Open Graphs and Monoidal Theories

String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. The distinguishing feature of these diagrams is that edges need not be connected to vertices at both ends, and these unconnected ends can be interpreted as the inputs and outputs of a diagram. In this paper, we give a concrete construction for string diagrams using a special kind of typed graph called an open-graph. While the category of open-graphs is not itself adhesive, we introduce the notion of a selective adhesive functor, and show that such a functor embeds the category of open-graphs into the ambient adhesive category of typed graphs. Using this functor, the category of open-graphs inherits "enough adhesivity" from the category of typed graphs to perform double-pushout (DPO) graph rewriting. A salient feature of our theory is that it ensures rewrite systems are "type-safe" in the sense that rewriting respects the inputs and outputs. This formalism lets us safely encode the interesting structure of a computational model, such as evaluation dynamics, with succinct, explicit rewrite rules, while the graphical representation absorbs many of the tedious details. Although topological formalisms exist for string diagrams, our construction is discreet, finitary, and enjoys decidable algorithms for composition and rewriting. We also show how open-graphs can be parametrised by graphical signatures, similar to the monoidal signatures of Joyal and Street, which define types for vertices in the diagrammatic language and constraints on how they can be connected. Using typed open-graphs, we can construct free symmetric monoidal categories, PROPs, and more general monoidal theories. Thus open-graphs give us a handle for mechanised reasoning in monoidal categories.Comment: 31 pages, currently technical report, submitted to MSCS, waiting review

Best-First Rippling

Rippling is a form of rewriting that guides search by only performing steps that reduce the syntactic differences between formulae. Termination is normally ensured by a measure that is decreases with each rewrite step. Because of this restriction, rippling will fail to prove theorems about, for example, mutual recursion as steps that temporarily increase the differences are necessary. Best-first rippling is an extension to rippling where the restrictions have been recast as heuristic scores for use in best-first search. If nothing better is available, previously illegal steps can be considered, making best-first rippling more flexible than ordinary rippling. We have implemented best-first rippling in the IsaPlanner system together with a mechanism for caching proof-states that helps remove symmetries in the search space, and machinery to ensure termination based on term embeddings. Our experiments show that the implementation of best-first rippling is faster on average than IsaPlanner’s version of traditional depth-first rippling, and solves a range of problems where ordinary rippling fails

Arkansas Landlord Selection of Land-Leasing Contract Type and Terms

Land leasing is a major source of the land input to production agriculture. Responses from a survey of landlords leasing crop land in Arkansas are analyzed to better understand those factors motivating landlords in the type of lease they select and the terms of those leases. Probit models are estimated to determine the relative importance of variables representing credit constraint, agency problem, and risk aversion factors. Regression models then estimate the impact of site, landlord, and tenant characteristics on contract terms – the percentage of crop and cost sharing arrangements between landlord and tenant. Probit results suggest credit constraint factors influence lease-type selection. Risk aversion, managerial ability, and social capital factors are also supported. Regression models show that land and crop characteristics are significant determinants of contract terms.Land leasing, Probit, Contract, Production agriculture, Land Economics/Use,

A Proof Planning Framework For Isabelle

Centre for Intelligent Systems and their ApplicationsProof planning is a paradigm for the automation of proof that focuses on encoding intelligence to guide the proof process. The idea is to capture common patterns of reasoning which can be used to derive abstract descriptions of proofs known as proof plans. These can then be executed to provide fully formal proofs. This thesis concerns the development and analysis of a novel approach to proof planning that focuses on an explicit representation of choices during search. We embody our approach as a proof planner for the generic proof assistant Isabelle and use the Isar language, which is human-readable and machine-checkable, to represent proof plans. Within this framework we develop an inductive theorem prover as a case study of our approach to proof planning. Our prover uses the difference reduction heuristic known as rippling to automate the step cases of the inductive proofs. The development of a flexible approach to rippling that supports its various modifications and extensions is the second major focus of this thesis. Here, our inductive theorem prover provides a context in which to evaluate rippling experimentally. This work results in an efficient and powerful inductive theorem prover for Isabelle as well as proposals for further improving the efficiency of rippling. We also draw observations in order to direct further work on proof planning. Overall, we aim to make it easier for mathematical techniques, and those specific to mechanical theorem proving, to be encoded and applied to problems