We also provide as a side effect a purely analog, machine-independent characterization of P and Computable Analysis.

I will present some recent work on the universality of polynomial differential equations. We show that when we impose no restrictions at all on the system, it is possible to build a fixed equation that is universal in the sense it can approximate arbitrarily well any continuous curve over R, simply by changing the initial condition of the system.

If time allows, I will also mention some recent application of this work to show that chemical reaction networks are strongly Turing complete with the differential semantics.

ABSTRACT. The automata-theoretic approach to LTL model checking relies on translations of LTL formulae into automata. Efficiency of and techniques used in these translations differ depending on the target automata class. For instance, deterministic automata were traditionally produced using Safra's construction, which is quite complex and practically not very efficient. In this work, we present a unified translation of LTL formulas into deterministic Rabin automata, limit-deterministic Büchi automata, and nondeterministic Büchi automata. The translations yield automata of asymptotically optimal size (double or single exponential, respectively). All three translations are derived from one single Master Theorem of purely logical nature. The Master Theorem decomposes the language of a formula into a positive boolean combination of languages that can be translated into automata by elementary means. In particular, Safra's, ranking, and breakpoint constructions used in other translations are not needed.