Colored E-Graph: Equality Reasoning with Conditions

E-graphs are a prominent data structure that has been increasing in popularity in recent years due to their expanding range of applications in various formal reasoning tasks. Often, they are used for equality saturation, a process of deriving consequences through repeatedly applying universally quantified equality formulas via term rewriting. They handle equality reasoning over a large spaces of terms, but are severely limited in their handling of case splitting and other types of logical cuts, especially when compared to other reasoning techniques such as sequent calculi and resolution. The main difficulty is when equality reasoning requires multiple inconsistent assumptions to reach a single conclusion. Ad-hoc solutions, such as duplicating the e-graph for each assumption, are available, but they are notably resource-intensive. We introduce a key observation is that each duplicate e-graph (with an added assumption) corresponds to coarsened congruence relation. Based on that, we present an extension to e-graphs, called Colored E-Graphs, as a way to represent all of the coarsened congruence relations in a single structure. A colored e-graph is a memory-efficient equivalent of multiple copies of an e-graph, with a much lower overhead. This is attained by sharing as much as possible between different cases, while carefully tracking which conclusion is true under which assumption. Support for multiple relations can be thought of as adding multiple "color-coded" layers on top of the original e-graph structure, leading to a large degree of sharing. In our implementation, we introduce optimizations to rebuilding and e-matching. We run experiments and demonstrate that our colored e-graphs can support hundreds of assumptions and millions of terms with space requirements that are an order of magnitude lower, and with similar time requirements

Initial report on Object Spreadsheets

There is a growing demand for data-driven web applications that help automate organizational and business processes of low to medium complexity by letting users view and update structured data in controlled ways. We present Object Spreadsheets, an end-user development tool that combines a spreadsheet interface with a rich data model to help the process administrators build the logic for such applications themselves. Its all-in-one interface with immediate feedback has the potential to bring more complex tasks within reach of end-user developers, compared to existing approaches. Our data model is based on the structure of entity-relationship models and directly supports nested variable-size collections and object references, which are common in web applications but poorly accommodated by traditional spreadsheets. Object Spreadsheets has a formula language suited to the data model and supports stored procedures to specify the forms of updates that application users may make. Formulas can be used to assemble data in the exact structure in which it is to be shown in the application UI, simplifying the task of UI building; we intend for Object Spreadsheets to be integrated with a UI builder to provide a complete solution for application development. We describe our prototype implementation and several example applications we built to demonstrate the applicability of the tool

SMT Sampling via Model-Guided Approximation

We investigate the domain of satisfiable formulas in satisfiability modulo theories (SMT), in particular, automatic generation of a multitude of satisfying assignments to such formulas. Despite the long and successful history of SMT in model checking and formal verification, this aspect is relatively under-explored. Prior work exists for generating such assignments, or samples, for Boolean formulas and for quantifier-free first-order formulas involving bit-vectors, arrays, and uninterpreted functions (QF_AUFBV). We propose a new approach that is suitable for a theory T of integer arithmetic and to T with arrays and uninterpreted functions. The approach involves reducing the general sampling problem to a simpler instance of sampling from a set of independent intervals, which can be done efficiently. Such reduction is carried out by expanding a single model - a seed - using top-down propagation of constraints along the original first-order formula

AmiGo: Computational Design of Amigurumi Crochet Patterns

We propose an approach for generating crochet instructions (patterns) from an input 3D model. We focus on Amigurumi, which are knitted stuffed toys. Given a closed triangle mesh, and a single point specified by the user, we generate crochet instructions, which when knitted and stuffed result in a toy similar to the input geometry. Our approach relies on constructing the geometry and connectivity of a Crochet Graph, which is then translated into a crochet pattern. We segment the shape automatically into chrochetable components, which are connected using the join-as-you-go method, requiring no additional sewing. We demonstrate that our method is applicable to a large variety of shapes and geometries, and yields easily crochetable patterns.Comment: 11 pages, 10 figures, SCF 202

Solving geometry problems using a combination of symbolic and numerical reasoning.

Abstract. We describe a framework that combines deductive, numeric, and inductive reasoning to solve geometric problems. Applications include the generation of geometric models and animations, as well as problem solving in the context of intelligent tutoring systems. Our novel methodology uses (i) deductive reasoning to generate a partial program from logical constraints, (ii) numerical methods to evaluate the partial program, thus creating geometric models which are solutions to the original problem, and (iii) inductive synthesis to read off new constraints that are then applied to one more round of deductive reasoning leading to the desired deterministic program. By the combination of methods we were able to solve problems that each of the methods was not able to solve by itself. The number of nondeterministic choices in a partial program provides a measure of how close a problem is to being solved and can thus be used in the educational context for grading and providing hints. We have successfully evaluated our methodology on 18 Scholastic Aptitude Test geometry problems, and 11 ruler/compass-based geometry construction problems. Our tool solved these problems using an average of a few seconds per problem

On the Automated Verification of Web Applications with Embedded SQL

A large number of web applications is based on a relational database together with a program, typically a script, that enables the user to interact with the database through embedded SQL queries and commands. In this paper, we introduce a method for formal automated verification of such systems which connects database theory to mainstream program analysis. We identify a fragment of SQL which captures the behavior of the queries in our case studies, is algorithmically decidable, and facilitates the construction of weakest preconditions. Thus, we can integrate the analysis of SQL queries into a program analysis tool chain. To this end, we implement a new decision procedure for the SQL fragment that we introduce. We demonstrate practical applicability of our results with three case studies, a web administrator, a simple firewall, and a conference management system

Deriving divide-and-conquer dynamic programming algorithms using solver-aided transformations

We introduce a framework allowing domain experts to manipulate computational terms in the interest of deriving better, more efficient implementations.It employs deductive reasoning to generate provably correct efficient implementations from a very high-level specification of an algorithm, and inductive constraint-based synthesis to improve automation. Semantic information is encoded into program terms through the use of refinement types. In this paper, we develop the technique in the context of a system called Bellmania that uses solver-aided tactics to derive parallel divide-and-conquer implementations of dynamic programming algorithms that have better locality and are significantly more efficient than traditional loop-based implementations. Bellmania includes a high-level language for specifying dynamic programming algorithms and a calculus that facilitates gradual transformation of these specifications into efficient implementations. These transformations formalize the divide-and conquer technique; a visualization interface helps users to interactively guide the process, while an SMT-based back-end verifies each step and takes care of low-level reasoning required for parallelism. We have used the system to generate provably correct implementations of several algorithms, including some important algorithms from computational biology, and show that the performance is comparable to that of the best manually optimized code.National Science Foundation (U.S.) (CCF-1139056)National Science Foundation (U.S.) (CCF- 1439084)National Science Foundation (U.S.) (CNS-1553510

Verified lifting of stencil computations

This paper demonstrates a novel combination of program synthesis and verification to lift stencil computations from low-level Fortran code to a high-level summary expressed using a predicate language. The technique is sound and mostly automated, and leverages counter-example guided inductive synthesis (CEGIS) to find provably correct translations. Lifting existing code to a high-performance description language has a number of benefits, including maintainability and performance portability. For example, our experiments show that the lifted summaries can enable domain specific compilers to do a better job of parallelization as compared to an off-the-shelf compiler working on the original code, and can even support fully automatic migration to hardware accelerators such as GPUs. We have implemented verified lifting in a system called STNG and have evaluated it using microbenchmarks, mini-apps, and real-world applications. We demonstrate the benefits of verified lifting by first automatically summarizing Fortran source code into a high-level predicate language, and subsequently translating the lifted summaries into Halide, with the translated code achieving median performance speedups of 4.1X and up to 24X for non-trivial stencils as compared to the original implementation.United States. Department of Energy. Office of Science (Award DE-SC0008923)United States. Department of Energy. Office of Science (Award DE-SC0005288

Object spreadsheets: a new computational model for end-user development of data-centric web applications

Spreadsheets offer many advantages as the computational and data-storage engine for applications that are authored by end users. Paradoxically, however, their main failing in this regard is their computational model. Despite being used in almost all cases to represent data that is essentially relational (with some hierarchical structuring), the spreadsheet model treats the two-dimensional grid as largely unstructured, with formulas linking cells in an ad hoc way. This paper reports on a quest to rethink the spreadsheet model. The model we propose supports not only conventional flat tables, but also nested variable-size lists and object references. It includes a formula language suited to the data model and procedures to specify updates. The model has been implemented in a tool called Object Spreadsheets, which is intended for the development of data-centric web applications. We describe several example applications we built using the tool to demonstrate its applicability.Wistron CorporationNational Science Foundation (U.S.

A simple inductive synthesis methodology and its applications

Given a high-level specification and a low-level programming language, our goal is to automatically synthesize an efficient program that meets the specification. In this paper, we present a new algorithmic methodology for inductive synthesis that allows us to do this. We use Second Order logic as our generic high level specification logic. For our low-level languages we choose small application-specific logics that can be immediately translated into code that runs in expected linear time in the worst case. We explain our methodology and provide examples of the synthesis of several graph classifiers, e.g, linear-time tests of whether the input graph is connected, acyclic, etc. In another set of applications we automatically derive many finite differencing expressions equivalent to ones that Paige built by hand in his thesis [Pai81]. Finally we describe directions for automatically combining such automatically generated building blocks to synthesize efficient code implementing more complicated specifications. The methods in this paper have been implemented in Python using the SMT solver Z3 [dMB]