A. C. Cem Say

Qualitative system identification: deriving structure from behavior

Abstract Qualitative reasoning programs (which perform simulation, comparative analysis, data interpretation, etc.) either take the model of the physical system to be considered as input, or compose it using a library of model fragments and input information about how to combine them. System identification is the task of creating models of systems, using data about their behaviors. We present the qualitative system identification algorithm QSI, which takes as input a set of qualitative behaviors of a physical system, and produces as output a constraint model of the system. QSIs output is guaranteed to produce its input when simulated. Furthermore, the QSI-made models usually contain meaningful “deep” parameters of the system which do not appear in the input behaviors. Various aspects of QSI and its applicability to diagnosis, as well as the model fragment formulation problem, are discussed.

Sound and complete qualitative simulation is impossible

State-of-the-art qualitative simulators (for instance, QSIM) are known to be sound; no trajectory which is the solution of a concrete equation matching the input can be missing from the output. A simulator which is seen to be incomplete, that is, which produces a spurious prediction for a particular input, can usually be augmented with an additional filter which eliminates that particular class of spurious behaviors, and the question of whether a simulator with purely qualitative input which never predicts spurious behaviors can ever be achieved by adding new filters in this way has remained unanswered until now. We prove that such a sound and complete qualitative simulation algorithm does not exist.

Quantum Finite Automata: A Modern Introduction

We present five examples where quantum finite automata (QFAs) outperform their classical counterparts. This may be useful as a relatively simple technique to introduce quantum computation concepts to computer scientists. We also describe a modern QFA model involving superoperators that is able to simulate all known QFA and classical finite automaton variants.

A new family of nonstochastic languages

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QUANTUM COUNTER AUTOMATA

The question of whether quantum real-time one-counter automata (rtQ1CAs) can outperform their probabilistic counterparts has been open for more than a decade. We provide an affirmative answer to this question, by demonstrating a non-context-free language that can be recognized with perfect soundness by a rtQ1CA. This is the first demonstration of the superiority of a quantum model to the corresponding classical one in the real-time case with an error bound less than 1. We also introduce a generalization of the rtQ1CA, the quantum one-way one-counter automaton (1Q1CA), and show that they too are superior to the corresponding family of probabilistic machines. For this purpose, we provide general definitions of these models that reflect the modern approach to the definition of quantum finite automata, and point out some problems with previous results. We identify several remaining open problems.

Proving the Power of Postselection

It is a widely believed, though unproven, conjecture that the capability of postselection increases the language recognition power of both probabilistic and quantum polynomial-time computers. It is also unknown whether polynomial-time quantum machines with postselection are more powerful than their probabilistic counterparts with the same resource restrictions. We approach these problems by imposing additional constraints on the resources to be used by the computer, and are able to prove for the first time that postselection does augment the computational power of both classical and quantum computers, and that quantum does outperform probabilistic in this context, under simultaneous time and space bounds in a certain range. We also look at postselected versions of space-bounded classes, as well as those corresponding to error-free and one-sided error recognition, and provide classical characterizations. It is shown that NL would equal RL if the randomized machines had the postselection capability.

Finite Automata with Advice Tapes

We define a model of advised computation by finite automata where the advice is provided on a separate tape. We consider several variants of the model where the advice is deterministic or randomized, the input tape head is allowed real-time, one-way, or two-way access, and the automaton is classical or quantum. We prove several separation results among these variants, and establish the relationships between this model and the previously studied ways of providing advice to finite automata.

Computation with narrow CTCs

We examine some variants of computation with closed timelike curves (CTCs), where various restrictions are imposed on the memory of the computer, and the information carrying capacity and range of the CTC. We give full characterizations of the classes of languages recognized by polynomial time probabilistic and quantum computers that can send a single classical bit to their own past. Such narrow CTCs are demonstrated to add the power of limited nondeterminism to deterministic computers, and lead to exponential speedup in constant-space probabilistic and quantum computation.

The Weakness of CTC Qubits and the Power of Approximate Counting

We present results in structural complexity theory concerned with the following interrelated topics: computation with postselection/restarting, closed timelike curves (CTCs), and approximate counting. The first result is a new characterization of the lesser known complexity class BPPpath in terms of more familiar concepts. Precisely, BPPpath is the class of problems that can be efficiently solved with a nonadaptive oracle for the approximate counting problem. Similarly, PP equals the class of problems that can be solved efficiently with nonadaptive queries for the related approximate difference problem. Another result is concerned with the computational power conferred by CTCs, or equivalently, the computational complexity of finding stationary distributions for quantum channels. Using the preceding characterization of PP, we show that any poly(n)-time quantum computation using a CTC of O(log n) qubits may as well just use a CTC of 1 classical bit. This result essentially amounts to showing that one can find a stationary distribution for a poly(n)-dimensional quantum channel in PP.

FINITE AUTOMATA WITH ADVICE TAPES

We define a model of advised computation by finite automata where the advice is provided on a separate tape. We consider several variants of the model where the advice is deterministic or randomized, the input tape head is allowed real-time, one-way, or two-way access, and the automaton is classical or quantum. We prove several separation results among these variants, demonstrate an infinite hierarchy of language classes recognized by automata with increasing advice lengths, and establish the relationships between this and the previously studied ways of providing advice to finite automata.