Pierre Mckenzie

• Université de Montréal

We prove that the reachability problem for two-dimensional vector addition systems with states is NL-complete or PSPACE-complete, depending on whether the numbers in the input are encoded in unary or binary. As a key underlying technical result, we show that, if a configuration is reachable, then there exists a witnessing path whose sequence of tra...

Cost register automata (CRAs) are one-way finite automata whose transitions have the side effect that a register is set to the result of applying a state-dependent semiring operation to a pair of registers. Here it is shown that CRAs over the tropical semiring \((\mathbb {N}\cup \{\infty \},\min ,+)\) can simulate polynomial time computation, provi...

The Parikh automaton model equips a finite automaton with integer registers and imposes a semilinear constraint on the set of their final settings. Here the theories of typed monoids and of rational series are used to characterize the language classes that arise algebraically. Complexity bounds are derived, such as containment of the unambiguous Pa...

Most decidability results concerning well-structured transition systems apply to the finitely branching variant. Yet some models (inserting automata, ω-Petri nets, …) are naturally infinitely branching. Here we develop tools to handle infinitely branching WSTS by exploiting the crucial property that in the (ideal) completion of a well-quasi-ordered...

The well-quasi-ordering (i.e., a well-founded quasi-ordering such that all antichains are finite) that defines well-structured transition systems (WSTS) is shown not to be the weakest hypothesis that implies decidability of the coverability problem. We show coverability decidable for monotone transition systems that only require the absence of infi...

The well-quasi-ordering (i.e., a well-founded quasi-ordering such that all antichains are finite) that defines well-structured transition systems (WSTS) is shown not to be the weakest hypothesis that implies decidability of the coverability problem. We show coverability decidable for monotone transition systems that only require the absence of infi...

Determining the complexity of the reachability problem for vector addition systems with states (VASS) is a long-standing open problem in computer science. Long known to be decidable, the problem to this day lacks any complexity upper bound whatsoever. In this paper, reachability for two-dimensional VASS is shown PSPACE-complete. This improves on a...

The problem of determining whether several finite automata accept a word in common is closely related to the well-studied membership problem in transformation monoids. We raise the issue of limiting the number of final states in the automata intersection problem. For automata with two final states, we show the problem to be \({\oplus}\)L-complete o...

Most decidability results concerning well-structured transition systems apply to the finitely branching variant. Yet some models (inserting automata, ω-Petri nets, ⋯) are naturally infinitely branching. Here we develop tools to handle infinitely branching WSTS by exploiting the crucial property that in the (ideal) completion of a well-quasi-ordered...

The class of languages captured by Constrained Automata (CA) that are unambiguous is shown to possess more closure properties than the provably weaker class captured by deterministic CA. Problems decidable for deterministic CA are nonetheless shown to remain decidable for unambiguous CA, and testing for regularity is added to this set of decidable...

The Parikh automaton model equips a finite automaton with integer registers and imposes a semilinear constraint on the set of their final settings. Here the theory of typed monoids is used to characterize the language classes that arise algebraically. Complexity bounds are derived, such as containment of the unambiguous Parikh automata languages in...

The Parikh finite word automaton model (PA) was introduced and studied by Klaedtke and Rueß. Here, we present some expressiveness properties of a restriction of the deterministic affine PA recently introduced, and use them as a tool to show that the bounded languages recognized by PA are the same as those recognized by deterministic PA. Moreover, t...

The Parikh finite word automaton (PA) was introduced and studied in 2003 by Klaedtke and Rueß. Natural variants of the PA arise from viewing a PA equivalently as an automaton that keeps a count of its transitions and semilinearly constrains their numbers. Here we adopt this view and define the affine PA, that extends the PA by having each transitio...

The effect of severely tightening the uniformity of Boolean circuit families is investigated. The impact on NC 1 and its subclasses is shown to depend on the characterization chosen for the class, while classes such as P appear to be more robust. Tightly uniform subclasses of NC 1 whose separation may be within reach of current techniques emerge.

The class of languages captured by Constrained Automata (CA) that are unambiguous is shown to possess more closure properties than the provably weaker class captured by deterministic CA. Problems decidable for deterministic CA are nonetheless shown to remain decidable for unambiguous CA, and testing for regularity is added to this set of decidable...

The Parikh finite word automaton model (PA) was introduced and studied by Klaedtke and Ruess in 2003. Here, by means of related models, it is shown that the bounded languages recognized by PA are the same as those recognized by deterministic PA. Moreover, this class of languages is the class of bounded languages whose set of iterations is semilinea...

The Parikh finite word automaton (PA) was introduced and studied by Klaedtke and Ruess in 2003. Natural variants of the PA arise from viewing a PA equivalently as an automaton that keeps a count of its transitions and semilinearly constrains their numbers. Here we adopt this view and define the affine PA (APA), that extends the PA by having each tr...

The Parikh finite word automaton (PA) was introduced and studied by Klaedtke and Rueß [16]. Natural variants of the PA arise from viewing a PA equivalently as an automaton that keeps a count of its transitions and semilinearly constrains their numbers. Here we adopt this view and define the affine PA (APA), that extends the PA by having each transi...

Martin Gardner in the early 1970’s described the game of RaceTrack [M. Gardner, Mathematical games—Sim, Chomp and Race Track: new games for the intellect (and not for Lady Luck), Scientific American, 228(1):108–115, Jan. 1973]. Here we study the complexity of deciding whether a RaceTrack player has a winning strategy. We first prove that the comple...

We introduce the Tree Evaluation Problem, show that it is in logDCFL (and hence in P), and study its branching program complexity in the hope of eventually proving a superlogarithmic space lower bound. The input to the problem is a rooted, balanced d-ary tree of height h, whose internal nodes are labeled with d-ary functions on [k] = {1,...,k}, and...

A d-gem is a { + , − ,×}-circuit having very few ×-gates and computing from {x} ∪ ℤ a univariate polynomial of degree d having d distinct integer roots. We introduce d-gems because they could help factoring integers and because their existence for infinitely many d would blatantly disprove a variant of the Blum-Cucker-Shub-Smale conjecture. A natur...

We study the branching program complexity of the tree evaluation problem, introduced in [3] as a candidate for separating NL from LogCFL. The input to the problem is a rooted, balanced d-ary tree of height h, whose internal nodes are labelled with d-ary functions on [k] = {1, . . . , k}, and whose leaves are labelled with elements of [k]. Each node...

Imposing an extensional uniformity condition on a non-uniform circuit complexity class \(\mathcal{C}\) means simply intersecting \(\mathcal{C}\) with a uniform class \(\mathcal{L}\). By contrast, the usual intensional uniformity conditions require that a resource-bounded machine be able to exhibit the circuits in the circuit family defining \(\math...

In the interactive communication model, two parties and possess respective private but correlated inputs and , and wants to learn from while minimizing the communication required for the worst possible input pair . Our contribution is the analysis of four nonzero-error models in this correlated data setting. In the private coin randomized model, bo...

Imposing an extensional uniformity condition on a non-uniform circuit complexity class C means simply intersecting C with a uniform class L. By contrast, the usual intensional uniformity conditions require that a resource-bounded machine be able to exhibit the circuits in the circuit family defining C. We say that (C,L) has the "Uniformity Duality...

The problem of testing membership in the subset of the natural numbers produced at the output gate of a {È, Ç, -, +, \bigcup, \bigcap, ^-, +, \times} combinational circuit is shown to capture a wide range of complexity classes. Although the general problem remains open, the case {È, Ç, +, \bigcup, \bigcap, +, \times} is shown NEXPTIME-complete, the...

Klondike is the well-known 52-card Solitaire game available on almost every computer. The problem of determining whether an n-card Klondike initial configuration can lead to a win is shown NP-complete. The problem remains NP-complete when only three suits are allowed instead of the usual four. When only two suits of opposite color are available, th...

We propose a new model of restricted branching programs which we call incremental branching programs. We show that syntactic incremental branching programs capture previously studied structured models of computation for the problem GEN, namely marking machines [S. A. Cook, J. Comput. Syst. Sci. 9, 308–316 (1974; Zbl 0306.68026)] and Poon’s extensio...

On http://www.cs.wisc.edu/techreports/2005/TR1523.pdf, A. Lal and D. von Melkebeek noted that the proof of Theorem 6.1 in our paper [ibid. 66, 549–566 (2003; Zbl 1054.68101)] is not correct the way it is. However, the result holds and the proof can be corrected by adding the following sentences after Rule 2: “An exception to the rule is applied in...

Without Abstract

First-order translations have recently been characterized as the maps computed by aperiodic single-valued non-deterministic finite transducers (NFTs). It is shown here that this characterization lifts to "V-translations" and "V-single-valued-NFTs", where V is an arbitrary monoid pseudovariety that is closed under reversal. More strikingly, two-way...

The problem of evaluating a circuit whose wires carry values from a fixed finite monoid M and whose non-input gates perform the monoid's operation is a natural extension to the well studied word problem over M, known to characterize NC 1 and most of its subclasses in terms of the algebraic properties of M [2, 5, 4, 18]. Here we investigate the circ...

We define the notion of an oracle branching program in order to investigate space-bounded computation. Within this new framework we examine the P-complete problem GEN which consists of determining membership in a subalgebra of a general (not necessarily associative) binary algebra (input as a multiplication table). Our work begins with the statemen...

We propose a new model of restricted branching programs which we call {em incremental branching programs}. We show that {em syntactic} incremental branching programs capture previously studied structured models of computation for the problem GEN, namely marking machines [Cook74]. and Poon's extension [Poon93] of jumping automata on graphs [CookRack...

Transition systems defined from recursive functions IN^p->IN^p are introduced and named WSNs, or well-structured nets. Such nets sit conveniently between Petri net extensions and general transition systems. In the first part of this paper, we study decidability properties of WSN classes obtained by imposing natural restrictions on their defining fu...

We prove that the graph isomorphism problem restricted to trees and to colored graphs with color multiplicities 2 and 3 is many-one complete for several complexity classes within NC2. In particular we show that tree isomorphism, when trees are encoded as strings, is NC1-hard under AC0-reductions. NC1-completeness thus follows from Buss's NC1 upper...

The problem of testing membership in the subset of the natural numbers produced at the output gate of a ∪, ∩,- ,+, * combinational circuit is shown to capture a wide range of complexity classes. Although the general problem remains open, the case ∪, ∩,+, * is shown NEXPTIME-complete, the cases ∪, ∩,- , *, ∪, ∩, *, ∪, ∩,+ are shown PSPACE-complete,...

Thérien and Wilke characterized the Until hierarchy of linear temporal logic in terms of aperiodic monoids. Here, a temporal operator able to count modulo q is introduced. Temporal logic augmented with such operators is found decidable as it is shown to express precisely the solvable regular languages. Natural hierarchies are shown to arise when mo...

Tensor calculus over semirings is shown relevant to complexity theory in unexpected ways. First, evaluating well-formed tensor formulas with explicit tensor entries is shown complete for \bigoplusP\bigoplusP, for NP, and for #P as the semiring varies. Indeed the permanent of a matrix is shown expressible as the value of a tensor formula in much t...

The Cayley group membership problem (CGM) is to input a groupoid (binary algebra) G given as a multiplication table, a subset X of G, and an element t of G and to determine whether t can be expressed as a product of elements of X. For general groupoids CGM is P-complete, and for associative algebras (semigroups) it is NL-complete. Here we investiga...

This paper addresses the problems of counting proof trees (as introduced by Venkateswaran and Tompa) and counting proof circuits, a related but seemingly more natural question. These problems lead to a common generalization of straight-line programs which we call polynomial replacement systems. We contribute a classication of these systems and we i...

We prove tight lower bounds, of up to n ffl , for the monotone depth of functions in monotone-P. As a result we achieve the separation of the following classes. 1. monotone-NC 6= monotone-P. 2. For every i 1, monotone-NC i 6= monotone-NC i+1 . 3. More generally: For any integer function D(n), up to n ffl (for some ffl ? 0), we give an explicit exam...

This paper describes the simulation of an S(n) space-bounded deterministic Turing machine by a reversible Turing machine operating in space S(n). It thus answers a question posed by C. H. Bennett [SIAM J. Comput. 18, No. 4, 766-776 (1989; Zbl 0676.68010)] and refutes the conjecture, made by M. Li and P. Vitányi [Proc. R. Soc. Lond., Ser. A 452, No....

First-order translations have recently been characterized as the maps computed by aperiodic single-valued nondeterministic finite transducers (NFTs). It is shown here that this characterization lifts to “V-translations” and “V-single-valued-NFTs”, where V is an arbitrary monoid pseudovariety. More strikingly, 2-way V-machines are introduced, and th...

We consider variants of alternating auxiliary stack automata and characterize their computational power when the number of alternations is bounded by a constant or unlimited. In this way we get new characterizations of NP, the polynomial hierarchy, PSpace, and bounded query classes like NL 〈 NP[1]〉 and Θ2 P = P NP[O(logn)], in a uniform framework.

We study the computational complexity of solving equations and of determining the satisfiability of programs over a fixed finite monoid. We partially answer an open problem of [4] by exhibiting quasi-polynomial time algorithms for a subclass of solvable non-nilpotent groups and relate this question to a natural circuit complexity conjecture. In the...

We study the computational complexity of solving equations and of determining the satisfiability of programs over a fixed finite monoid. We partially answer an open problem of [4] by exhibiting quasi-polynomial time algorithms for a sub-class of solvable non-nilpotent groups and relate this question to a natural circuit complexity conjecture.

Simpler proofs that DAuxPDA-TIME(polynomial) equals LOGDCFL and that SAC 1 equals LOGCFL are given which avoid Sudborough's multi-head automata [Sud78]. The first characterization of LOGDCFL in terms of polynomial proof-tree-size is obtained, using circuits built from the multiplex select gates of [FLR96]. The classes L and NC 1 are also characteri...

. We locate the complexities of evaluating, of inverting, and of testing membership in the image of, morphisms h : Sigma ! Delta . By and large, we show these problems complete for classes within NL. Then we develop new properties of finite codes and of finite sets of words, which yield image membership subproblems that are closely tied to the unam...

We prove that the tree isomorphism problem, when trees are encoded as strings, is NC 1 -hard under DLOGTIME-reductions. NC 1 -completeness thus follows from Buss's recent NC 1 upper bound. By contrast, we prove that testing isomorphism of two trees encoded as pointer lists is L-complete. 1 Introduction GI, the graph isomorphism problem, is one of t...

Transition systems defined from recursive functions IN p ! IN p are introduced and named WSN, or well-structured nets. Such "minimally Petri net-like" transition systems sit conveniently between Petri net extensions and general transition systems. In a first part, we study decidability properties of WSN classes obtained by imposing natural restrict...

D. Therien and T. Wilke (1996) characterized the Until hierarchy of linear temporal logic in terms of aperiodic monoids. Here, a temporal operator able to count modulo q is introduced. Temporal logic augmented with such operators is found decidable as it is shown to express precisely the solvable regular languages. Natural hierarchies are shown to...

S On the Power of Randomized Branching Programs Farid Ablayev Kazan University (joint work with Marek Karpinski, Universitat Bonn) We define a notion of randomized branching programs in a natural way similar to the notion of randomized circuits. We present two explicit boolean functions f n : f0; 1g 4n ! f0; 1g and g n : f0; 1g n ! f0; 1g such that...

We locate the complexities of evaluating, of inverting, and of testing membership in the image of, morphisms h : \Sigma ! \Delta . By and large, we show these problems complete for classes within NL. Then we develop new properties of finite codes and of finite sets of words, which yield image membership subproblems that are closely tied to the unam...

We define the counting classes #NC1, GapNC1, PNC1 and C=NC1. We prove that boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We observe that #NC1 subseteq #L and that C=NC1 subs...

We define the counting classes #NC1,GapNC1,PNC1, andC=NC1. We prove that boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We observe that #NC1⊆#L, thatPNC1⊆L, and thatC=NC1⊆L....

Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers in first-order logic with linear order. Our work extends the elaborate theory relating monoi...

Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers. Our work extends the elaborate theory relating monoidal quantifiers to NC1 and its subclass...

We prove that the tree isomorphism problem, when trees are encoded as strings, is NC¹-hard under DLOGTIME-reductions. NC¹-completeness thus follows from Buss's recent NC¹ upper bound. By contrast, we prove that testing isomorphism of two trees encoded as pointer lists is L-complete

This paper describes the simulation of an S(n) space-bounded deterministic Turing machine by a reversible Turing machine operating in space S(n). It thus answers a question posed by C. Bennett (1989) and refutes the conjecture, made by M. Li and P. Vitanyi (1996), that any reversible simulation of an irreversible computation must obey Bennett's rev...

This paper describes the simulation of an S(n) spacebounded deterministic Turing machine by a reversible Turing machine operating in space S(n). It thus answers a question posed by Bennett in 1989 and refutes the conjecture, made by Li and Vitanyi in 1996, that any reversible simulation of an irreversible computation must obey Bennett's reversible...

We prove that boundedness and reachability tree finiteness are undecidable for systems of two identical automata communicating via two perfect unbounded one-way FIFO channels and constructed solely from cycles about their initial states. Using a form of mutual exclusion for such systems, we prove further that undecidability holds even when the iden...

The problem of evaluating a circuit whose wires carry values from a finite monoid M and whose gates perform the monoid operation provides a meaningful generalization to the well-studied problem of evaluating a word over M. Evaluating words over monoids is closely tied to the fine structure of the complexity class NC1, and in this paper analogous ti...

The computation tree of a nondeterministic machineMwith inputxgives rise to aleaf stringformed by concatenating the outcomes of all the computations in the tree in lexicographical order. We may characterize problems by considering, for a particular “leaf language”Y, the set of allxfor which the leaf string ofMis contained inY. In this way, in the c...

Leaf languages were used in the context of polynomial time computation to capture complexity classes and to study machine-independent relativizations. In this paper, the expressibility of the leaf language mechanism is investigated in the contexts of logarithmic space and of logarithmic time computation

It is shown that the formula and circuit evaluation problems in the nonassociative context capture natural complexity classes up to NP, thus extending the known result that the word problem over a groupoid is LOGCFL-complete. The problem of multiplying together matrices whose elements are taken from an algebraic structure more general than a semiri...

The problem of testing membership in aperiodic or “group-free” transformation monoids is the natural counterpart to the well-studied membership problem in permutation groups. The class A of all finite aperiodic monoids and the class G of all finite groups are two examples of varieties, the fundamental complexity units in terms of which finite monoi...

We consider the complexity of various computational problems over nonassociative algebraic structures. Specifically, we look at the problem of evaluating circuits, formulas, and words, over both nonassociative structures themselves and over matrices with elements in these structures. Extending past work, we show that such problems can characterize...

Concepts from the algebraic theory of finite automata are carried over to the program-over-monoid setting which underlies Barrington's algebraic characterization of the complexity classNC 1. Sets of languages accepted by polynomial-length programs over finite monoids drawn from a given monoid variety V emerge as fundamental language classes: as V r...

Groupoids are used instead of monoids to extend D.A. Barrington's (1988) successful polynomial length program over a monoid computation model to characterize complexity classes TC and LOGCFL. Further allowing groupoid families instead of fixed groupoids, deterministic and nondeterministic logarithmic space are also characterized. Several language c...

Barrington's “polynomal-length program over a monoid” is a model of computation which has been studied intensively in connection with the structure of the complexity class NC1 [Barrington (1986), Barrington and Thérien (1987, 1988), McKenzie and Thérien (1989), Péladeau (1989)]. Here two extensions of the model are considered. First, with the use o...

Without Abstract

We define the notion of an oracle branching program in order to investigate space-bounded computation. Within this new framework we examine the P-complete problem GEN which consists of determining membership in a subalgebra of a general (not necessarily associative) binary algebra (input as a multiplication table). Our work begins with the statemen...

A number of basic problems involving solvable and nilpotent permutation groups are shown to have fast parallel solutions. Testing solvability is in NC as well as, for solvable groups, finding order, testing membership, finding centralizers, finding centers, finding the derived series and finding a composition series. Additionally, for nilpotent gro...

The authors classify Abelian permutation group problems with respect to their parallel complexity. For such groups specified by generating permutations the authors show that testing membership, computing order and testing isomorphism are NC¹-equivalent to (and therefore have essentially the same parallel complexity as) determining solvability of a...

We exhibit several problems complete for deterministic logarithmic space under NC1 (i.e., log depth) reducibility. The list includes breadth-first search and depth-first search of an undirected tree, connectivity of undirected graphs known to be made up of one or more disjoint cycles, undirected graph acyclicity, and several problems related to rep...

We exhibit several problems complete for deterministic logarithmic space under NC¹ (i.e., log depth) reducibility. The list includes breadth-first search and depth-first search of an undirected tree, connectivity of undirected graphs known to be made up of one or more disjoint cycles, undirected graph acyclicity, and several problems related to rep...

We develop fast parallel solutions to a number of basic problems involving solvable and nilpotent permutation groups. Testing solvability is in NC, and RNC includes, for solvable groups, finding order, testing membership, finding the derived series and finding a composition series. Additionally, for nilpotent groups, one can, in RNC, find the cente...

Thesis (Ph.D.)--University of Toronto, 1984. Bibliography: p. 75-80. Photocopy.

We show that the permutation group membership problem can be solved in depth (logn)3 on a Monte Carlo Boolean circuit of polynomial size in the restricted case in which the group is abelian. We also show that this restricted problem is NC1-hard for NSPACE(logn).

The problem FThd(k)FT^{h}_{d}(k) consists in computing the value in [k] = {1,...,k} taken by the root of a balanced d-ary tree of height h whose internal nodes are labelled with d-ary functions on [k] and whose leaves are labelled with elements of [k]. We propose FThd(k){FT^{h}_{d}(k)} as a good candidate for witnessing L \subsetneqLogDCFL{\mathb...

Motivated by the integer factoring problem, we define a d-gem as a {+, −, ×}-circuit having at most d product gates and computing, from {x}∪Z, a degree d polynomial having d distinct roots in Z, where d is the minimum length of an addition chain for d. For n ≤ 4 we exhibit 2 n -gems having the additional property of being skew, that is, one input t...

The problem of evaluating a circuit whose wires carry values from a fixed finite monoid M and whose non-input gates perform the monoid’s operation is a natural extension to the well studied word problem over M, known to characterize NC 1 and most of its subclasses in terms of the algebraic properties of M. Here we investigate the circuit evaluation...

Die Fachgruppe AFS (früher Fachgruppe 0.1.5) der Gesellschaft für Informatik veranstaltet seit 1991 einmal im Jahr ein Treffen der Fachgruppe im Rahmen eines Theorietags, der traditionell eineinhalb Tage dauert. Seit dem Jahr 1996 wird dem eigentlichen Theorietag noch ein eintägiger Workshop zu speziellen Themen der theoretischen Informatik vorange...