Semantics of Higher-Order Recursion Schemes

Higher-order recursion schemes are recursive equations defining new operations from given ones called "terminals". Every such recursion scheme is proved to have a least interpreted semantics in every Scott's model of \lambda-calculus in which the terminals are interpreted as continuous operations. For the uninterpreted semantics based on infinite \lambda-terms we follow the idea of Fiore, Plotkin and Turi and work in the category of sets in context, which are presheaves on the category of finite sets. Fiore et al showed how to capture the type of variable binding in \lambda-calculus by an endofunctor H\lambda and they explained simultaneous substitution of \lambda-terms by proving that the presheaf of \lambda-terms is an initial H\lambda-monoid. Here we work with the presheaf of rational infinite \lambda-terms and prove that this is an initial iterative H\lambda-monoid. We conclude that every guarded higher-order recursion scheme has a unique uninterpreted solution in this monoid

Inapproximability of the independent set polynomial in the complex plane

We study the complexity of approximating the independent set polynomial $Z_G(\lambda)$ of a graph $G$ with maximum degree $\Delta$ when the activity $\lambda$ is a complex number. This problem is already well understood when $\lambda$ is real using connections to the $\Delta$-regular tree $T$. The key concept in that case is the "occupation ratio" of the tree $T$. This ratio is the contribution to $Z_T(\lambda)$ from independent sets containing the root of the tree, divided by $Z_T(\lambda)$ itself. If $\lambda$ is such that the occupation ratio converges to a limit, as the height of $T$ grows, then there is an FPTAS for approximating $Z_G(\lambda)$ on a graph $G$ with maximum degree $\Delta$. Otherwise, the approximation problem is NP-hard. Unsurprisingly, the case where $\lambda$ is complex is more challenging. Peters and Regts identified the complex values of $\lambda$ for which the occupation ratio of the $\Delta$-regular tree converges. These values carve a cardioid-shaped region $\Lambda_\Delta$ in the complex plane. Motivated by the picture in the real case, they asked whether $\Lambda_\Delta$ marks the true approximability threshold for general complex values $\lambda$. Our main result shows that for every $\lambda$ outside of $\Lambda_\Delta$, the problem of approximating $Z_G(\lambda)$ on graphs $G$ with maximum degree at most $\Delta$ is indeed NP-hard. In fact, when $\lambda$ is outside of $\Lambda_\Delta$ and is not a positive real number, we give the stronger result that approximating $Z_G(\lambda)$ is actually #P-hard. If $\lambda$ is a negative real number outside of $\Lambda_\Delta$, we show that it is #P-hard to even decide whether $Z_G(\lambda)>0$, resolving in the affirmative a conjecture of Harvey, Srivastava and Vondrak. Our proof techniques are based around tools from complex analysis - specifically the study of iterative multivariate rational maps

Tree Regular Model Checking for Lattice-Based Automata

Tree Regular Model Checking (TRMC) is the name of a family of techniques for analyzing infinite-state systems in which states are represented by terms, and sets of states by Tree Automata (TA). The central problem in TRMC is to decide whether a set of bad states is reachable. The problem of computing a TA representing (an over- approximation of) the set of reachable states is undecidable, but efficient solutions based on completion or iteration of tree transducers exist. Unfortunately, the TRMC framework is unable to efficiently capture both the complex structure of a system and of some of its features. As an example, for JAVA programs, the structure of a term is mainly exploited to capture the structure of a state of the system. On the counter part, integers of the java programs have to be encoded with Peano numbers, which means that any algebraic operation is potentially represented by thousands of applications of rewriting rules. In this paper, we propose Lattice Tree Automata (LTAs), an extended version of tree automata whose leaves are equipped with lattices. LTAs allow us to represent possibly infinite sets of interpreted terms. Such terms are capable to represent complex domains and related operations in an efficient manner. We also extend classical Boolean operations to LTAs. Finally, as a major contribution, we introduce a new completion-based algorithm for computing the possibly infinite set of reachable interpreted terms in a finite amount of time.Comment: Technical repor

The complex of formal operations on the Hochschild chains of commutative algebras

We compute the homology of the complex of formal operations on the Hochschild complex of differential graded commutative algebras as defined by Wahl and prove that these can be built as infinite sums of operations obtained from Loday's shuffle operations, Connes' boundary operator and the shuffle product.Comment: 19 page