The Complexity of Satisfiability for Sub-Boolean Fragments of ALC

The standard reasoning problem, concept satisfiability, in the basic description logic ALC is PSPACE-complete, and it is EXPTIME-complete in the presence of unrestricted axioms. Several fragments of ALC, notably logics in the FL, EL, and DL-Lite family, have an easier satisfiability problem; sometimes it is even tractable. All these fragments restrict the use of Boolean operators in one way or another. We look at systematic and more general restrictions of the Boolean operators and establish the complexity of the concept satisfiability problem in the presence of axioms. We separate tractable from intractable cases.Comment: 17 pages, accepted (in short version) to Description Logic Workshop 201

Boolean Variation and Boolean Logic BackPropagation

The notion of variation is introduced for the Boolean set and based on which Boolean logic backpropagation principle is developed. Using this concept, deep models can be built with weights and activations being Boolean numbers and operated with Boolean logic instead of real arithmetic. In particular, Boolean deep models can be trained directly in the Boolean domain without latent weights. No gradient but logic is synthesized and backpropagated through layers

Information as Distinctions: New Foundations for Information Theory

The logical basis for information theory is the newly developed logic of partitions that is dual to the usual Boolean logic of subsets. The key concept is a "distinction" of a partition, an ordered pair of elements in distinct blocks of the partition. The logical concept of entropy based on partition logic is the normalized counting measure of the set of distinctions of a partition on a finite set--just as the usual logical notion of probability based on the Boolean logic of subsets is the normalized counting measure of the subsets (events). Thus logical entropy is a measure on the set of ordered pairs, and all the compound notions of entropy (join entropy, conditional entropy, and mutual information) arise in the usual way from the measure (e.g., the inclusion-exclusion principle)--just like the corresponding notions of probability. The usual Shannon entropy of a partition is developed by replacing the normalized count of distinctions (dits) by the average number of binary partitions (bits) necessary to make all the distinctions of the partition

Optical computing by injection-locked lasers

A programmable optical computer has remained an elusive concept. To construct a practical computing primitive equivalent to an electronic Boolean logic, one should find a nonlinear phenomenon that overcomes weaknesses present in many optical processing schemes. Ideally, the nonlinearity should provide a functionally complete set of logic operations, enable ultrafast all-optical programmability, and allow cascaded operations without a change in the operating wavelength or in the signal encoding format. Here we demonstrate a programmable logic gate using an injection-locked Vertical-Cavity Surface-Emitting Laser (VCSEL). The gate program is switched between the AND and the OR operations at the rate of 1 GHz with Bit Error Ratio (BER) of 10e-6 without changes in the wavelength or in the signal encoding format. The scheme is based on nonlinearity of normalization operations, which can be used to construct any continuous complex function or operation, Boolean or otherwise.Comment: 47 pages, 7 figures in total, 2 tables. Intended for submission to Nature Physics within the next two week

Beyond Boolean logic: exploring representation languages for learning complex concepts

We study concept learning for semantically-motivated, set-theoretic concepts. We first present an experiment in which we show that subjects learn concepts which cannot be represented by a simple Boolean logic. We then present a computational model which is similarly capable of learning these concepts,and show that it provides a good fit to human learning curves. Additionally, we compare the performance of several potential representation languages which are richer than Boolean logic in predicting human response distributions. Keywords: Rule-based concept learning; probabilistic model;semantics

A Complementary Resistive Switch-based Crossbar Array Adder

Redox-based resistive switching devices (ReRAM) are an emerging class of non-volatile storage elements suited for nanoscale memory applications. In terms of logic operations, ReRAM devices were suggested to be used as programmable interconnects, large-scale look-up tables or for sequential logic operations. However, without additional selector devices these approaches are not suited for use in large scale nanocrossbar memory arrays, which is the preferred architecture for ReRAM devices due to the minimum area consumption. To overcome this issue for the sequential logic approach, we recently introduced a novel concept, which is suited for passive crossbar arrays using complementary resistive switches (CRSs). CRS cells offer two high resistive storage states, and thus, parasitic sneak currents are efficiently avoided. However, until now the CRS-based logic-in-memory approach was only shown to be able to perform basic Boolean logic operations using a single CRS cell. In this paper, we introduce two multi-bit adder schemes using the CRS-based logic-in-memory approach. We proof the concepts by means of SPICE simulations using a dynamical memristive device model of a ReRAM cell. Finally, we show the advantages of our novel adder concept in terms of step count and number of devices in comparison to a recently published adder approach, which applies the conventional ReRAM-based sequential logic concept introduced by Borghetti et al.Comment: 12 pages, accepted for IEEE Journal on Emerging and Selected Topics in Circuits and Systems (JETCAS), issue on Computing in Emerging Technologie

Experimental Demonstrations of Native Implementation of Boolean Logic Hamiltonian in a Superconducting Quantum Annealer

Experimental demonstrations of quantum annealing with native implementation of Boolean logic Hamiltonians are reported. As a superconducting integrated circuit, a problem Hamiltonian whose set of ground states is consistent with a given truth table is implemented for quantum annealing with no redundant qubits. As examples of the truth table, NAND and NOR are successfully fabricated as an identical circuit. Similarly, a native implementation of a multiplier comprising six superconducting flux qubits is also demonstrated. These native implementations of Hamiltonians consistent with Boolean logic provide an efficient and scalable way of applying annealing computation to so-called circuit satisfiability problems that aim to find a set of inputs consistent with a given output over any Boolean logic functions, especially those like factorization through a multiplier Hamiltonian. A proof-of-concept demonstration of a hybrid computing architecture for domain-specific quantum computing is described.Comment: 12 pages, 11 figure