Barry Trager - Academia.edu (original) (raw)

Papers by Barry Trager

Research paper thumbnail of Computing with polynomials given byblack boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators

Journal of Symbolic Computation, Mar 1, 1990

Algorithms are developed that adopt a novel implicit representation for multivariate polynomials ... more Algorithms are developed that adopt a novel implicit representation for multivariate polynomials and rational functions with rational coefficients, that of black boxes for their evaluation. We show that within this representation the polynomial greatest common divisor and factorization problems, as well as the problem of extracting the numerator and denominator of a rational function, can all be solved in random polynomial-time. Since we can convert black boxes efficiently to sparse format, problems with sparse solutions, e.g., sparse polynomial factorization and sparse multivariate rational function interpolation, are also in random polynomial time. Moreover, the black box representation is one of the most space efficient implicit representations that we know. Therefore, the output programs can be easily distributed over a network of processors for further manipulation, such as sparse interpolation.

Research paper thumbnail of Comments on Integration of Algebraic Functions

Research paper thumbnail of Algebraic factoring and rational function integration

Research paper thumbnail of Reduced circuit implementation of encoder and syndrome generator

OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information), May 27, 2014

Research paper thumbnail of CertRL: formalizing convergence proofs for value and policy iteration in Coq

Reinforcement learning algorithms solve sequential decision-making problems in probabilistic envi... more Reinforcement learning algorithms solve sequential decision-making problems in probabilistic environments by optimizing for long-term reward. The desire to use reinforcement learning in safety-critical settings inspires a recent line of work on formally constrained reinforcement learning; however, these methods place the implementation of the learning algorithm in their Trusted Computing Base. The crucial correctness property of these implementations is a guarantee that the learning algorithm converges to an optimal policy. This paper begins the work of closing this gap by developing a Coq formalization of two canonical reinforcement learning algorithms: value and policy iteration for finite state Markov decision processes. The central results are a formalization of the Bellman optimality principle and its proof, which uses a contraction property of Bellman optimality operator to establish that a sequence converges in the infinite horizon limit. The CertRL development exemplifies how the Giry monad and mechanized metric coinduction streamline optimality proofs for reinforcement learning algorithms. The CertRL library provides a general framework for proving properties about Markov decision processes and reinforcement learning algorithms, paving the way for further work on formalization of reinforcement learning algorithms.

Research paper thumbnail of Error correcting code with chip kill capability and power saving enhancement

OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information), Aug 30, 2011

Research paper thumbnail of Formalization of a Stochastic Approximation Theorem

arXiv (Cornell University), Feb 11, 2022

Stochastic approximation algorithms are iterative procedures which are used to approximate a targ... more Stochastic approximation algorithms are iterative procedures which are used to approximate a target value in an environment where the target is unknown and direct observations are corrupted by noise. These algorithms are useful, for instance, for root-finding and function minimization when the target function or model is not directly known. Originally introduced in a 1951 paper by Robbins and Monro, the field of Stochastic approximation has grown enormously and has come to influence application domains from adaptive signal processing to artificial intelligence. As an example, the Stochastic Gradient Descent algorithm which is ubiquitous in various subdomains of Machine Learning is based on stochastic approximation theory. In this paper, we give a formal proof (in the Coq proof assistant) of a general convergence theorem due to Aryeh Dvoretzky [21] (proven in 1956) which implies the convergence of important classical methods such as the Robbins-Monro and the Kiefer-Wolfowitz algorithms. In the process, we build a comprehensive Coq library of measure-theoretic probability theory and stochastic processes.

Research paper thumbnail of Integration of Algebraic Functions

Research paper thumbnail of Transformer Models for Type Inference in the Simply Typed Lambda Calculus: A Case Study in Deep Learning for Code

arXiv (Cornell University), Mar 15, 2023

Despite a growing body of work at the intersection of deep learning and formal languages, there h... more Despite a growing body of work at the intersection of deep learning and formal languages, there has been relatively little systematic exploration of transformer models for reasoning about typed lambda calculi. This is an interesting area of inquiry for two reasons. First, typed lambda calculi are the lingua franc of programming languages. A set of heuristics that relate various typed lambda calculi to effective neural architectures would provide a systematic method for mapping language features (e.g., polymorphism, subtyping, inheritance, etc.) to architecture choices. Second, transformer models are widely used in deep learning architectures applied to code, but the design and hyperparameter space for them is large and relatively unexplored in programming language applications. Therefore, we suggest a benchmark that allows us to explore exactly this through perhaps the simplest and most fundamental property of a programming language: the relationship between terms and types. Consequently, we begin this inquiry of transformer architectures for typed lambda calculi by exploring the effect of transformer warm-up and optimizer selection in the task of type inference: i.e., predicting the types of lambda calculus terms using only transformers. We find that the optimization landscape is difficult even in this simple setting. One particular experimental finding is that optimization by Adafactor converges much faster compared to the optimization by Adam and RAdam. We conjecture that such different performance of optimizers might be related to the difficulties of generalization over formally generated dataset. * Also affiliated with the University of Illinois at Urbana-Champaign. † Also affiliated with IHES.

Research paper thumbnail of General Probability in Coq

Research paper thumbnail of Good reduction of plane curves

ACM Communications in Computer Algebra, Aug 14, 2015

Research paper thumbnail of How to make AXIOM into a scratchpad

It is shown that transverse homoclinic intersections such as the ones described by the Melnikov t... more It is shown that transverse homoclinic intersections such as the ones described by the Melnikov theory can be computed by a local analysis of the complex-time singularities of the solutions. This provides a new algorithmic procedure to compute homoclinic intersections in n-dimensions once the homoclinic manifold is known. It also gives new insights on the singularity structure of integrable and nonintegrable systems. test for integrability. However, it is not a flexible tool for nonintegrable systems and does not give any insight on the real-time dynamics of these systems [3, 4]. The two approaches, Painlev6's and Melnikov's, are combined here to give a better picture of both the homoclinic intersections and the singularity structure for nonintegrable systems. It introduces a new way of computing the Melnikov vector in n-dimensions. The algorithmic procedure described in this paper is based on an extended paper by the authors in which all the results are proven in a more general setting [5].

Research paper thumbnail of A language for computational algebra

This paper reports ongoing research at th e IBM Research Center on the development of a languag e... more This paper reports ongoing research at th e IBM Research Center on the development of a languag e with extensible parameterized types and generic operator s for computational algebra. The language provides a n abstract data type mechanism for defining algorithm s which work in as general a setting as possible. The language is based on the notions of domains and categories. Domains represent algebraic structures. Categorie s designate collections of domains having common operations with stated mathematical properties. Domain s and categories are computed objects which may b e dynamically assigned to variables, passed as arguments , and returned by functions. Although the language ha s been carefully tailored for the application of algebrai c computation, it actually provides a very general abstrac t data type mechanism. Our notion of a category to group domains with common properties appears novel amon g programming languages (cf. image functor of RUSSELL) and leads to a very powerful notion of abstract algorithm s missing from other work on data types known to th e authors. 1. Introduction. This paper describes a language wit h parameterized types and generic operators particularl y suited to computational algebra. A flexible framewor k is given for building algebraic structures and definin g algorithms which work in as general a setting a s possible. This section will be an overview of our mai n concepts: domains " and " categories In section 2, w e give more precise definitions and some examples. Section 3 contains our conclusions .

Research paper thumbnail of Proceedings of the 1998 international symposium on Symbolic and algebraic computation

International Symposium on Symbolic and Algebraic Computation, Aug 1, 1998

Research paper thumbnail of Computation of topological invariants for real projective surfaces with isolated singularities

HAL (Le Centre pour la Communication Scientifique Directe), Jun 3, 2013

Given a real algebraic surface S in RP 3 , we propose a procedure to determine the topology of S ... more Given a real algebraic surface S in RP 3 , we propose a procedure to determine the topology of S and to compute non-trivial topological invariants for the pair (RP 3 , S) under the hypothesis that the real singularities of S are isolated. In particular, starting from an implicit equation of the surface, we compute the number of connected components of S, their Euler characteristics and the labelled 2-adjacency graph of the surface.

Research paper thumbnail of A new class of array codes for memory storage

In this article we describe a class of error control codes called "diff-MDS" codes that are custo... more In this article we describe a class of error control codes called "diff-MDS" codes that are custom designed for highly resilient computer memory storage. The error scenarios of concern range from simple single bit errors, to memory chip failures and catastrophic memory module failures. Our approach to building codes for this setting relies on the concept of expurgating a parity code that is easy to decode for memory module failures so that a few additional small errors can be handled as well, thus preserving most of the decoding complexity advantages of the original code while extending its original intent. The manner in which we expurgate is carefully crafted so that the strength of the resulting code is comparable to that of a Reed-Solomon code when used for this particular setting. An instance of this class of algorithms has been incorporated in IBM's zEnterprise mainframe offering, setting a new industry standard for memory resiliency.

Research paper thumbnail of Integration of simple radical extensions

Lecture Notes in Computer Science, 1979

I. Intzodu cUon Risch's landmarh paper [Ris89] presen ted the first decision proced~re for th... more I. Intzodu cUon Risch's landmarh paper [Ris89] presen ted the first decision proced~re for the integration of elementary functions. In that paper he required that the functions appearing in the integrand be algebraically independent. Shortly afterwards in [Risalg] and [RisTO] he relaxed that restriction and outlined a complete decision procedure for the integration of elemeniary functions in finite terms. Unfortunately his algorithms for dealing ~th algebraic functions required considerably more complex machinery than his earlier ones for purely transcendental functions. ~qeses' implementation of the earlier approach in ~%IACS~f]v!~ [MAC??] demonstrated its practicality, whereas the same has yet to be done for Risch's more recent approach.

Research paper thumbnail of Computing with polynomials given by black boxes for their evaluations: greatest common divisors, factorization, separation of numerators and denominators

Research paper thumbnail of Towards a Unification of Logic and Information Theory

arXiv (Cornell University), Jan 25, 2023

We examine the problem of efficient transmission of logical statements from a sender to a receive... more We examine the problem of efficient transmission of logical statements from a sender to a receiver under a diverse set of initial conditions for the sender and receiver's beliefs and on the goal for the communication. From the standpoint of our work, two different collections of logical statements are equivalent if there anything that can be proved from one collection can also be deduced from the other collection. Distinguishing between these two collections is thus unnecessary from the standpoint of our work and leads to communication cost efficiencies. In order to develop an example of an information theory for the transmission of logical statements, we focus on a simple logical system equivalent to propositional logic where a collection of logical statements can be alternately depicted as a collection of multivariate polynomial equations with coefficients and variables in a finite field. We then apply classical concepts from information theory, notably concepts for rate-distortion theory, to develop closed form expressions for the cost of communicating these logical statements. We additionally provide a theory of linear codes for implementing these communication systems that produces systems that are asymptotically bit-cost optimal in some settings. It is our belief that the scope for improving beyond our limited exploration is vast, including treating more sophisticated logical systems such as first order logic, studying different types of communication constraints and creating practical algorithms for attaining the Shannon limits.

Research paper thumbnail of Verification of Galois field based circuits by formal reasoning based on computational algebraic geometry

Formal Methods in System Design, 2014

Algebraic error correcting codes (ECC) are widely used to implement reliability features in moder... more Algebraic error correcting codes (ECC) are widely used to implement reliability features in modern servers and systems and pose a formidable verification challenge. We present a novel methodology and techniques for provably correct design of ECC logics. The methodology is comprised of a design specification method that directly exposes the ECC algorithm's underlying math to a verification layer, encapsulated in a tool "BLUEVERI", which establishes the correctness of the design conclusively by using an apparatus of computational algebraic geometry (Buchberger's algorithm for Gröbner basis construction). We present results from its application to example circuits to demonstrate the effectiveness of the approach. The methodology has been successfully applied to prove correctness of large error correcting circuits on IBM's POWER systems to protect memory storage and processor to memory communication, as well as a host of smaller error correcting circuits.

Research paper thumbnail of Computing with polynomials given byblack boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators

Journal of Symbolic Computation, Mar 1, 1990

Algorithms are developed that adopt a novel implicit representation for multivariate polynomials ... more Algorithms are developed that adopt a novel implicit representation for multivariate polynomials and rational functions with rational coefficients, that of black boxes for their evaluation. We show that within this representation the polynomial greatest common divisor and factorization problems, as well as the problem of extracting the numerator and denominator of a rational function, can all be solved in random polynomial-time. Since we can convert black boxes efficiently to sparse format, problems with sparse solutions, e.g., sparse polynomial factorization and sparse multivariate rational function interpolation, are also in random polynomial time. Moreover, the black box representation is one of the most space efficient implicit representations that we know. Therefore, the output programs can be easily distributed over a network of processors for further manipulation, such as sparse interpolation.

Research paper thumbnail of Comments on Integration of Algebraic Functions

Research paper thumbnail of Algebraic factoring and rational function integration

Research paper thumbnail of Reduced circuit implementation of encoder and syndrome generator

OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information), May 27, 2014

Research paper thumbnail of CertRL: formalizing convergence proofs for value and policy iteration in Coq

Reinforcement learning algorithms solve sequential decision-making problems in probabilistic envi... more Reinforcement learning algorithms solve sequential decision-making problems in probabilistic environments by optimizing for long-term reward. The desire to use reinforcement learning in safety-critical settings inspires a recent line of work on formally constrained reinforcement learning; however, these methods place the implementation of the learning algorithm in their Trusted Computing Base. The crucial correctness property of these implementations is a guarantee that the learning algorithm converges to an optimal policy. This paper begins the work of closing this gap by developing a Coq formalization of two canonical reinforcement learning algorithms: value and policy iteration for finite state Markov decision processes. The central results are a formalization of the Bellman optimality principle and its proof, which uses a contraction property of Bellman optimality operator to establish that a sequence converges in the infinite horizon limit. The CertRL development exemplifies how the Giry monad and mechanized metric coinduction streamline optimality proofs for reinforcement learning algorithms. The CertRL library provides a general framework for proving properties about Markov decision processes and reinforcement learning algorithms, paving the way for further work on formalization of reinforcement learning algorithms.

Research paper thumbnail of Error correcting code with chip kill capability and power saving enhancement

OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information), Aug 30, 2011

Research paper thumbnail of Formalization of a Stochastic Approximation Theorem

arXiv (Cornell University), Feb 11, 2022

Stochastic approximation algorithms are iterative procedures which are used to approximate a targ... more Stochastic approximation algorithms are iterative procedures which are used to approximate a target value in an environment where the target is unknown and direct observations are corrupted by noise. These algorithms are useful, for instance, for root-finding and function minimization when the target function or model is not directly known. Originally introduced in a 1951 paper by Robbins and Monro, the field of Stochastic approximation has grown enormously and has come to influence application domains from adaptive signal processing to artificial intelligence. As an example, the Stochastic Gradient Descent algorithm which is ubiquitous in various subdomains of Machine Learning is based on stochastic approximation theory. In this paper, we give a formal proof (in the Coq proof assistant) of a general convergence theorem due to Aryeh Dvoretzky [21] (proven in 1956) which implies the convergence of important classical methods such as the Robbins-Monro and the Kiefer-Wolfowitz algorithms. In the process, we build a comprehensive Coq library of measure-theoretic probability theory and stochastic processes.

Research paper thumbnail of Integration of Algebraic Functions

Research paper thumbnail of Transformer Models for Type Inference in the Simply Typed Lambda Calculus: A Case Study in Deep Learning for Code

arXiv (Cornell University), Mar 15, 2023

Despite a growing body of work at the intersection of deep learning and formal languages, there h... more Despite a growing body of work at the intersection of deep learning and formal languages, there has been relatively little systematic exploration of transformer models for reasoning about typed lambda calculi. This is an interesting area of inquiry for two reasons. First, typed lambda calculi are the lingua franc of programming languages. A set of heuristics that relate various typed lambda calculi to effective neural architectures would provide a systematic method for mapping language features (e.g., polymorphism, subtyping, inheritance, etc.) to architecture choices. Second, transformer models are widely used in deep learning architectures applied to code, but the design and hyperparameter space for them is large and relatively unexplored in programming language applications. Therefore, we suggest a benchmark that allows us to explore exactly this through perhaps the simplest and most fundamental property of a programming language: the relationship between terms and types. Consequently, we begin this inquiry of transformer architectures for typed lambda calculi by exploring the effect of transformer warm-up and optimizer selection in the task of type inference: i.e., predicting the types of lambda calculus terms using only transformers. We find that the optimization landscape is difficult even in this simple setting. One particular experimental finding is that optimization by Adafactor converges much faster compared to the optimization by Adam and RAdam. We conjecture that such different performance of optimizers might be related to the difficulties of generalization over formally generated dataset. * Also affiliated with the University of Illinois at Urbana-Champaign. † Also affiliated with IHES.

Research paper thumbnail of General Probability in Coq

Research paper thumbnail of Good reduction of plane curves

ACM Communications in Computer Algebra, Aug 14, 2015

Research paper thumbnail of How to make AXIOM into a scratchpad

It is shown that transverse homoclinic intersections such as the ones described by the Melnikov t... more It is shown that transverse homoclinic intersections such as the ones described by the Melnikov theory can be computed by a local analysis of the complex-time singularities of the solutions. This provides a new algorithmic procedure to compute homoclinic intersections in n-dimensions once the homoclinic manifold is known. It also gives new insights on the singularity structure of integrable and nonintegrable systems. test for integrability. However, it is not a flexible tool for nonintegrable systems and does not give any insight on the real-time dynamics of these systems [3, 4]. The two approaches, Painlev6's and Melnikov's, are combined here to give a better picture of both the homoclinic intersections and the singularity structure for nonintegrable systems. It introduces a new way of computing the Melnikov vector in n-dimensions. The algorithmic procedure described in this paper is based on an extended paper by the authors in which all the results are proven in a more general setting [5].

Research paper thumbnail of A language for computational algebra

This paper reports ongoing research at th e IBM Research Center on the development of a languag e... more This paper reports ongoing research at th e IBM Research Center on the development of a languag e with extensible parameterized types and generic operator s for computational algebra. The language provides a n abstract data type mechanism for defining algorithm s which work in as general a setting as possible. The language is based on the notions of domains and categories. Domains represent algebraic structures. Categorie s designate collections of domains having common operations with stated mathematical properties. Domain s and categories are computed objects which may b e dynamically assigned to variables, passed as arguments , and returned by functions. Although the language ha s been carefully tailored for the application of algebrai c computation, it actually provides a very general abstrac t data type mechanism. Our notion of a category to group domains with common properties appears novel amon g programming languages (cf. image functor of RUSSELL) and leads to a very powerful notion of abstract algorithm s missing from other work on data types known to th e authors. 1. Introduction. This paper describes a language wit h parameterized types and generic operators particularl y suited to computational algebra. A flexible framewor k is given for building algebraic structures and definin g algorithms which work in as general a setting a s possible. This section will be an overview of our mai n concepts: domains " and " categories In section 2, w e give more precise definitions and some examples. Section 3 contains our conclusions .

Research paper thumbnail of Proceedings of the 1998 international symposium on Symbolic and algebraic computation

International Symposium on Symbolic and Algebraic Computation, Aug 1, 1998

Research paper thumbnail of Computation of topological invariants for real projective surfaces with isolated singularities

HAL (Le Centre pour la Communication Scientifique Directe), Jun 3, 2013

Given a real algebraic surface S in RP 3 , we propose a procedure to determine the topology of S ... more Given a real algebraic surface S in RP 3 , we propose a procedure to determine the topology of S and to compute non-trivial topological invariants for the pair (RP 3 , S) under the hypothesis that the real singularities of S are isolated. In particular, starting from an implicit equation of the surface, we compute the number of connected components of S, their Euler characteristics and the labelled 2-adjacency graph of the surface.

Research paper thumbnail of A new class of array codes for memory storage

In this article we describe a class of error control codes called "diff-MDS" codes that are custo... more In this article we describe a class of error control codes called "diff-MDS" codes that are custom designed for highly resilient computer memory storage. The error scenarios of concern range from simple single bit errors, to memory chip failures and catastrophic memory module failures. Our approach to building codes for this setting relies on the concept of expurgating a parity code that is easy to decode for memory module failures so that a few additional small errors can be handled as well, thus preserving most of the decoding complexity advantages of the original code while extending its original intent. The manner in which we expurgate is carefully crafted so that the strength of the resulting code is comparable to that of a Reed-Solomon code when used for this particular setting. An instance of this class of algorithms has been incorporated in IBM's zEnterprise mainframe offering, setting a new industry standard for memory resiliency.

Research paper thumbnail of Integration of simple radical extensions

Lecture Notes in Computer Science, 1979

I. Intzodu cUon Risch's landmarh paper [Ris89] presen ted the first decision proced~re for th... more I. Intzodu cUon Risch's landmarh paper [Ris89] presen ted the first decision proced~re for the integration of elementary functions. In that paper he required that the functions appearing in the integrand be algebraically independent. Shortly afterwards in [Risalg] and [RisTO] he relaxed that restriction and outlined a complete decision procedure for the integration of elemeniary functions in finite terms. Unfortunately his algorithms for dealing ~th algebraic functions required considerably more complex machinery than his earlier ones for purely transcendental functions. ~qeses' implementation of the earlier approach in ~%IACS~f]v!~ [MAC??] demonstrated its practicality, whereas the same has yet to be done for Risch's more recent approach.

Research paper thumbnail of Computing with polynomials given by black boxes for their evaluations: greatest common divisors, factorization, separation of numerators and denominators

Research paper thumbnail of Towards a Unification of Logic and Information Theory

arXiv (Cornell University), Jan 25, 2023

We examine the problem of efficient transmission of logical statements from a sender to a receive... more We examine the problem of efficient transmission of logical statements from a sender to a receiver under a diverse set of initial conditions for the sender and receiver's beliefs and on the goal for the communication. From the standpoint of our work, two different collections of logical statements are equivalent if there anything that can be proved from one collection can also be deduced from the other collection. Distinguishing between these two collections is thus unnecessary from the standpoint of our work and leads to communication cost efficiencies. In order to develop an example of an information theory for the transmission of logical statements, we focus on a simple logical system equivalent to propositional logic where a collection of logical statements can be alternately depicted as a collection of multivariate polynomial equations with coefficients and variables in a finite field. We then apply classical concepts from information theory, notably concepts for rate-distortion theory, to develop closed form expressions for the cost of communicating these logical statements. We additionally provide a theory of linear codes for implementing these communication systems that produces systems that are asymptotically bit-cost optimal in some settings. It is our belief that the scope for improving beyond our limited exploration is vast, including treating more sophisticated logical systems such as first order logic, studying different types of communication constraints and creating practical algorithms for attaining the Shannon limits.

Research paper thumbnail of Verification of Galois field based circuits by formal reasoning based on computational algebraic geometry

Formal Methods in System Design, 2014

Algebraic error correcting codes (ECC) are widely used to implement reliability features in moder... more Algebraic error correcting codes (ECC) are widely used to implement reliability features in modern servers and systems and pose a formidable verification challenge. We present a novel methodology and techniques for provably correct design of ECC logics. The methodology is comprised of a design specification method that directly exposes the ECC algorithm's underlying math to a verification layer, encapsulated in a tool "BLUEVERI", which establishes the correctness of the design conclusively by using an apparatus of computational algebraic geometry (Buchberger's algorithm for Gröbner basis construction). We present results from its application to example circuits to demonstrate the effectiveness of the approach. The methodology has been successfully applied to prove correctness of large error correcting circuits on IBM's POWER systems to protect memory storage and processor to memory communication, as well as a host of smaller error correcting circuits.