Solving Degenerate Sparse Polynomial Systems Faster

Consider a system F of n polynomial equations in n unknowns, over an algebraically closed field of arbitrary characteristic. We present a fast method to find a point in every irreducible component of the zero set Z of F. Our techniques allow us to sharpen and lower prior complexity bounds for this problem by fully taking into account the monomial term structure. As a corollary of our development we also obtain new explicit formulae for the exact number of isolated roots of F and the intersection multiplicity of the positive-dimensional part of Z. Finally, we present a combinatorial construction of non-degenerate polynomial systems, with specified monomial term structure and maximally many isolated roots, which may be of independent interest.Comment: This is the final journal version of math.AG/9702222 (``Toric Generalized Characteristic Polynomials''). This final version is a major revision with several new theorems, examples, and references. The prior results are also significantly improve

On Characterizing the Data Access Complexity of Programs

Technology trends will cause data movement to account for the majority of energy expenditure and execution time on emerging computers. Therefore, computational complexity will no longer be a sufficient metric for comparing algorithms, and a fundamental characterization of data access complexity will be increasingly important. The problem of developing lower bounds for data access complexity has been modeled using the formalism of Hong & Kung's red/blue pebble game for computational directed acyclic graphs (CDAGs). However, previously developed approaches to lower bounds analysis for the red/blue pebble game are very limited in effectiveness when applied to CDAGs of real programs, with computations comprised of multiple sub-computations with differing DAG structure. We address this problem by developing an approach for effectively composing lower bounds based on graph decomposition. We also develop a static analysis algorithm to derive the asymptotic data-access lower bounds of programs, as a function of the problem size and cache size

Liveness-Driven Random Program Generation

Randomly generated programs are popular for testing compilers and program analysis tools, with hundreds of bugs in real-world C compilers found by random testing. However, existing random program generators may generate large amounts of dead code (computations whose result is never used). This leaves relatively little code to exercise a target compiler's more complex optimizations. To address this shortcoming, we introduce liveness-driven random program generation. In this approach the random program is constructed bottom-up, guided by a simultaneous structural data-flow analysis to ensure that the generator never generates dead code. The algorithm is implemented as a plugin for the Frama-C framework. We evaluate it in comparison to Csmith, the standard random C program generator. Our tool generates programs that compile to more machine code with a more complex instruction mix.Comment: Pre-proceedings paper presented at the 27th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2017), Namur, Belgium, 10-12 October 2017 (arXiv:1708.07854

FO-definable transformations of infinite strings

The theory of regular and aperiodic transformations of finite strings has recently received a lot of interest. These classes can be equivalently defined using logic (Monadic second-order logic and first-order logic), two-way machines (regular two-way and aperiodic two-way transducers), and one-way register machines (regular streaming string and aperiodic streaming string transducers). These classes are known to be closed under operations such as sequential composition and regular (star-free) choice; and problems such as functional equivalence and type checking, are decidable for these classes. On the other hand, for infinite strings these results are only known for $\omega$-regular transformations: Alur, Filiot, and Trivedi studied transformations of infinite strings and introduced an extension of streaming string transducers over $\omega$-strings and showed that they capture monadic second-order definable transformations for infinite strings. In this paper we extend their work to recover connection for infinite strings among first-order logic definable transformations, aperiodic two-way transducers, and aperiodic streaming string transducers

Efficient Monitoring of ??-languages

We present a technique for generating efficient monitors for Omega-regular-languages. We show how Buchi automata can be reduced in size and transformed into special, statistically optimal nondeterministic finite state machines, called binary transition tree finite state machines (BTT-FSMs), which recognize precisely the minimal bad prefixes of the original omega-regular-language. The presented technique is implemented as part of a larger monitoring framework and is available for download

Specifying and Executing Optimizations for Parallel Programs

Compiler optimizations, usually expressed as rewrites on program graphs, are a core part of all modern compilers. However, even production compilers have bugs, and these bugs are difficult to detect and resolve. The problem only becomes more complex when compiling parallel programs; from the choice of graph representation to the possibility of race conditions, optimization designers have a range of factors to consider that do not appear when dealing with single-threaded programs. In this paper we present PTRANS, a domain-specific language for formal specification of compiler transformations, and describe its executable semantics. The fundamental approach of PTRANS is to describe program transformations as rewrites on control flow graphs with temporal logic side conditions. The syntax of PTRANS allows cleaner, more comprehensible specification of program optimizations; its executable semantics allows these specifications to act as prototypes for the optimizations themselves, so that candidate optimizations can be tested and refined before going on to include them in a compiler. We demonstrate the use of PTRANS to state, test, and refine the specification of a redundant store elimination optimization on parallel programs.Comment: In Proceedings GRAPHITE 2014, arXiv:1407.767

Hilbert's Program Then and Now

Hilbert's program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to "dispose of the foundational questions in mathematics once and for all, "Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, "finitary" means, one should give proofs of the consistency of these axiomatic systems. Although Godel's incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial successes, and generated important advances in logical theory and meta-theory, both at the time and since. The article discusses the historical background and development of Hilbert's program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s.Comment: 43 page