Spiking neural P (SNP) system is a computational framework that operates through parallel and distributed mechanisms, drawing inspiration from the architectural organization and operational principles of neural networks in the human brain. However, it does not consider the self-inhibition regulation function of neurons after activation in the human brain. This study constructs a self-inhibition SNP (SISNP) system that, upon consecutive neuronal activations, triggers the expression of intrinsic inhibitory signals within the same neuron. This process reduces or abolishes the neuron’s ability to respond to subsequent stimuli. It is also demonstrated that the SISNP system possesses Turing machine equivalence in generating and accepting numbers, as well as in performing functional computation, which provides some interpretability. The computational efficiency of the SISNP system is demonstrated by constructing a solution to the SAT problem. The SISNP system offers more refined control mechanisms, enhances the self-regulatory capacity, and improves computational stability.
We study the problem of finding an element of rank O(log n) among n elements with high probability in the presence of persistent random comparison faults. In this fault model, the outcome of comparing two elements is wrong with a constant error probability p<12, comparing the same pair of elements multiple times always gets the same outcome, and the comparison faults between different pairs of elements are independent. The state-of-the-art methods attain O(nlog n) time for p < 1/4 and nO(1) time for 1/4 ≤ p < 1/2. In this paper, we develop an O(n)-time algorithm that, for any constant error probability p<12, finds an element of rank O(log n) among n elements with high probability. We also show that no algorithm can find an element of rank o(log n) with high probability. Moreover, the O(n) time is optimal since Ω(n) sample elements are necessary to guarantee the existence of an element of rank O(log n) with high probability.
Graph search paradigms such as Maximum Cardinality Search (MCS) play a fundamental role in the algorithmic study of chordal graphs. A classical question in this area is the End-Vertex Problem (EVP), which asks whether a given vertex can appear last in a graph search ordering of a graph. More generally, the Partial Search Order Problem (PSOP) asks whether there exists an S-ordering consistent with a given partial order on the vertex set, for a graph search paradigm S.We investigate the PSOP for MCS on chordal graphs and provide efficient algorithmic solutions that unify and extend several previously studied problems. First, we present a simple linear-time algorithm for the EVP of MCS on chordal graphs. Our approach avoids complex data structures by introducing a new variant of MCS, called MCS-Ultra, which incorporates a carefully designed bonus allocation scheme to guide the search order while preserving linear-time complexity.Building on this perspective, we then study the PSOP of MCS on chordal graphs. We develop a polynomial-time dynamic programming algorithm that resolves an open problem posed by Scheffler [WG 2022]. The algorithm is based on a structural decomposition on clique graphs, called the layer structure, and a detailed analysis of the structural properties of MCS orderings in chordal graphs.
We provide the first generalized game characterization of van Glabbeek’s linear-time–branching-time spectrum with silent steps. In our quantitative formulation, one multi-dimensional energy game can be used to characterize and decide a wide array of weak behavioral equivalences between stability-respecting branching bisimilarity and weak trace equivalence in one go. Deciding all notions this way takes time exponential in the number of states. To establish correctness, we relate attacker-winning energy budgets and distinguishing sublanguages of Hennessy–Milner logic that we characterize by eight dimensions of formula expressiveness.
Two-way finite automata with quantum and classical states (2QCFA's), introduced by Ambainis and Watrous, extend deterministic two-way finite automata with a constant-size quantum register and provide a natural hybrid model of computation. In this paper, we introduce two-way nondeterministic finite automata with quantum and classical states (2QCNFA's), extending this framework to the nondeterministic setting. We show that 2QCNFA's are closed under union, intersection, and marked concatenation, and that they can recognize certain comparison languages whose status for 2QCFA's remains open. We further initiate the study of decision problems for this model by proving that 2QCNFA's can recognize the set of invalid computations of a deterministic Turing machine, which implies that the universality problem is undecidable. In addition, we investigate minimization and descriptional-complexity questions, showing that there is no approximation algorithm for minimization between 2QCNFA's and classical models such as DFAs and context-free grammars, and that the corresponding trade-offs are non-recursive.
We discuss the status of the question of the number of atoms in a measure algebra first introduced by Levin and V'yugin and later studied by V'yugin in subsequent work. Although it has been claimed in print that there is a countably infinite number of atoms in this measure algebra, the only published proof of this result is incomplete and a more recent unpublished proof does not appear to resolve the issue.
Quantum entanglement can be employed in some distributed communication tasks to reduce computation and communication overheads, yielding speedups over their classical counterparts. However, existing quantum Byzantine protocols often entail excessive communication and entanglement resource consumption. Conversely, classical schemes face insecurity with the impending reality of quantum computers. To address these challenges, we present a novel quantum Detectable Weak Byzantine Agreement (DWBA) protocol with high practicability, security and optimal fault-tolerance bound. For all n players, our protocol can complete the consensus of infinite classical information via fixed n+1-entangled qubits and a digest function, which requires infinite entanglement resources in previous schemes. Moreover, our protocol can tolerate arbitrary t (t < n) faulty players without necessitating any additional initial setup information (e.g., digital signature, private random coins). In terms of efficiency, our protocol only requires O(1) rounds and O((n−t)2) communication complexity, constituting an order of magnitude improvement over previous protocols. In addition, the DWBA problem solved by our protocol is inherently stronger than the 3-player Detectable Byzantine Agreement (DBA) problem solved by previous protocols, and in a weaker but sufficiently practical model, our protocol can achieve Weak Byzantine Agreement (WBA).
The clearing problem in the context of financial networks, is the computational challenge of evaluating a firms' exposure to systemic risk, i.e, a firms' ratio of assets by liabilities termed as clearing recovery rate. When only simple debt contracts are involved, the problem is tractable, while under the presence of complex derivative contracts called credit default swaps (CDSes), the problem is FIXP-complete and finding an approximate e-solution is PPAD-hard for a small, yet unspecified, constant e. We present a direct reduction from the PURE-CIRCUIT problem to the clearing problem involving CDSes and establish an improved explicit inapproximability bound for e-solutions. We prove root that finding e-solutions is PPAD-hard for e <= 2-43. The intractability persists even when central clearing authorities are introduced. Additionally we provide an optimisation-based framework for computing exact solutions to the problem with central clearing authorities and a simple polynomial-time algorithm when these authorities are coupled with covered CDSes.
The exchanged hypercube EH(s, t) is a cost-efficient variant of the hypercube. It preserves desirable properties such as strong connectivity, small diameter, and recursive decomposition while significantly reducing interconnection complexity. In this paper, we analyze star-based fault-tolerance parameters of EH(s, t). We first determine the exact values of the star-structure connectivity and star-substructure connectivity of EH(s, t). In particular, in the substructure case, the size of the star is allowed to reach its theoretical maximum value, thereby extending the results of Liu et al. [Comput. J. 67 (2024) 527-536]. Next, we investigate the r-component structure connectivity and r-component substructure connectivity, determining the 3-component star-structure and star-substructure connectivity of EH(s, t). We then study the recently proposed h-step-limited star connectivity, establishing the exact 0, 1, and 2-step-limited star connectivity of EH(s, t). We further generalize this notion to the r-component h-step-limited star connectivity. For this generalized parameter, we determine the 3-component 1-step-limited and 2-step-limited star connectivity of EH(s, t). These results provide a systematic understanding of star-type failures in exchanged hypercubes. They cover both classical star structures and step-limited star structures. All results also apply to the dual cube DCn = EH(n, n).
Several quantum Markov chains and temporal quantum logics have been considered in the literature. In this paper we propose a discrete time classical-quantum state quantum Markov chain (cqQMC) with super-operators over a finite dimensional Hilbert space H. The cqQMC is related to previous proposals but yet with a different semantics in that states are pairs of a classical and a quantum state. We define a quantum temporal logic (QCTL) where propositions are pairs of a classical state and a projection over H. We demonstrate how a cqQMC can be translated to a discrete time Markov chain (DTMC). The translation is sound and complete in that model checking any QCTL formula for a cqQMC can equivalently be carried out by checking the same formula for the inferred DTMC. Taking advantage of the finite classical control we show that even though a cqQMC may have infinitely many states then under certain restrictions the inferred DTMC is finite, and hence model checking is decidable.
Numerical spiking neural P (NSNP) systems serve as a Turing complete numerical paradigm within the realm of spiking neural P (SNP) systems, whereby continuous production functions act as activation functions to describe the firing behavior of a neuron, and a neuron can fire irrespective of the magnitude of the weighted sum of inputs fed into the production function. However, NSNP systems lack the firing threshold feature, a distinguishing one not only of SNP systems but also of artificial and biological neural networks. In this work, aiming to develop NSNP systems that incorporate the firing threshold feature without compromising their computational power, a new class-variant of NSNP systems, termed NSNP systems with thresholds (NSNPT systems), is considered; here, each neuron is associated with a threshold, and a neuron can fire only if its production value computed by the production function is not less than the threshold. We examine the computational power of NSNPT systems by demonstrating the Turing completeness of these systems both in the context of number generating or number accepting, and even the latter can be achieved when each neuron is restricted to contain only one linear function of a single variable. Finally, a matrix representation and matrix-based simulation algorithm are developed to experimentally validate NSNPT systems.
Let G = (V, E) be a graph and let K|V| denote the complete graph on the same vertex set as G. A potential edge set of G is a subset of E(K|V|) such that the subgraph it induced in K|V| forms a set of paths. A graph G is k-edge Hamiltonian-connected (k-EHC) if, for any potential edge set E with 1 <= |E| <= k, G U E contains a Hamiltonian cycle that includes every edge of E. This notion generalizes the concept of Hamiltonian connectedness. Determining whether a graph is k-EHC is NP-complete, even for k = 2. In [J. Graph Theory 69 (2012) 241-250], a characterization of 2-EHC graphs was provided. This paper expands upon that result by characterizing graphs that are k-EHC for k >= 1. As an application of this characterization, we further demonstrate that a series of network classes based on the enhanced structure of hypercubes, known as augmented cubes, meet the k-EHC property for k = 2, 3.
Spiking neural P systems (SNP systems), a class of parallel distributed computational models inspired by biological neurons, have become an important research direction in biocomputing in recent years due to their biological interpretability and low-power computing. There are many studies on the expressive power of SNP system variants, but their computational efficiency is not high in terms of resource overhead. Inspired by the biological winner-take-all (WTA) computation mechanism, this study proposes spiking neural P systems incorporating WTA mechanism (WTASNP systems). Competing neuron nodes are introduced into the WTASNP systems to realize the competitive selection and inhibition control mechanisms of the WTA computation. After competition, only the winner neuron is allowed to emit spikes, while the loser neurons are inhibited. It is proved that the WTASNP systems have Turing universality. Furthermore, it reduces the computational resource requirements for solving the NP-complete SAT problem with SNP systems from O(n2) or O(2n) complexity levels down to linearly solvable O(n). The WTASNP systems not only effectively preserve the preamble spike information, but also inhibits the loser neuron spike issuance by the WTA computational mechanism of competing neurons, reduces redundant computation to avoid neuron over-excitation, and improves the computational efficiency and expressive power.
Inspired by Zheng and Weihrauch's arithmetic(al) hierarchy of real numbers, we investigate a hierarchy of computability levels for functions from R -> R. In particular, we seek to capture the classes of estimable, lower semicomputable, and upper semicomputable functions, understand how they are connected, and whether they can be generalized to important larger classes. We establish the lattice structure of our hierarchy along with its properties under various analytical and algebraic operations that are useful for calculations, particularly function composition. We also draw a connection to the arithmetic hierarchy of sets of natural numbers, extending classical results such as Shoenfield's limit lemma by allowing oracles with infinite input. Our main contribution is to make real hypercomputation results of Ziegler and Brattka accessible to computer scientists who are not computable analysts, which has direct applications to areas such as algorithmic information theory, descriptive set theory, and scientific computing.
Alternating-time Temporal Logic ATL is a key framework for reasoning about strategic and coalitional ability. Built into standard ATL are several assumptions, including neverending interaction (the system never stops), joint action determinism (the next state is uniquely determined by the current state and actions chosen by all agents) and independence of choices (an agent's choice does not depend on choices of other agents). Recently, combinations of these assumptions have been relaxed in the case of Coalition Logic-the next time fragment of ATL-resulting in eight different logics. In this paper we do the same for full ATL. We define a generalized semantics and prove some key semantic properties, laying the groundwork for axiomatic completeness. Along the way, we provide proofs of some semantic properties of standard ATL that are currently missing in the literature. In particular, we show that the interpretations based on positional and historical strategies are equivalent, in the general setting. This is "well known" for standard ATL, but perhaps surprisingly, it turns out that there is no proof of it in the literature. We also show that the semantics based on strategies and co-strategies are equivalent. All of this hinges on a fixed point characterization of "globally" and "until", which we provide (the latter case is missing even for full ATL in the literature).
The maximum length of the shortest string accepted by an n-state two-way finite automaton is currently known to be at least of the order Omega(1.626n) and at most ( 2n )-1 = O(root 1 n 4n). In this paper, n+1 a family of n-state automata with shortest accepted strings of length 34 & sdot;2n-1 is constructed, thus improving the lower bound. Also a modest improvement to the upper bound is made: the length of the shortest accepted string is at most ( 2n )-n. For the special case of direction-determinate n+1 automata (those that always remember in the current state whether the last move was to the left or to the right), the maximum length of the shortest accepted string is determined precisely as (n)-1= Theta(1 & LeftFloor;n root n2n). 2 & RightFloor;
We explore two extensions of house allocation (HA) by enabling the allocation of each object (with limited or unlimited copies) to multiple agents. One extension is associated with matroid constraints, called the Matroid House Allocation Problem (MHA), while the other is associated with knapsack constraints, and is called the Knapsack House Allocation Problem (KHA). We investigate the highly general setting for both problems, where agents possess weights or priorities and can express indifference towards objects. Our focus is on designing (universally) truthful and Pareto optimal mechanisms to compute a maximum weighted matching. We propose a tight 2-approximate deterministic mechanism that is both truthful and Pareto optimal for the Matroid House Allocation Problem (MHA). Additionally, we develop a randomized mechanism that is universally truthful and Pareto optimal, with an approximation ratio of e-1 for the same problem. This ratio of e e e-1 is the best achievable among universally truthful and Pareto optimal mechanisms, assuming the mechanism is also non-bossy. These results represent significant advancements over previous findings and are achieved through different techniques. Furthermore, we apply our MHA mechanisms to the Online Bipartite Matching Problem and Job Recruitment Problems by incorporating matroid constraints, resulting in optimal algorithms for both problems. For the Knapsack House Allocation Problem (KHA), we design a (4, 2)-approximate mechanism that is universally truthful and Pareto optimal for cases with equal capacities. Additionally, we provide a universally truthful mechanism with a constant approximation for KHA. Moreover, we develop a 4-approximate mechanism for equal capacity KHA with strict preferences.
