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[EQUATION] Thus, we have what one can call a differential [MATH] -Lie algebra . The general ethos of deformation theory is that a differential Lie algebra always controls the deformation of some structure. The case at hand is no different. |
Theorem 7.1 Deformations of an almost commutative Q-algebra of degree [MATH] are controlled by the differential [MATH] -Lie algebra [MATH] . Specifically, [MATH] [MATH] , is a homological [MATH] -derivation of [MATH] -degree [MATH] and weight [MATH] if and only if |
(i) [MATH] [MATH] , and (ii) [MATH] Proof. The condition on the [MATH] -degree and weight is obvious. The second part requires only a short calculation: |
[EQUATION] Thus, if we insist on [MATH] , then we require that [MATH] In other words, we require that [MATH] satisfy the required Maurer–Cartan equation. |
Remark 7.2 For the case of classical Lie algebroids, the reader can consult Ji . Note that Crainic & Moerdijk also define a differential graded Lie algebra controlling the deformation theory of Lie algebroids and that this is isomorphic to the differential graded Lie algebra studied by Ji. However, the use of Q-manifol... |
and Grabowska, Grabowski & P. Urbański . The deformation theory of VB-algebroids has been explored by La Pastina & Vitagliano Definition 7.3 |
Let [MATH] be an almost commutative Q-algebra of degree [MATH] . Then a [MATH] -derivation [MATH] is said to be an infinitesimal symmetry or just a symmetry , if and only if |
[EQUATION] Equivalently, [MATH] is a symmetry if and only if [MATH] . This follows trivially from [MATH] . Thus, we will refer to a symmetry [MATH] as an inner symmetry if and only if there exists a section [MATH] such that [MATH] . In reverse, every section leads to an inner symmetry in an obvious way. |
Proposition 7.4 The set of symmetries of [MATH] is a [MATH] -Lie algebra with respect to the [MATH] -commutator bracket. Proof. It is evident, due to the linearity of the [MATH] -commutator bracket, that constant multiples and sums of symmetries are again symmetries. Thus, we naturally have the structure of a [MATH] -g... |
Note that we have insisted that symmetries be of weight zero. The reason for this is to ensure that [MATH] is closed under the action of any symmetry. We have not made any assumption about the [MATH] -degree. |
Proposition 7.5 Let [MATH] be an almost commutative Q-algebra of degree [MATH] and let [MATH] be an infinitesimal symmetry. Then |
[EQUATION] for all [MATH] and [MATH] Proof. The proof follows from direct calculation using the [MATH] -Jacobi identity repeatedly. Explicitly, |
[EQUATION] Then from the condition that we have a symmetry, we obtain the desired result. Example 7.6 Examining the de Rham complex [MATH] of any almost commutative algebra and in particular the Cartan calculus ( 2.2 ), we see that the Lie derivative [MATH] for any [MATH] is an infinitesimal symmetry. |
Acknowledgements We cordially thank Prof. Tomasz Brzeziński and Prof. Richard Szabo for their interest and valuable advice. We furthermore, thank the anonymous referee for their invaluable comments and suggestions. |
# Source: arxiv 1806.02932 # Title: Program Synthesis Through Reinforcement Learning Guided Tree Search # Sections: all # Downloaded: 2026-03-03T02:30:40.855805+00:00 |
Program Synthesis Through Reinforcement Learning Guided Tree Search Abstract Program Synthesis is the task of generating a program from a provided specification. Traditionally, this has been treated as a search problem by the programming languages (PL) community and more recently as a supervised learning problem by the... |
Introduction Within both the programming languages and machine learning communities, there has been a renaissance in program synthesis the task of automatically generating computer code from a user-provided specification. Due to increased computational power and the rise of deep neural networks, this once intractable p... |
including improving programmer efficiency by automating routine tasks providing an intuitive interface for non-experts without programming knowledge |
, and reducing bugs and improving runtime efficiency for performance-critical code Within the programming languages(PL) community, program synthesis is typically solved using enumerative search – finding correct programs for a given specification by naïvely enumerating candidates until a satisfying program is found |
. This approach is made tractable by narrowing the search space through integrating deductive components into the search process |
, or by modifying the language of interest into an equivalent language with a narrower search space, and searching within that space |
. Another common approach is to reduce synthesis tasks to finding a satisfying assignment to a Boolean formula via a SAT solver , but this merely pushes the enumerative search into the solver. |
Researchers in the machine learning(ML) community have attacked this problem from a different direction – rather than searching naïvely through a restricted space, correct programs can be efficiently found within a larger search space by intelligently searching or sampling from that space using a learned model of how s... |
. Conceptually, this approach should compose well with PL techniques – intelligently searching through a small search space is much easier than intelligently searching through a large search space. Why then are these techniques not commonly used together? |
Most existing work in the ML community relies heavily on the structure of the domain specific language(DSL) being synthesized to achieve good performance, and can’t generalize to new languages easily. These methods often require language features such as full differentiability of the language |
or a large training set of specification and satisfying program pairs . These requirements make it difficult to combine existing ML-based methods with techniques developed by the PL community, which require very different properties in a DSL for effective synthesis. Further, most existing methods have heavily focused o... |
Reinforcement Learning Guided Tree Search Our approach, reinforcement learning guided tree search (RLGTS) , illustrated in figure , seeks to combine the benefits of both search-based and ML-based methods, and allow for the combination of techniques from both research communities to further enhance performance. We propo... |
and using reinforcement learning(RL) to learn to solve a program given only a set of input/output examples for that program, a language specification, and a reward function for the quality of a given program. In our RL-based approach, we interpret the program state and current partial program as an environment, and lin... |
RLGTS does not depend on the availability of training data for a given language, and makes no assumptions about the structure of the language other than that the language allows for partial programs to be executed and evaluated. Further, our RL-based approach can be combined with other program synthesis methods easily ... |
In summary, we make the following contributions: 1. We introduce reinforcement learning guided tree search, an approach to program synthesis that interprets program generation as a reinforcement learning task. |
2. We describe an implementation of RLGTS on a subset of the RISC-V assembly language, and created an RL agent for this task by combining a Q-network-based policy with a simple search tree method. |
3. We demonstrate improvements in the fraction of programs solved of up to 100% and 800% compared to RL-only and enumerative search-only baselines respectively on a synthetic dataset of random programs. Furthermore, we compare RLGTS to a Markov chain Monte Carlo(MCMC) based method that has been used to great success in... |
, and show superior performance on more challenging benchmark programs, solving up to 400% more programs within a fixed program evaluation limit and remaining competitive in total performance even when that limit is increased 50x for MCMC. |
Methods 2.1 Reinforcement Learning Model of Synthesis Here we describe our formulation of the general program synthesis task as a multi-step Markov decision process, solvable via reinforcement learning. |
In the standard terminology , a fully-observed MDP is a process having some state [MATH] for timesteps [MATH] . At each state [MATH] an action [MATH] from among [MATH] possible actions is emitted, with some unknown function [MATH] determining the following state |
[MATH] from among some (typically large) state space. Actions [MATH] are selected by an agent [MATH] with a policy for selecting actions parametrized by learned parameters |
[MATH] , commonly a neural network. This policy is trained to maximize the expected cumulative reward value [MATH] emitted by some reward function [MATH] , with [MATH] , with time decay factor [MATH] |
Based on these definitions, we represent program synthesis as follows. A program of length [MATH] is a set of actions [MATH] . Each action [MATH] represents a single line of code(e.g. "ADD f0 f1 f2") applied to state [MATH] , which represents the memory state(the values of all variables) after execution of all previous... |
In our instantiation on RISC-V, we allow for multiple input/output examples, so the program state is a tuple consisting of the state of multiple executions. Thusly, our initial state is a tuple comprised of all the input states of the examples, and our desired output state is a tuple consisting of all the output states... |
It is worth noting that while this formulation does not by default include non-linear programs containing control flow, it can theoretically be extended to support them, as well as programs containing other elements of modern programming languages as the MDP representation of a process is Turing complete |
though we leave practical exploration of these topics to further research. 2.2 Reward Function For our reward function [MATH] , given a set of input-output state pairs [MATH] with [MATH] pairs, we use two components. First, a metric for the correctness of the next state for the state [MATH] produced from each example [... |
[EQUATION] with [MATH] as the number of variables used in this program, and [MATH] as a hyperparameter weight on this component of the reward function. We express correctness as a fraction of [MATH] to normalize across output values of different magnitudes. To this we add a term penalizing program length to encourage t... |
[EQUATION] where [MATH] is the length of the program before taking action [MATH] . Because the [MATH] term can become large if [MATH] is far from [MATH] , we combine these terms and scale the resulting values such that large values of [MATH] become close to 0 as |
[EQUATION] with [MATH] a hyperparameter weight controlling the scaling of the reward values. We define the space of [MATH] pairs for a given set of |
[MATH] as a program space , the discrete reward landscape which our RL agent seeks to maximize. Our formulation of program synthesis as an MDP is quite general and can be applied for many different reward functions provided by the user, such as that described above. However, we make a key assumption about the reward fu... |
We also demonstrate empirically in our experiments that this gradient is present for many short floating point arithmetic programs in RISC-V. |
2.3 RL Model For reinforcement learning, we use a standard dueling double Q-learning algorithm, which learns to predict future rewards for a given state and action using the loss function |
[EQUATION] given agent Q-function [MATH] with parameters [MATH] and target Q-function parameters [MATH] , as per Hasselt et al . We set the decay term on future rewards [MATH] to 0.99. The objective of the Q function trained using equation is to predict the expected future reward [MATH] that will result from taking eac... |
[MATH] . During synthesis, we then use an [MATH] -greedy policy of either taking action [MATH] or else taking a random action with probability [MATH] |
The input to the agent consists of two parts, the state [MATH] encoding current and target values for each variable and a sequence of 1-hot vectors encoding previous lines of the program that produced |
[MATH] [MATH] . A schematic diagram of our network architecture is shown in Figure . It consists of two input modules, for [MATH] and [MATH] . The |
[MATH] module is a two-layer fully connected(FC) neural network, while the [MATH] module is a two-layer LSTM operating on input sequences of up to length |
[MATH] , the maximum program length. These modules are concatenated and fed into two more FC layers, followed by a two-layer value stream computing [MATH] the expected future reward value from being in the current state, and a separate two-layer advantage stream computing [MATH] the advantage of each action |
[MATH] above or below [MATH] . These modules are combined as per Wang et al. to compute a Q-value per action as [EQUATION] with each action |
[MATH] representing a single line of code expressible in the language. ReLU non-linearities were used in the FC layers, with LSTM non-linearities following the standard arrangement of sigmoid and tanh functions |
2.4 Tree Search Notably unlike most well-studied RL applications, where an agent capable of reliably reaching high reward states across many rollouts of the agent is desired, in program synthesis once our agent discovers a solution to a given program we have completed synthesis for that program and can stop training an... |
In line with this objective, we combined our Q-function with a simple prioritized search tree algorithm to aid in efficient exploration. The task of the Q-function then becomes to predict a Q-score for each unexplored edge of the tree, the edge representing a possible action [MATH] to append to the program stored at a ... |
[MATH] , the edge that the current Q-function thinks will yield the highest reward among unexplored edges, evaluating the program [MATH] and adding a new node to the tree with additional unexplored edges. We alternate between sampling unexplored edges(using [MATH] -greedy sampling) and training the Q-network, with 100 ... |
. By taking this approach, we encourage exploration by only evaluating each unique [MATH] combination once, and guarantee our worst-case performance to be that of enumerative search, memory and computational resources permitting. While we do not explore it further here, this approach also allows RL-based synthesis to b... |
Experiments 3.1 Experiment Setup To analyze the performance of our instantiation of RLGTS, we synthesize programs using a common subset of the RISC-V assembly programming language |
. We select a core subset of instructions on floating point values, shown in Figure , which excludes control flow, memory reads/writes, and “magic number” inputs for simplicity. This domain is interesting because there are few previously published methods that can perform better than naïve enumerative search other than... |
To benchmark our performance, we construct a dataset of synthetic programs by generating random programs with specified attributes. For each program in the evaluation set, we specify the number of lines, number of allowed instructions, number of allowed variables, and number of examples, and synthesize a random program... |
To train our network, we use a learning rate of 0.001, and sampled batches of 64 experiences from an experience buffer storing all previously observed states using proportional prioritized experience replay as per Schaul et al. |
, with [MATH] and [MATH] , linearly annealing [MATH] to 1.0 after 10,000 iterations. We updated the target Q network parameters by setting [MATH] every 100 training iterations. We set the reward function hyperparameters [MATH] and [MATH] to 5 and 100 respectively for all methods. While our system contains a number of h... |
Each method was allowed to run on each program in the benchmark suite until a satisfying program was found, 20,000 programs were proposed, 16 GB of memory were consumed, or 8 hours on one 2.4GHz Broadwell CPU and one Nvidia p100 GPU had been consumed. In our comparisons, we focus on sample efficiency instead of clock t... |
3.2 Baselines We compare RLGTS to several baselines. In the simplest case, we run a breadth-first enumerative search algorithm, which has expected program solve time of approximately [MATH] for a program space with [MATH] possible programs of length equal to the GT program. |
Second, we compare to a simple multi-armed bandit model as the simplest form of RL-trainable model, representing each decision [MATH] defining a program as an independent random variable sampled based on a learned probability [MATH] trained using the REINFORCE algorithm |
Next, we compare to a Q-learning baseline as an ablation of our full system, using the same network architecture and training procedure, but lacking the search tree of the full system. |
Lastly, we compare to the approach used in Stoke, a heuristic-driven stochastic search based system and currently the state of the art for optimal synthesis of RISC-V programs |
. To accommodate the simplifications we made to the RISC-V language, we re-implement Stoke’s MCMC search to allow it to search over the RISC-V space and reward function that our method uses. We refer to this baseline as "MCMC." We selected a value for the MCMC [MATH] hyperparameter by testing on a holdout validation pr... |
3.3 Program Length Because program length is a major determinant of search space complexity, we characterized each method’s behavior as a function of the length of ground truth program. Figure shows the results on our synthetic benchmark for lengths between 3 and 10 lines. RLGTS solves at least twice as many programs a... |
A fraction of the programs generated have a convex reward function space and can be trivially solved by convex descent using a best-first search algorithm. Specifically, for a program to be considered convex, among the rewards of all possible length 1 programs the first line of a satisfying program ranks highest, among... |
Search Depth Limit While RLGTS readily outstrips the naïve search and multi-armed bandit baselines for various lengths and search depths, we found during testing that MCMC search performance on this benchmark is highly dependent on the difference between the true program length and the search depth limit on maximum pro... |
. Because of this sensitivity, we compare RLGTS against MCMC for fixed differences between ground-truth program length and maximum search depth, We show the results of this comparison in figure . While MCMC performs competitively with RLGTS on longer programs when the target program length is known, its performance deg... |
3.4 Action Space Complexity In addition to program length, we also explore the number of single-line programs expressible as a factor for performance. Figure shows performance for RLGTS and MCMC, as well as a best-first search baseline, as a function of the number of instructions allowed, ranging from 2 instructions to... |
Discussion Here, we have presented reinforcement learning guided tree search, a new approach for program synthesis powered by reinforcement learning. This approach is general and flexible, with up to 400% better performance than the state of the art in traditional search-based methods on cases where search achieves a n... |
Acknowledgments We would like to thank Kyle Genova, David Walker, Olga Russakovsky, and Aarti Gupta for useful discussions and feedback. We would also like to thank Maciej Balog for information and advice relating to the Deepcoder language and system. |
# Source: arxiv 1806.03042 # Title: Braided Morita equivalence for finite-dimensional semisimple and cosemisimple Hopf algebras # Sections: all # Downloaded: 2026-03-03T02:41:33.379959+00:00 |
Braided Morita equivalence for finite-dimensional semisimple and cosemisimple Hopf algebras Dedicated to Professor Masahiko Suzuki on the occasion of his 65th birthday |
Abstract Braided Morita invariants of finite-dimensional semisimple and cosemisimple Hopf algebras with braidings are constructed by refining the polynomial invariants introduced by the author. The invariants are computed for the duals of Suzuki’s braided Hopf algebras, and as an application of that, the braided Morita... |
Introduction On the classification of Hopf algebras over a field [MATH] two problems are now actively progressed. One is the classification up to isomorphism under some restriction like dimension fixed, or semisimple, or pointed. Another is the classification up to monoidal Morita equivalence, that is based on a catego... |
We write [MATH] for the braiding structure and [MATH] for the braided monoidal category [MATH] Two quasitriangular Hopf algebras [MATH] and [MATH] are called braided Morita equivalent if the braided categories [MATH] and [MATH] are equivalent as |
[MATH] -linear braided monoidal categories. There are a few results of classification of quasitriangular Hopf algebras up to braided Morita equivalence |
The eigenvalues of [MATH] -matrices and the Brauer groups in a braided monoidal category are well-known as braided Morita invariants |
In the author introduced some monoidal Morita invariant of semisimple and cosemisimple Hopf algebras of finite dimension. It is given as a polynomial in one variable, which constructed from the data of the braidings and the absolutely simple modules. By refining the invariant on braidings we have braided Morita invaria... |
, which fit into a Hopf algebra extension [MATH] , where [MATH] is the cyclic group of order [MATH] , and [MATH] is the dihedral group of order [MATH] In particular, the [MATH] -dimensional Kac-Paljutkin algebra |
, denoted by [MATH] , is contained in the family of Suzuki’s Hopf algebras. As an application of the computation results of our polynomial invariants, we determine the braided Morita equivalence classes over [MATH] |
In closely connection with the above consideration, the coribbon elements of Suzuki’s braided Hopf algebras are determined. Actually, although they have studied in |
by the author, the proof of Lemma 8 and the statement of Theorem 5 in contain several mistakes. I noticed them by a detailed note |
sent from Sommerhäuser. We modify arguments in and show the correct results on that with thanks to him. Another proof of the revised version of Theorem 5 in |
is also given by using the spherical structures of Suzuki’s Hopf algebras. This paper is organized as follows. In Section 2 we review the definition of (co)ribbon Hopf algebras, and introduce braided Morita invariants of semisimple and cosemisimple quasitriangular Hopf algebras of finite dimension. In Section 3 we revi... |
We give the revised results on determination of the coribbon elements of Suzuki’s braided Hopf algebras. In Section 4 we compute the polynomial invariants defined in Section 2 for the duals of Suzuki’s braided Hopf algebras. In the final section we compute the Hopf algebra automorphism group for [MATH] and determine th... |
Throughout this paper [MATH] denotes a field. For a bialgebra or a Hopf algebra [MATH] , denoted by [MATH] [MATH] and [MATH] the comultiplication, the counit and the antipode of [MATH] , respectively. We use Sweedler’s notation such as [MATH] for [MATH] For general facts on Hopf algebras or monoidal categories, refer t... |
and Kassel’s book Acknowledgments I express my sincere gratitude to Professor Yorck Sommerhäuser for careful reading my paper and for reminding me that there are incorrect descriptions in it. I would like to thank Professor Hiroyuki Yamane for giving an opportunity to speak in this conference and write a paper in the p... |
Braided Morita invariants of quasitriangular Hopf algebras 2.1 Definitions of braided and coribbon Hopf algebras The notion of a quasitriangular bialgebra or a quasitriangular Hopf algebra is introduced by Drinfeld |
It is a pair of a bialgebra or a Hopf algebra [MATH] over [MATH] and an invertible element [MATH] satisfying some suitable conditions. Such an [MATH] is called a universal [MATH] -matrix of [MATH] |
Lemma 2.1 Drinfeld , Radford ). Let [MATH] be a quasitriangular Hopf algebra. Then [MATH] the antipode [MATH] is bijective, [MATH] |
[MATH] [MATH] [MATH] [MATH] [MATH] Furthermore, if we write [MATH] in the form [MATH] and set [MATH] then the following conditions are satisfied. |
[MATH] [MATH] is invertible, and [MATH] for all [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] Here, [MATH] The element [MATH] is called the Drinfeld element of [MATH] |
An element [MATH] is called a ribbon element of a quasitriangular bialgebra [MATH] and the triplet [MATH] is called a ribbon bialgebra |
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