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July 1986 - June 1988
July 1988 - present
Education
August 1979 - June 1985
Publications
Publications (240)
The Workshop ‘Mathematical Logic: Proof Theory, Constructive Mathematics’ focused on proof-theoretic research on the foundations of mathematics, on the extraction of explicit computational content from given proofs in core areas of ordinary mathematics using proof-theoretic methods as well as on topics in proof complexity. The workshop contributed...
We discuss substitution rules that allow the substitution of formulas for formula variables. A substitution rule was first introduced by Frege. More recently, substitution is studied in the setting of propositional logic. We state theorems of Urquhart’s giving lower bounds on the number of steps in the substitution Frege system for propositional lo...
This article is motivated by seeking lower bounds on OBDD(∧, w, r) refutations, namely, OBDD refutations that allow weakening and arbitrary reorderings. We first work with 1 - NBP ∧ refutations based on read-once nondeterministic branching programs. These generalize OBDD(∧, r) refutations. There are polynomial size 1 - NBP(∧) refutations of the pig...
The Workshop ‘Mathematical Logic:Proof Theory, Constructive Mathematics’ focused on proofs both as formal derivations in deductive systems as well as on the extraction of explicit computational content from given proofs in core areas of ordinary mathematics using proof-theoretic methods. The workshop contributed to the following research strands:
The paper describes the use of dual-rail MaxSAT systems to solve Boolean satisfiability (SAT), namely to determine if a set of clauses is satisfiable. The MaxSAT problem is the problem of satisfying the maximum number of clauses in an instance of SAT. The dual-rail encoding adds extra variables for the complements of variables, and allows encoding...
We give a combinatorial analysis (using edge expansion) of a variant of the iterative expander construction due to Reingold, Vadhan, and Wigderson [44], and show that this analysis can be formalized in the bounded arithmetic system VNC1 (corresponding to the “NC1 reasoning”). As a corollary, we prove the assumption made by Jeřábek [28] that a const...
This paper studies propositional proof systems in which lines are sequents of decision trees or branching programs - deterministic and nondeterministic. The systems LDT and LNDT are propositional proof systems in which lines represent deterministic or non-deterministic decision trees. Branching programs are modeled as decision dags. Adding extensio...
We study the proof complexity of RAT proofs and related systems including BC, SPR and PR which use blocked clauses and (subset) propagation redundancy. These systems arise in satisfiability (SAT) solving, and allow inferences which preserve satisfiability but not logical implication. We introduce a new inference SR using substitution redundancy. We...
The 2-D Tucker search problem is shown to be PPA-hard under many-one reductions; therefore it is complete for PPA. The same holds for k-D Tucker for all k≥2. This corrects a claim in the literature that the Tucker search problem is in PPAD.
The proof system of Dual-Rail MaxSAT (DRMaxSAT) was recently shown to be capable of efficiently refuting families of formulas that are well-known to be hard for resolution, concretely when the MaxSAT solving approach is either MaxSAT resolution or core-guided algorithms. Moreover, DRMaxSAT based on MaxSAT resolution was shown to be stronger than ge...
We study the proof complexity of RAT proofs and related systems including BC, SPR, and PR which use blocked clauses and (subset) propagation redundancy. These systems arise in satisfiability (SAT) solving, and allow inferences which preserve satisfiability but not logical implication. We introduce a new inference SR using substitution redundancy. W...
This paper studies the complexity of constant depth propositional proofs in the cedent and sequent calculus. We discuss the relationships between the size of tree-like proofs, the size of dag-like proofs, and the heights of proofs. The main result is to correct a proof construction in an earlier paper about transformations from proofs with polyloga...
The workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of core mathematics and theoretical computer science as well as homotopy type theory and logical aspects of computational complexity.
We consider tautologies expressing equivalence‐chain properties in the spirit of Thapen and Krajíček, which are candidates for exponentially separating depth k and depth Frege proof systems. We formulate a special case where the initial member of the equivalence chain is fully specified and the equivalence‐chain implications are actually equivalenc...
The Cobham Recursive Set Functions (CRSF) provide an analogue of polynomial time computation which applies to arbitrary sets. We give three new equivalent characterizations of CRSF. The first is algebraic, using subset-bounded recursion and a form of Mostowski collapse. The second is our main result: the CRSF functions are shown to be precisely the...
Monte Carlo simulation software plays a critical role in PET system design. Performing complex, repeated Monte Carlo simulations can be computationally prohibitive, as even a single simulation can require a large amount of time and a computing cluster to complete. Here we introduce Gray, a Monte Carlo simulation software for PET systems. Gray explo...
Conflict-driven clause learning (CDCL) is at the core of the success of modern SAT solvers. In terms of propositional proof complexity, CDCL has been shown as strong as general resolution. Improvements to SAT solvers can be realized either by improving existing algorithms, or by exploiting proof systems stronger than CDCL. Recent work proposed an a...
We prove that propositional translations of the Kneser-Lovász theorem have polynomial size extended Frege proofs and quasi-polynomial size Frege proofs for all fixed values of k. We present a new counting-based combinatorial proof of the Kneser-Lovász theorem based on the Hilton-Milner theorem; this avoids the topological arguments of prior proofs...
We introduce new stable, natural merge sort algorithms, called 2-merge sort and $\alpha$-merge sort. We prove upper and lower bounds for several merge sort algorithms, including Timsort, Shiver's sort, $\alpha$-stack sorts, and our new 2-merge and $\alpha$-merge sorts. The upper and lower bounds have the forms $c \cdot n \log m$ and $c \cdot n \log...
We give a uniform proof of the theorems of Yao and Beigel–Tarui representing ACC predicates as constant depth circuits with (Formula presented.) gates and a symmetric gate. The proof is based on a relativized, generalized form of Toda’s theorem expressed in terms of closure properties of formulas under bounded universal, existential and modular cou...
We study consistency search problems for Frege and extended Frege proofs—namely the NP search problems of finding syntactic errors in Frege and extended Frege proofs of contradictions. The input is a polynomial time function, or an oracle, describing a proof of a contradiction; the output is the location of a syntactic error in the proof. The consi...
An injection structure A = (A, f) is a set A together with a one-place one-to-one function f. A is an FST injection structure if A is a regular set, that is, the set of words accepted by some finite automaton, and f is realized by a finite-state transducer.We initiate the study of FST injection structures. We show that the model checking problem fo...
We extend results of Bonet, Buss and Pitassi on Bondy’s Theorem and of Nozaki, Arai and Arai on Bollobás’ Theorem by proving that Frankl’s Theorem on the trace of sets has quasipolynomial size Frege proofs. For constant values of the parameter
t
, we prove that Frankl’s Theorem has polynomial size AC
0
-Frege proofs from instances of the pigeonhole...
This paper introduces the Cobham Recursive Set Functions (CRSF) as a version of polynomial time computable functions on general sets, based on a limited (bounded) form of ∈-recursion. This is inspired by Cobham's classic definition of polynomial time functions based on limited recursion on notation. We introduce a new set composition function, and...
Jěŕabek introduced fragments of bounded arithmetic which are axiomatized with weak surjective pigeonhole principles and support a robust notion of approximate counting. We extend these fragments to accommodate modular counting quantifiers. These theories can formalize and prove the relativized versions of Toda’s theorem on the collapse of the polyn...
We introduce the safe recursive set functions based on a Bellantoni–Cook style subclass of the primitive recursive set functions. We show that the functions computed by safe recursive set functions under a list encoding of finite strings by hereditarily finite sets are exactly the polynomial growth rate functions computed by alternating exponential...
We discuss recent results on the propositional proof complexity of Frege proof systems, including some recently discovered quasipolynomial size proofs for the pigeonhole principle and the Kneser-Lovász theorem. These are closely related to formalizability in bounded arithmetic.
We prove that the propositional translations of the Kneser-Lov\'asz theorem
have polynomial size extended Frege proofs and quasi-polynomial size Frege
proofs. We present a new counting-based combinatorial proof of the
Kneser-Lov\'asz theorem that avoids the topological arguments of prior proofs
for all but finitely many cases for each k. We introdu...
This paper defines a new notion of bounded computable randomness for certain
classes of sub-computable functions which lack a universal machine. In
particular, we define such versions of randomness for primitive recursive
functions and for PSPACE functions. These new notions are robust in that there
are equivalent formulations in terms of (1) Marti...
We study the long-standing open problem of giving ∀Σ b 1 separa-tions for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeřábek's theories for approximate counting and their subtheories. We show that the ∀Σ b 1 Herbrandized ordering pri...
The Stone tautologies are known to have polynomial size resolution
refutations and require exponential size regular refutations. We prove that the
Stone tautologies also have polynomial size proofs in both pool resolution and
the proof system of regular tree-like resolution with input lemmas (regRTI).
Therefore, the Stone tautologies do not separat...
ImmermanNeil. Upper and lower bounds for first order expressibility. Journal of computer and system sciences, vol. 25 (1982), pp. 76–98. ImmermanNeil. Relational queries computable in polynomial time. Information and control, vol. 68 (1986), pp. 86–104. ImmermanNeil. Languages that capture complexity classes. SIAM journal on computing, vol. 16 (198...
This article concerns the second-order systems U12 and V12 of bounded arithmetic, which have proof-theoretic strengths corresponding to polynomial-space and exponential-time computation. We formulate improved witnessing theorems for these two theories by using S12 as a base theory for proving the correctness of the polynomial-space or exponential-t...
The workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexity
Alternation trading proofs are motivated by the goal of separating NP from complexity classes such as Logspace or NL; they have been used to give super-linear runtime bounds for deterministic and co-nondeterministic sublinear space algorithms which solve the Satisfiability problem. For algorithms which use n
o(1) space, alternation trading proofs c...
We introduce martingales defined by probabilistic strategies, in which
randomness is used to decide whether to bet. We show that
different criteria for the success of
computable probabilistic strategies can be used to characterize
ML-randomness, computable randomness, and partial computable randomness.
Our characterization of ML-randomness partiall...
A shuffle of two strings is formed by interleaving the characters into a new
string, keeping the characters of each string in order. A string is a square if
it is a shuffle of two identical strings. There is a known polynomial time
dynamic programming algorithm to determine if a given string z is the shuffle
of two given strings x,y; however, it ha...
We prove that the graph tautology formulas of Alekhnovich, Johannsen,
Pitassi, and Urquhart have polynomial size pool resolution refutations that use
only input lemmas as learned clauses and without degenerate resolution
inferences. We also prove that these graph tautology formulas can be refuted by
polynomial size DPLL proofs with clause learning,...
Justification Logic studies epistemic and provability phenomena by introducing justi-fications/proofs into the language in the form of justification terms. Pure justification logics serve as counterparts of traditional modal epistemic logics, and hybrid logics combine epistemic modalities with justification terms. The computational complex-ity of p...
This textbook, first published in 2003, emphasises the fundamentals and the mathematics underlying computer graphics. The minimal prerequisites, a basic knowledge of calculus and vectors plus some programming experience in C or C++, make the book suitable for self study or for use as an advanced undergraduate or introductory graduate text. The auth...
We present sharpened lower bounds on the size of
cut free proofs for first-order logic. Prior lower bounds for
eliminating cuts from a proof established
superexponential lower bounds as a stack of exponentials, with the height
of the stack proportional to the maximum depth d
of the formulas in the original proof.
Our results remove the constant of...
We introduce a translation of the simply typed λ-calculus into C++, and give a mathematical proof of the correctness of this translation. For this purpose we develop a suitable fragment of C++ together with a denotational semantics. We ...
We present methods for removing top-level cuts from a sequent calculus or Tait-style proof without significantly increasing the space used for storing the proof. For propositional logic, this requires con-verting a proof from tree-like to dag-like form, but it most doubles the number of lines in the proof. For first-order logic, the proof size can...
We establish the unexpected power of conflict driven clause learning (CDCL) proof search by proving that the sets of unsatisfiable clauses obtained from the guarded graph tautology principles of Alekhnovich, Johannsen, Pitassi and Urquhart have polynomial size pool resolution refutations that use only input lemmas as learned clauses. We further sho...
The paper corrects and strengthens earlier upper bounds on the size of free-cut elimination. Free-cut elimination allows cut elimination to be carried out in the presence of non-logical axioms. The correction requires that the notion of a free-cut be modifled so that a cut formula is anchored provided that all of its introductions are anchored, ins...
Total NP search problems (TFNP problems) typically have their totality guaranteed by some combinatorial property. This paper proves that if there is a polynomial time Turing reduction between two TFNP problems, then there are quasipolynomial size, polylogarithmic height, constant depth, free-cut free propositional (Frege) proofs of the combina-tori...
This article is an abridged and revised version of a 1996 McGill University technical report [15]. The technical report was
based on lectures delivered by the author at a workshop in Holetown, Barbados and on the authors prepared overhead transparencies.
The audience at this workshop wrote scribe notes which then formed the technical report [15]. T...
We give the first systematic study of strong isomorphism reductions,
a notion of reduction more appropriate than polynomial time reduction
when, for example, comparing the computational complexity of the
isomorphim problem for different classes of structures.
We show that the partial ordering of its degrees is
quite rich. We analyze its relationshi...
This paper characterizes alternation trading based proofs that the satisfiability problem is not in the time and space bounded class DTISP(n(c), n(epsilon)), for various values c < 2 and epsilon < 1. We characterize exactly what can be proved for epsilon is an element of o(1) with currently known methods, and prove the conjecture of Williams that t...
The workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexity.
We reflne the constructions of Ferrante-Rackofi and Solovay on iter- ated deflnitions in flrst-order logic and their expressibility in with polyno- mial size formulas. These constructions introduce additional quantiflers; however, we show that these extra quantiflers range over only flnite sets and can be eliminated. We prove optimal upper and lowe...
A pool resolution proof is a dag-like resolution proof which admits a depth-first traversal tree in which no variable is used as a resolution variable twice on any branch. The problem of determining whether a given dag-like resolution proof is a valid pool resolution proof is shown to be NP-complete. Propositional resolution has been the foundation...
The complexity class of Π p k -Polynomial Local Search (PLS) prob-lems with Π p -goal is introduced, and is used to give new characterisa-tions of definable search problems in fragments of Bounded Arithmetic. The characterisations are established via notations for propositional proofs obtained by translating Bounded Arithmetic proofs using the Pari...
This is a survey of work on proof complexity and proof search, as motivated by the P versus NP problem. We discuss propositional proof complexity, Cook's program, proof automatizability, proof search, algorithms for satisfiability, and the state of the art of our (in)ability to separate P and NP .
The complexity class of \Pi^p_k-polynomial local search (PLS) problems is introduced and is used to give new witnessing theorems for fragments of bounded arithmetic. For 1 • ik + 1, the \Sigma^p_i -deflnable functions of T^{k+1}_2 are characterized in terms of ƒ p k-PLS problems. These ƒpk-PLS problems can be deflned in a weak base theory such as S...
We introduce new features for the broad phase algorithm sweep and prune that increase scalability for large virtual reality environments and allow for efficient AABB insertion and removal to support dy- namic object creation and destruction. We introduce a novel seg- mented interval list structure that allows AABB insertion and re- moval without re...
From the Publisher:
This book is aimed at the advanced undergraduate level or introductory graduate level and can also be used for self-study. Prerequisites include basic knowledge of calculus and vectors. The OpenGL programming portions require knowledge of programming in C or C++. The more important features of OpenGL are covered in the book, but...
Justification Logic studies epistemic and provability phenomena by introducing justifications/proofs into the language in the form of justification terms. Pure justification logics serve as counterparts of traditional modal epistemic logics, and hybrid logics combine epistemic modalities with justification terms. The computational complexity of pur...
Resolution refinements called w-resolution trees with lemmas (WRTL) and with input lemmas (WRTI) are introduced. Dag-like resolution is equivalent to both WRTL and WRTI when there is no regularity con- dition. For regular proofs, an exponential separation between regular dag-like resolution and both regular WRTL and regular WRTI is given. It is pro...
Let R be a resolution refutation, given as a sequence of clauses without explicit description of the underlying dag. Then, it is NP-complete to decide whether R is a regular resolution refutation.
This paper considers theories of bounded arithmetic that are predicative in the sense of Nelson, that is, theories that are
interpretable in Robinson’s Q.We give a nearly exact characterization of functions that can be total in predicative bounded
theories. As an upper bound, any such function has a polynomial growth rate and its bit-graph is in no...
The workshop Mathematical Logic: Proof Theory, Constructive Mathematics was held April 6-12, 2008 and had the following main aims:
To promote the interaction of proof theory with core areas of mathematics. This, in particular, refers to uses of proof theoretic techniques (most notably various forms of functional and realizability interpretations) t...
GRAY (High Energy Photon Ray Tracer) is a Monte-Carlo ray-driven high energy photon transport engine for mainly PET and SPECT applications that supports complex mesh based primitives for source distributions, phantom shapes, and detector geometries. Monte-Carlo modeling is critical for system design evaluation and image reconstruction development....
We prove that the problem of determining the minimum propositional proof length is NP-hard to approximate within any constant
factor. These results hold for all Frege systems, for all extended Frege systems, for resolution and Horn resolution, and
for the sequent calculus and the cut-free sequent calculus. Also, if NP is not in
QP = DTIME(nlogO(1)...
A grid graph has rectangularly arranged vertices with edges permitted only between orthogonally adjacent vertices. The st-connectivity principle states that it is not possible to have a red path of edges and a green path of edges which connect diagonally opposite corners of the grid graph unless the paths cross somewhere.We prove that the propositi...
We shall describe two general methods for proving lower bounds on the lengths of proofs in propositional calculus and give examples of such lower bounds. One of the methods is based on interactive proofs where one player is claiming that he has a falsifying assignment for a tautology and the second player is trying to convict him of a lie. The seco...
The workshop Mathematical Logic: Proof Theory, Type Theory and Constructive Mathematics , was held March 20th–March 26th, 2005 and had several aims.
To promote interaction between traditional proof theory and a more structural mathematical proof theory. It is hoped to encourage the application-oriented to consider their tools more abstractedly and...
We introduce a framework for collision detection between a pair of rigid polyhedra. Given the initial and final positions and orientations of two objects, the algorithm determines whether they collide, and if so, when and where. All collisions, including collisions at different times, are computed at once along with “properness” values that measure...
We introduce two methods for the inverse kinematics of multibodies with multiple end effectors. The first method clamps the distance of the target positions. Experiments show this is effective in reducing oscillation when target positions are unreachable. The second method is an extension of damped least squares called selectively damped least squa...
We develop a new, self-contained proof that the expected number of generations required for gene allele fixation or extinction in a population of size n is O ( n ) under general assumptions. The proof relies on a discrete Markov chain analysis. We further develop an algorithm to compute expected fixation or extinction time to any desired precision....
Contents Lecture #1, Robert Ellis 4 1 Introduction to Propositional Logic 4 2 Propositional Proof Systems 6 Lecture #2, Sashka Davis 10 3 Introduction to Frege Proof Systems 10 4 The Completeness and Implicational Completeness Theorems 12 5 Observations 13 6 P-simulate 13 Lecture #3, Reid Andersen 15 7 p-Simulation 15 8 Extended Frege Sytems 16 Lec...
We discuss the Paris-Wilkie translation from bounded arithmetic proofs to bounded depth propositional proofs in both relativized and non-relativized forms. We describe normal forms for proofs in bounded arithmetic, and a definition of Σ ′-depth for PK-proofs that makes the translation from bounded arithmetic to propositional logic particularly tran...
This paper proves exponential separations between depth d-LK and depth -LK for every utilizing the order induction principle. As a consequence, we obtain an exponential separation between depth d-LK and depth (d+1)-LK for d∈N. We investigate the relationship between the sequence-size, tree-size and height of depth d-LK-derivations for , and describ...
This is a introduction to the Jacobian transpose method, the pseudoinverse method, and the damped least squares methods for inverse kinematics (IK). The mathematical foundations of these methods are presented, with an analysis based on the singular value decomposition.
We prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to DNFs with small terms. We use this to prove lower bounds for the Res(k) propositional proof system, an extension of resolution which works with k- DNFs instead of clauses. We also obtain an exponential separation between d...
We develop a new, self-contained proof that the expected number of genera-tions required for gene allele fixation or extinction in a population of size n is O(n) under general assumptions. The proof relies on a discrete Markov chain analysis. We further develop an algorithm to compute expected fixa-tion/extinction time to any desired precision. Our...
Introduction to circuit complexity. Theorems of Shannon and Lupanov giving upper and lower bounds of circuit complexity of almost all Boolean functions. January 14-21. Matt Clegg's notes on Spira's theorem relating depth and formula size Krapchenko's lower bound on formula size over AND/OR/NOT January 23-28. Frank Baeuerle's notes on Neciporuk's th...
We prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to DNFs with small conjunctions. We use this to prove lower bounds for the Res(k) propositional proof system, an extension of resolution which works with k-DNFs instead of clauses. We also obtain an exponential separation bet...
We introduce a framework for collision detection between a pair of rigid polyhedra. Given the initial and final positions and orientations of two objects, the algorithm determines whether the bodies collide, and if so, when and where. All the collisions, including collisions at di#erent times are computed at once. This kind of collision information...
We prove a quadratic upper bound for the depth of cut free proofs in propositional intuitionistic logic formalized with Gentzen's sequent calculus. We discuss bounds on the necessary number of reuses of left implication rules. We exhibit an example showing that this quadratic bound is optimal. As a corollary, this gives a new proof that proposi-tio...
This paper investigates provability and non-provability of well-foundedness of ordinal notations in weak theories of bounded arithmetic. We define a notion of well-foundedness on bounded domains. We show that T21 and S22 can prove the well-foundedness on bounded domains of the ordinal notations below E0 and Γ0. As a corollary, the class of polynomi...
We extend results of A. Haken to give an exponential lower bound on the size of resolution proofs for propositional formulas encoding a generalized pigeonhole principle. These propositional formulas express the fact that there is no one-one mapping from c n objects to n objects when c > 1 . As a corollary, resolution proof systems do not p-simulate...
This paper presents new results on axiomatizations for fragments of Bounded Arithmetic which improve upon the author's dissertation. It is shown that (# i+1 )-PIND and strong # i -replacement are consequences of S 2 . Also # i+1 -IND is a consequence of T 2 . The latter result is proved by showing that S i+1 -conservative over 2 . Furthermore, S i+...