1,048 research outputs found
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
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