For a fixed digraph G, the Constraint Satisfaction Problem with the template G, or CSP(G) for short, is the problem of deciding whether a given input digraph H admits a homomorphism to G. The dichotomy conjecture of Feder and Vardi states that CSP(G), for any choice of G, is solvable in polynomial time or NP-complete. This paper confirms the conjecture for a class of oriented trees called speci...

We show that the Schreier sets Sα (α < ω1) satisfy the following dichotomy property. For every hereditary collection F of finite subsets of N, either there exists infinite M = (mi)1 ⊆ N such that Sα(M) = {{mi : i ∈ E} : E ∈ Sα} ⊆ F , or there exist infinite M = (mi)1 , N ⊆ N such that F [N ](M) = {{mi : i ∈ F} : F ∈ F and F ⊂ N} ⊆ Sα.

Holant problems capture a class of Sum-of-Product computations such as counting matchings. It is inspired by holographic algorithms and is equivalent to tensor networks, with counting CSP being a special case. A classification for Holant problems is more difficult to prove, not only because it implies a classification for counting CSP, but also due to the deeper reason that there exist more int...

Constrained encoding has several applications in the synthesis of finite state machines (FSMs). E.g., it can be used to generate asynchronous FSM state assignment that guarantees a critical hazard-free implementation, or to generate synchronous FSM state assignment with minimum PLA implementation. This paper presents ZEDICHO, an original zero-suppressed binary decision diagram (ZBDD) based algo...

We draw attention to a clear dichotomy between small-world networks exhibiting exponential neighborhood growth, and fractal-like networks, where neighborhoods grow according to a power law. This distinction is observed in a number of real-world networks, and is related to the degree correlations and geographical constraints. We conclude by pointing out that the status of human social networks i...

We study the algorithmic complexity of determining whether a system of polynomial equations over a finite algebra admits a solution. We prove that the problem has a dichotomy in the class of finite groupoids with an identity element. By developing the underlying idea further, we present a dichotomy theorem in the class of finite algebras that admit a non-trivial idempotent Maltsev condition. Th...

In this paper, a class of MIMO nonlinear systems are studied. Some frequency domain conditions are established for the property of dichotomy. These kinds of systems can also be viewed as a class of interconnected systems composed of SISO systems through some linear and nonlinear interconnections. A class of nonlinear input and output interconnections are presented. The corresponding condition f...

This essay rejoins Merinda Simmons’s protection of Russell McCutcheon’s critic vs. caretaker dichotomy in her response to my “Can a Critic be a Caretaker Too? Religion, Conflict and Conflict Transformation” (JAAR 2011). While Simmons aims to preserve McCutcheon’s binary as a purportedly benignly unavoidable opposition, I expose the perils of epistemic anti-realism at the heart of that dichotomy...