Comparing the vulnerability of different coal industrial symbiosis networks under economic fluctuations

We establish a vulnerability analytical framework of CISN, and illustrate the impact path of economic fluctuation on CISN performance. Based on this, we propose an improved cascading failure model with directed weighted network, and design five network performance indicators (i.e., relative value of cascading failure, average path length, relative value of maximal connected sub-graphs, network efficiency, and structure entropy). Taking three coal eco-industrial parks in China as cases, we simulate and compare the impacts on CISN vulnerability (i.e., equality-based, dependent-based, and nested-based CISNs) of economic fluctuation. The results indicate that the interaction between economic fluctuation and network structure is the key factor in determining system vulnerability. Concerning overall vulnerability, equality-based CISN is highest, dependent-based CISN is next, and nested-based CISN is lowest. Regarding disturbance type, the changes in the five performance indicators of the three types of CISN are more intense under energy price shocks than with declining demand. Moreover, the cascading failure scale of equality-based CISN is greatest with declining demand, while the other two kinds of CISN’s is greatest under energy price shocks. Concerning disturbance intensity, equality-based CISN shows initial value sensitivity to economic fluctuation, and nested-based CISN has the strongest tolerances for economic fluctuation. From the network performance perspective, the performance of nested-based CISN is superior to that of dependent-based and equality-based CISNs. Due to longer average path length and lower network efficiency, the failure diffusion trend of equality-based CISN shows the curve of Type-S, and the diffusion rate is smooth and slow. Contrariwise, the initial diffusion rate of dependent-based CISN is the highest, indicating that the loss of system efficiency can somewhat improve the system’s anti-risk ability

Finitary Higher Inductive Types in the Groupoid Model

A higher inductive type of level 1 (a 1-hit) has constructors for points and paths only, whereas a higher inductive type of level 2 (a 2-hit) has constructors for surfaces too. We restrict attention to finitary higher inductive types and present general schemata for the types of their point, path, and surface constructors. We also derive the elimination and equality rules from the types of constructors for 1-hits and 2-hits. Moreover, we construct a groupoid model for dependent type theory with 2-hits and point out that we obtain a setoid model for dependent type theory with 1-hits by truncating the groupoid model

Guarded Cubical Type Theory: Path Equality for Guarded Recursion

This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with coinductive types. We wish to implement GDTT with decidable type-checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions. CTT is a variation of Martin-L\"of type theory in which the identity type is replaced by abstract paths between terms. CTT provides a computational interpretation of functional extensionality, is conjectured to have decidable type checking, and has an implemented type-checker. Our new type theory, called guarded cubical type theory, provides a computational interpretation of extensionality for guarded recursive types. This further expands the foundations of CTT as a basis for formalisation in mathematics and computer science. We present examples to demonstrate the expressivity of our type theory, all of which have been checked using a prototype type-checker implementation, and present semantics in a presheaf category.Comment: 17 pages, to be published in proceedings of CSL 201

Homotopy Type Theory in Lean

We discuss the homotopy type theory library in the Lean proof assistant. The library is especially geared toward synthetic homotopy theory. Of particular interest is the use of just a few primitive notions of higher inductive types, namely quotients and truncations, and the use of cubical methods.Comment: 17 pages, accepted for ITP 201

Notions of Anonymous Existence in Martin-L\"of Type Theory

As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-L\"of type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we give some simple characterizations of types that do have unique identity proofs. A key ingredient in these constructions are weakly constant endofunctions on identity types. We study such endofunctions on arbitrary types and show that they always factor through a propositional type, the truncated or squashed domain. Such a factorization is impossible for weakly constant functions in general (a result by Shulman), but we present several non-trivial cases in which it can be done. Based on these results, we define a new notion of anonymous existence in type theory and compare different forms of existence carefully. In addition, we show possibly surprising consequences of the judgmental computation rule of the truncation, in particular in the context of homotopy type theory. All the results have been formalized and verified in the dependently typed programming language Agda.Comment: 36 pages, to appear in the special issue of TLCA'13 (LMCS