23,236 research outputs found
Complexity Bounds for Ordinal-Based Termination
`What more than its truth do we know if we have a proof of a theorem in a
given formal system?' We examine Kreisel's question in the particular context
of program termination proofs, with an eye to deriving complexity bounds on
program running times.
Our main tool for this are length function theorems, which provide complexity
bounds on the use of well quasi orders. We illustrate how to prove such
theorems in the simple yet until now untreated case of ordinals. We show how to
apply this new theorem to derive complexity bounds on programs when they are
proven to terminate thanks to a ranking function into some ordinal.Comment: Invited talk at the 8th International Workshop on Reachability
Problems (RP 2014, 22-24 September 2014, Oxford
Using Program Synthesis for Program Analysis
In this paper, we identify a fragment of second-order logic with restricted
quantification that is expressive enough to capture numerous static analysis
problems (e.g. safety proving, bug finding, termination and non-termination
proving, superoptimisation). We call this fragment the {\it synthesis
fragment}. Satisfiability of a formula in the synthesis fragment is decidable
over finite domains; specifically the decision problem is NEXPTIME-complete. If
a formula in this fragment is satisfiable, a solution consists of a satisfying
assignment from the second order variables to \emph{functions over finite
domains}. To concretely find these solutions, we synthesise \emph{programs}
that compute the functions. Our program synthesis algorithm is complete for
finite state programs, i.e. every \emph{function} over finite domains is
computed by some \emph{program} that we can synthesise. We can therefore use
our synthesiser as a decision procedure for the synthesis fragment of
second-order logic, which in turn allows us to use it as a powerful backend for
many program analysis tasks. To show the tractability of our approach, we
evaluate the program synthesiser on several static analysis problems.Comment: 19 pages, to appear in LPAR 2015. arXiv admin note: text overlap with
arXiv:1409.492
Well-Founded Semantics for Extended Datalog and Ontological Reasoning
The DatalogĀ± family of expressive extensions of Datalog has recently been introduced as a new paradigm for query answering over ontologies, which captures and extends several common description logics. It extends plain Datalog by features such as existentially quantified rule heads and, at the same time, restricts the rule syntax so as to achieve decidability and tractability. In this paper, we continue the research on DatalogĀ±. More precisely, we generalize the well-founded semantics (WFS), as the standard semantics for nonmonotonic normal programs in the database context, to DatalogĀ± programs with negation under the unique name assumption (UNA). We prove that for guarded DatalogĀ± with negation under the standard WFS, answering normal Boolean conjunctive queries is decidable, and we provide precise complexity results for this problem, namely, in particular, completeness for PTIME (resp., 2-EXPTIME) in the data (resp., combined) complexity
Recursive Program Optimization Through Inductive Synthesis Proof Transformation
The research described in this paper involved developing transformation techniques which increase the efficiency of the noriginal program, the source, by transforming its synthesis proof into one, the target, which yields a computationally more efficient algorithm. We describe a working proof transformation system which, by exploiting the duality between mathematical induction and recursion, employs the novel strategy of optimizing recursive programs by transforming inductive proofs. We compare and contrast this approach with the more traditional approaches to program transformation, and highlight the benefits of proof transformation with regards to search, correctness, automatability and generality
Near-optimal Bootstrapping of Hitting Sets for Algebraic Models
The classical lemma of Ore-DeMillo-Lipton-Schwartz-Zippel
[Ore22,DL78,Zip79,Sch80] states that any nonzero polynomial of degree at most will evaluate to a nonzero value at some point on a
grid with . Thus, there is an explicit
hitting set for all -variate degree , size algebraic circuits of size
.
In this paper, we prove the following results:
- Let be a constant. For a sufficiently large constant and
all , if we have an explicit hitting set of size
for the class of -variate degree polynomials that are computable by
algebraic circuits of size , then for all , we have an explicit hitting
set of size for -variate circuits of
degree and size . That is, if we can obtain a barely non-trivial
exponent compared to the trivial sized hitting set even for
constant variate circuits, we can get an almost complete derandomization of
PIT.
- The above result holds when "circuits" are replaced by "formulas" or
"algebraic branching programs".
This extends a recent surprising result of Agrawal, Ghosh and Saxena [AGS18]
who proved the same conclusion for the class of algebraic circuits, if the
hypothesis provided a hitting set of size at most
(where is any constant). Hence, our work significantly weakens the
hypothesis of Agrawal, Ghosh and Saxena to only require a slightly non-trivial
saving over the trivial hitting set, and also presents the first such result
for algebraic branching programs and formulas.Comment: The main result has been strengthened significantly, compared to the
older version of the paper. Additionally, the stronger theorem now holds even
for subclasses of algebraic circuits, such as algebraic formulas and
algebraic branching program
- ā¦