Papers by Pierre McKenzie
In the interactive communication model, two distant parties possess correlated inputs such as str... more In the interactive communication model, two distant parties possess correlated inputs such as strings of bits, and the goal is for one party to learn his interlocutor's input while minimizing the communication. Our main contribution is to analyze the power of randomization in this correlated data setting. We show that the deterministic, amortized deterministic, private coin randomized, and public coin randomized models are all equivalent for a large class of problems arising from several practical applications. Furthermore, we conjecture that the private coin randomized model and the deterministic model are equivalent for every problem, and show that a proof of this statement solves the direct-sum problem for interactive communication.
Systems and Software Verification: Model-Checking Techniques and Tools
Model checking is a powerful approach for the formal verification of software. It automatically p... more Model checking is a powerful approach for the formal verification of software. It automatically provides complete proofs of correctness, or explains, via counter-examples, why a system is not correct. Here, the author provides a well written and basic introduction to the new ...
ACM Transactions on Computation Theory, 2012
We introduce the tree evaluation problem, show that it is in LogDCFL (and hence in P), and study ... more 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 whose leaves are labeled with elements of [k]. Each node obtains a value in [k] equal to its d-ary function applied to the values of its d children. The output is the value of the root. We show that the standard black pebbling algorithm applied to the binary tree of height h yields a deterministic k-way branching program with O(k h ) states solving this problem, and we prove that this upper bound is tight for h = 2 and h = 3. We introduce a simple semantic restriction called thrifty on k-way branching programs solving tree evaluation problems and show that the same state bound of Θ(k h ) is tight for all h ≥ 2 for deterministic thrifty programs. We introduce fractional pebbling for trees and show that this yields nondeterministic thrifty programs with Θ(k h/2+1 ) states solving the Boolean problem "determine whether the root has value 1", and prove that this bound is tight for h = 2, 3, 4. We also prove that this same bound is tight for unrestricted nondeterministic k-way branching programs solving the Boolean problem for h = 2, 3. * Versions of parts of this paper appeared in [BCMSW09] and [BCMSW.A09]
Equation Satisfiability and Program Satisfiability for Finite Monoids
Lecture Notes in Computer Science, 2000
We study the computational complexity of solving equations and of determining the satisfiability ... more 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 special case when M is aperiodic, we show
Journal of Computer and System Sciences, 2006
Journal of Computer and System Sciences, 2001
The Cayley group membership problem (CGM) is to input a groupoid (binary algebra) G given as a mu... more 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
Journal of Computer and System Sciences, 2000
This paper describes the simulation of an S(n) space-bounded deterministic Turing machine by a re... more 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 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 pebble game rules.
Journal of Computer and System Sciences, 2001
Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we l... more 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 NC 1 and its subclasses. In the absence of the BIT predicate, we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's "hardest context-free language" is LOGCFL-complete under quantifier-free BIT-free projections. We then prove that FO with unary groupoidal quantifiers is strictly more expressive with the BIT predicate than without. Considering a particular groupoidal quantifier, we prove that first-order logic with majority of pairs is strictly more expressive than first-order with majority of individuals. As a technical tool of independent interest, we define the notion of an aperiodic nondeterministic finite automaton and prove that FO translations are precisely the mappings computed by single-valued aperiodic nondeterministic finite transducers.
Information and Computation, 1991
computational complexity, 2007
The problem of testing membership in the subset of the natural numbers produced at the output gat... more 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, the case {∪, +} is shown NP-complete, the case {∩, +} is shown C = L-complete, and several other cases are resolved.
Lecture Notes in Computer Science, 2009
A d-gem is a {+, −, ×}-circuit having very few ×-gates and computing from {x} ∪ Z a univariate po... more 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 natural step towards validating the conjecture would thus be to rule out d-gems for large d. Here we construct d-gems for several values of d up to 55. Our 2 n -gems for n ≤ 4 are skew, that is, each {+, −}-gate adds an integer. We prove that skew 2 n -gems if they exist require n {+, −}-gates, and that these for n ≥ 5 would imply new solutions to the Prouhet-Tarry-Escott problem in number theory. By contrast, skew d-gems over the real numbers are shown to exist for every d.
Uploads
Papers by Pierre McKenzie