François Lemieux

Are inflationary predictions sensitive to very high energy physics

It has been proposed that the successful inflationary description of density perturbations on cosmological scales is sensitive to the details of physics at extremely high (trans-Planckian) energies. We test this proposal by examining how inflationary predictions depend on higher-energy scales within a simple model where the higher-energy physics is well understood. We find the best of all possible worlds: inflationary predictions are robust against the vast majority of high-energy effects, but can be sensitive to some effects in certain circumstances, in a way which does not violate ordinary notions of decoupling. This implies both that the comparison of inflationary predictions with CMB data is meaningful, and that it is also worth searching for small deviations from the standard results in the hopes of learning about very high energies.

Finite Loops Recognize Exactly the Regular Open Languages

In this paper, we characterize exactly the class of languages that are recognizable by finite loops, i.e. by cancellative binary algebras with an identity. This turns out to be the well-studied class of regular open languages. Our proof technique is interesting in itself: we generalize the operation of block product of monoids, which is so useful in the associative case, to the situation where the left factor in the product is non-associative.

The Complexity of Computing over Quasigroups

In [7] the notions of recognition by semigroups and by programs over semigroups were extended to groupoids. As a consequence of this transformation, the induced classes of languages became CFL instead of REG, in the first case, and SAC1 instead of NC1 in the second case. In this paper, we investigate the classes obtained when the groupoids are restricted to be quasigroups (i.e. the multiplication table forms a latin square). We prove that languages recognized by quasigroups are regular and that programs over quasigroups characterize NC1. We introduce the notions of linear recognition by semigroups and by programs over semigroups. This leads to a new characterization of the linear context-free languages and NL. Here again, when quasigroups are used, only languages in REG and NC1 can be obtained. We also consider the problem of evaluating a well-parenthesized expression over a finite loop (a quasigroup with an identity). This problem is in NC1 for any finite loop, and we give algebraic conditions for its completeness. In particular, we prove that it is sufficient that the loop be noasolvable, extending a well-known theorem of Barrington.

Groupoids that recognize only regular languages

Finite semigroups, i.e. finites sets equipped with a binary associative operation, have played a role in theoretical computer science for fifty years. They were first observed to be closely related to finite automata, hence, by the famous theorem of Kleene, to regular languages. It was later understood that this association is very deep and the theory of pseudo-varieties of Schutzenberger and Eilenberg [5] became the accepted framework in which to discuss computations realized by finite-state machines. It is today fair to say that semigroups and automata are so tightly intertwined that it makes little sense to study one without the other.

Star-Free Open Languages and Aperiodic Loops

It is known that recognition of regular languages by finite monoids can be generalized to context-free languages and finite groupoids, which are finite sets closed under a binary operation. A loop is a groupoid with a neutral element and in which each element has a left and a right inverse. It has been shown that finite loops recognize exactly those regular languages that are open in the group topology. In this paper, we study the class of aperiodic loops, which are those loops that contain no nontrivial group. We show that this class is stable under various definitions, and we prove some closure properties. We also prove that aperiodic loops recognize only star-free open languages and give some examples. Finally, we show that the wreath product principle can be applied to groupoids, and we use it to prove a decomposition theorem for recognizers of regular open languages.

Faithful Loops for Aperiodic E-Ordered Monoids

One of the main objectives of the algebraic theory of regular languages concerns the classification of regular languages based on Eilenbergs variety theorem [10]. This theorem states that there exists a bijection between varieties of regular languages and varieties of finite monoids. For example, the variety of star-free regular languages (the closure of finite languages under Boolean operations and concatenation) is related to the monoid variety of aperiodic monoids (those with no nontrivial subgroups)[21].