On Polynomial Recursive Sequences

We study the expressive power of polynomial recursive sequences, a nonlinear extension of the well-known class of linear recursive sequences. These sequences arise naturally in the study of nonlinear extensions of weighted automata, where (non)expressiveness results translate to class separations. A typical example of a polynomial recursive sequence is b_n = n!. Our main result is that the sequence u_n = n? is not polynomial recursive

On polynomial recursive sequences

International audienceWe study the expressive power of polynomial recursive sequences, a nonlinear extension of the well-known class of linear recursive sequences. These sequences arise naturally in the study of nonlinear extensions of weighted automata, where (non)expressiveness results translate to class separations. A typical example of a polynomial recursive sequence is $b_n=n!$. Our main result is that the sequence $u_n=n^n$ is not polynomial recursive

Bidimensional Linear Recursive Sequences and Universality of Unambiguous Register Automata

We study the universality and inclusion problems for register automata over equality data. We show that the universality and the inclusion problems can be solved with 2-EXPTIME complexity when the input automata are without guessing and unambiguous, improving on the currently best-known 2-EXPSPACE upper bound by Mottet and Quaas. When the number of registers of both automata is fixed, we obtain a lower EXPTIME complexity, also improving the EXPSPACE upper bound from Mottet and Quaas for fixed number of registers. We reduce inclusion to universality, and then we reduce universality to the problem of counting the number of orbits of runs of the automaton. We show that the orbit-counting function satisfies a system of bidimensional linear recursive equations with polynomial coefficients (linrec), which generalises analogous recurrences for the Stirling numbers of the second kind, and then we show that universality reduces to the zeroness problem for linrec sequences. While such a counting approach is classical and has successfully been applied to unambiguous finite automata and grammars over finite alphabets, its application to register automata over infinite alphabets is novel. We provide two algorithms to decide the zeroness problem for bidimensional linear recursive sequences arising from orbit-counting functions. Both algorithms rely on techniques from linear non-commutative algebra. The first algorithm performs variable elimination and has elementary complexity. The second algorithm is a refined version of the first one and it relies on the computation of the Hermite normal form of matrices over a skew polynomial field. The second algorithm yields an EXPTIME decision procedure for the zeroness problem of linrec sequences, which in turn yields the claimed bounds for the universality and inclusion problems of register automata.Comment: full version of the homonymous paper to appear in the proceedings of STACS'2

On Rational Recursive Sequences

We study the class of rational recursive sequences (ratrec) over the rational numbers. A ratrec sequence is defined via a system of sequences using mutually recursive equations of depth 1, where the next values are computed as rational functions of the previous values. An alternative class is that of simple ratrec sequences, where one uses a single recursive equation, however of depth k: the next value is defined as a rational function of k previous values. We conjecture that the classes ratrec and simple ratrec coincide. The main contribution of this paper is a proof of a variant of this conjecture where the initial conditions are treated symbolically, using a formal variable per sequence, while the sequences themselves consist of rational functions over those variables. While the initial conjecture does not follow from this variant, we hope that the introduced algebraic techniques may eventually be helpful in resolving the problem. The class ratrec strictly generalises a well-known class of polynomial recursive sequences (polyrec). These are defined like ratrec, but using polynomial functions instead of rational ones. One can observe that if our conjecture is true and effective, then we can improve the complexities of the zeroness and the equivalence problems for polyrec sequences. Currently, the only known upper bound is Ackermanian, which follows from results on polynomial automata. We complement this observation by proving a PSPACE lower bound for both problems for polyrec. Our lower bound construction also implies that the Skolem problem is PSPACE-hard for the polyrec class

Comparison-Free Polyregular Functions.

This paper introduces a new automata-theoretic class of string-to-string functions with polynomialgrowth. Several equivalent definitions are provided: a machine model which is a restricted variant ofpebble transducers, and a few inductive definitions that close the class of regular functions undercertain operations. Our motivation for studying this class comes from another characterization,which we merely mention here but prove elsewhere, based on a λ-calculus with a linear type system.As their name suggests, these comparison-free polyregular functions form a subclass of polyregularfunctions; we prove that the inclusion is strict. We also show that they are incomparable withHDT0L transductions, closed under usual function composition – but not under a certain “map”combinator – and satisfy a comparison-free version of the pebble minimization theorem.On the broader topic of polynomial growth transductions, we also consider the recently introducedlayered streaming string transducers (SSTs), or equivalently k-marble transducers. We prove that afunction can be obtained by composing such transducers together if and only if it is polyregular,and that k-layered SSTs (or k-marble transducers) are closed under “map” and equivalent to acorresponding notion of (k + 1)-layered HDT0L systems

Annual Report 2019-2020

LETTER FROM THE DEAN As I write this letter wrapping up the 2019-20 academic year, we remain in a global pandemic that has profoundly altered our lives. While many things have changed, some stayed the same: our CDM community worked hard, showed up for one another, and continued to advance their respective fields. A year that began like many others changed swiftly on March 11th when the University announced that spring classes would run remotely. By March 28th, the first day of spring quarter, we had moved 500 CDM courses online thanks to the diligent work of our faculty, staff, and instructional designers. But CDM’s work went beyond the (virtual) classroom. We mobilized our makerspaces to assist in the production of personal protective equipment for Illinois healthcare workers, participated in COVID-19 research initiatives, and were inspired by the innovative ways our student groups learned to network. You can read more about our response to the COVID-19 pandemic on pgs. 17-19. Throughout the year, our students were nationally recognized for their skills and creative work while our faculty were published dozens of times and screened their films at prestigious film festivals. We added a new undergraduate Industrial Design program, opened a second makerspace on the Lincoln Park Campus, and created new opportunities for Chicago youth. I am pleased to share with you the College of Computing and Digital Media’s (CDM) 2019-20 annual report, highlighting our collective accomplishments. David MillerDeanhttps://via.library.depaul.edu/cdmannual/1003/thumbnail.jp

Annual Report 2020-2021

LETTER FROM THE DEAN As I write this letter during the beginning of the 2021–22 academic year, we have started to welcome the majority of our students to campus— many for the very first time, and some for the first time in a year and a half. It has been wonderful to be together, in-person, again. Four quarters of learning and working remotely was challenging, to be sure, but I have been consistently amazed by the resilience, innovation, and hard work of our students, faculty, and staff, even in the most difficult of circumstances. This annual report, covering the 2020–21 academic year—one that was entirely virtual—highlights many of those examples: from a second place national ranking by our Security Daemons team to hosting a blockbuster virtual screenwriting conference with top talent; from gaming grants helping us reach historically excluded youth to alumni successes across our three schools. Recently, I announced that, after 40 years at DePaul and 15 years as the Dean of CDM, I will be stepping down from the deanship at the end of the 2021–22 academic year. I began my tenure at DePaul in 1981 as an assistant professor, with the founding of the Department of Computer Science, joining seven faculty members who were leaving the mathematics department for this new venture. It has been amazing to watch our college grow during that time. We now have more than 40 undergraduate and graduate degree programs, over 22,000 college alumni, and a catalog of nationally ranked programs. And we plan to keep going. If there is anything I’ve learned at CDM, it’s that a lot can be accomplished in a year (as this report shows), and I’m committed to working hard and continuing the progress we’ve made together in 2021–22. David MillerDeanhttps://via.library.depaul.edu/cdmannual/1004/thumbnail.jp

Revisiting the growth of polyregular functions: output languages, weighted automata and unary inputs

Polyregular functions are the class of string-to-string functions definable by pebble transducers (an extension of finite automata) or equivalently by MSO interpretations (a logical formalism). Their output length is bounded by a polynomial in the input length: a function computed by a $k$-pebble transducer or by a $k$-dimensional MSO interpretation has growth rate $O(n^k)$. Boja\'nczyk has recently shown that the converse holds for MSO interpretations, but not for pebble transducers. We give significantly simplified proofs of those two results, extending the former to first-order interpretations by reduction to an elementary property of $\mathbb{N}$-weighted automata. For any $k$, we also prove the stronger statement that there is some quadratic polyregular function whose output language differs from that of any $k$-fold composition of macro tree transducers (and which therefore cannot be computed by any $k$-pebble transducer). In the special case of unary input alphabets, we show that $k$ pebbles suffice to compute polyregular functions of growth $O(n^k)$. This is obtained as a corollary of a basis of simple word sequences whose ultimately periodic combinations generate all polyregular functions with unary input. Finally, we study polyregular and polyblind functions between unary alphabets (i.e. integer sequences), as well as their first-order subclasses.Comment: 27 pages, not submitted ye