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

Backbone Fragility and the Local Search Cost Peak

The local search algorithm WSat is one of the most successful algorithms for solving the satisfiability (SAT) problem. It is notably effective at solving hard Random 3-SAT instances near the so-called `satisfiability threshold', but still shows a peak in search cost near the threshold and large variations in cost over different instances. We make a number of significant contributions to the analysis of WSat on high-cost random instances, using the recently-introduced concept of the backbone of a SAT instance. The backbone is the set of literals which are entailed by an instance. We find that the number of solutions predicts the cost well for small-backbone instances but is much less relevant for the large-backbone instances which appear near the threshold and dominate in the overconstrained region. We show a very strong correlation between search cost and the Hamming distance to the nearest solution early in WSat's search. This pattern leads us to introduce a measure of the backbone fragility of an instance, which indicates how persistent the backbone is as clauses are removed. We propose that high-cost random instances for local search are those with very large backbones which are also backbone-fragile. We suggest that the decay in cost beyond the satisfiability threshold is due to increasing backbone robustness (the opposite of backbone fragility). Our hypothesis makes three correct predictions. First, that the backbone robustness of an instance is negatively correlated with the local search cost when other factors are controlled for. Second, that backbone-minimal instances (which are 3-SAT instances altered so as to be more backbone-fragile) are unusually hard for WSat. Third, that the clauses most often unsatisfied during search are those whose deletion has the most effect on the backbone. In understanding the pathologies of local search methods, we hope to contribute to the development of new and better techniques

Getting to the Root of Tree Soil Microbiome Sampling

Microbiomes play critical roles in host functioning and therefore there is increasing interest in the microbiome assembly of plants. However, sampling strategies for long-lived perennial trees need to be standardised to produce robust data that accurately represents the microbiome over time. This issue is currently unresolved because there is little evidence indicating which portion of perennial tree species (e.g., root region or surrounding soil) is the best to sample to produce the most accurate measure of microbiome communities. Our aim was to sample different compartments of a plant’s belowground microbiome to identify the optimal sampling strategy to account for the microbial community present. We found that the structure of the microbial community depends most strongly on the environment (site) and compartment of sample collected (bulk soil, rhizosphere, or rhizoplane), rather than the depth or cardinal direction of the sample. We also found that the microbial community increased in diversity with increased distance from the tree within the rhizoplane and rhizosphere. The data presented here provides systematic evidence for a pragmatic and robust sampling regime that was tested and validated across different environments and soil types while controlling for host genotype. This sampling regime will enable effective partitioning of root compartments when studying the microbiome associated with perennial tree species, allowing targeted questions about the microbiome to be explored with greater accuracy

Algorithmic aspects of theory blending

In Cognitive Science, conceptual blending has been proposed as an important cognitive mechanism that facilitates the creation of new concepts and ideas by constrained combination of available knowledge. It thereby provides a possible theoretical foundation for modeling high-level cognitive faculties such as the ability to understand, learn, and create new concepts and theories. This paper describes a logic-based framework which allows a formal treatment of theory blending, discusses algorithmic aspects of blending within the framework, and provides an illustrating worked out example from mathematics

A Rational Reconstruction and Extension of Recursion Analysis

The focus of this paper is the technique of recur8\on analysis. Recursion analysis is used by the Boyer-Moore Theorem Prover to choose an appropriate induction schema and variable to prove theorems by mathematical induction. A rational reconstruction of recursion analysis is outlined, using the technique of proof plans. This rational reconstruction suggests an extension of recursion analysis which frees the induction suggestion from the forms of recursion found in the conjecture. Preliminary results are reported of the automation of this rational reconstruction and extension using the CLAM-Oyster system

Case-Analysis for Rippling and Inductive Proof

Rippling is a heuristic used to guide rewriting and is typically used for inductive theorem proving. We introduce a method to support case-analysis within rippling. Like earlier work, this allows goals containing if-statements to be proved automatically. The new contribution is that our method also supports case-analysis on datatypes. By locating the case-analysis as a step within rippling we also maintain the termination. The work has been implemented in IsaPlanner and used to extend the existing inductive proof method. We evaluate this extended prover on a large set of examples from Isabelle’s theory library and from the inductive theorem proving literature. We find that this leads to a significant improvement in the coverage of inductive theorem proving. The main limitations of the extended prover are identified, highlight the need for advances in the treatment of assumptions during rippling and when conjecturing lemmas