Defining Recursive Predicates in Graph Orders

We study the first order theory of structures over graphs i.e. structures of the form ($\mathcal{G},\tau$) where $\mathcal{G}$ is the set of all (isomorphism types of) finite undirected graphs and $\tau$ some vocabulary. We define the notion of a recursive predicate over graphs using Turing Machine recognizable string encodings of graphs. We also define the notion of an arithmetical relation over graphs using a total order $\leq_t$ on the set $\mathcal{G}$ such that ($\mathcal{G},\leq_t$) is isomorphic to ($\mathbb{N},\leq$). We introduce the notion of a \textit{capable} structure over graphs, which is one satisfying the conditions : (1) definability of arithmetic, (2) definability of cardinality of a graph, and (3) definability of two particular graph predicates related to vertex labellings of graphs. We then show any capable structure can define every arithmetical predicate over graphs. As a corollary, any capable structure also defines every recursive graph relation. We identify capable structures which are expansions of graph orders, which are structures of the form ($\mathcal{G},\leq$) where $\leq$ is a partial order. We show that the subgraph order i.e. ($\mathcal{G},\leq_s$), induced subgraph order with one constant $P_3$ i.e. ($\mathcal{G},\leq_i,P_3$) and an expansion of the minor order for counting edges i.e. ($\mathcal{G},\leq_m,sameSize(x,y)$) are capable structures. In the course of the proof, we show the definability of several natural graph theoretic predicates in the subgraph order which may be of independent interest. We discuss the implications of our results and connections to Descriptive Complexity

Complexity Analysis of Precedence Terminating Infinite Graph Rewrite Systems

The general form of safe recursion (or ramified recurrence) can be expressed by an infinite graph rewrite system including unfolding graph rewrite rules introduced by Dal Lago, Martini and Zorzi, in which the size of every normal form by innermost rewriting is polynomially bounded. Every unfolding graph rewrite rule is precedence terminating in the sense of Middeldorp, Ohsaki and Zantema. Although precedence terminating infinite rewrite systems cover all the primitive recursive functions, in this paper we consider graph rewrite systems precedence terminating with argument separation, which form a subclass of precedence terminating graph rewrite systems. We show that for any precedence terminating infinite graph rewrite system G with a specific argument separation, both the runtime complexity of G and the size of every normal form in G can be polynomially bounded. As a corollary, we obtain an alternative proof of the original result by Dal Lago et al.Comment: In Proceedings TERMGRAPH 2014, arXiv:1505.06818. arXiv admin note: text overlap with arXiv:1404.619

Layered Fixed Point Logic

We present a logic for the specification of static analysis problems that goes beyond the logics traditionally used. Its most prominent feature is the direct support for both inductive computations of behaviors as well as co-inductive specifications of properties. Two main theoretical contributions are a Moore Family result and a parametrized worst case time complexity result. We show that the logic and the associated solver can be used for rapid prototyping and illustrate a wide variety of applications within Static Analysis, Constraint Satisfaction Problems and Model Checking. In all cases the complexity result specializes to the worst case time complexity of the classical methods