Equivalence Relations, Invariants, and Normal Forms

For an equivalence relation E on the words over some finite alphabet, we consider the following four problems, listed in order of increasing difficulty. Recognition: Decide whether two words are equivalent. Invariant: Calculate a function constant on precisely the equivalence classes. Normal form: Calculate a particular member of an equivalence class, given an arbitrary member. First member: Calculate the first member of an equivalence class, given an arbitrary member. We consider the questions whether p solutions for the easier problems yield NP solutions for the harder ones, or vice versa. We show that affirmative answers to several of these questions are equivalent to natural principles like NP=co-NP, NP ∩co-NP=P, and the shrinking principle for NP sets. We supplement known oracles with enough new ones to show that all the questions considered have negative answers relative to some oracles. In other words, these questions cannot be answered affirmatively by means of relativizable polynomial-time Turing reductions. Finally, we show that the analogous questions with "p" replaced by "Borel" have negative answers.KeywordsNormal FormEquivalence RelationInvariant ProblemRecursion TheoryNatural PrincipleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Two experiments used post-class formation within-class relational assessment test performances to evaluate whether participants demonstrated preference for certain members of an equivalence class based on the type of relation that existed between class members. In Experiment 1, two 5-node 7-member equivalence classes, consisting entirely of nonsense syllables, were established using the simultaneous protocol. Only 1 of the 6 participants in Experiment 1 formed classes. After class formation, the effects of the different relations between stimuli were evaluated using within-class relational assessment tests, and the 1 participant showed an absolute preference for transitive over equivalence relations, and for baseline over symmetrical relations. Experiment 2 was identical to Experiment 1, except that one of the nonsense syllable stimuli in each class was replaced by a pictorial stimulus. Under these conditions, classes were formed by 5 of 13 participants. During the relational assessment tests, the 5 participants who formed classes demonstrated almost exclusive preferences for transitive relations over equivalence relations and for trained baseline relations over symmetrical relations. Thus, this research demonstrates that the members of equivalence classes are differentially related to each other based on relational type.

The experimenter asked if documented equivalence class membership would influence the development of shared discriminative stimulus function established through simple successive discrimination training. In Experiment 1, equivalence classes were established with two sets of 9 stimuli. Common stimulus functions were then trained within or across the equivalence classes. Greater acquisition rates of the simple discriminations with stimuli drawn from within the equivalence classes were observed. In Experiment 2, a third stimulus set was added with which no equivalence relations were explicitly trained. The findings of Experiment 1 were replicated, but the Set 3 results were inconsistent across subjects. The outcomes of the two experiments demonstrate that equivalence classes have an effect on other behavioral relations which requires further investigation.

Hepaxpert-I is a medical expert system for interpretive analysis of hepatitis A and B serology findings. The knowledge base of HEPAXPERT-I contains 13 rules for hepatitis A and 106 rules for hepatitis B serology. After entering an input pattern of test results, a pattern matching algorithm based on indexing is employed as efficient look-up procedure for the interpretive text. Formally, a partition of the sets of possible serological finding patterns—64 for hepatitis A and 4,096 for hepatitis B serology—induced by an equivalence relation, divides the elements of the sets into disjoint subsets, the equivalence classes. Each equivalence class together with its associated interpretive text is represented as one rule that assigns to every member of the equivalence class one interpretive text. Since September 1, 1989, Hepaxpert-I has been routinely used at the hepatitis serology laboratory of the 2nd Department for Gastroenterology and Hepatology of the University of Vienna Medical School (Vienna General Hospital). By February 28, 1990, 3,408 interpretive reports had been automatically generated and sent out. Beforehand, the system had been validated with 24,242 serological test results. The acceptance of the generated reports by the physicians requiring the findings is extremely high.

We introduce the peak normal form for elements of the Baumslag–Solitar groups BS(p,q). This normal form is very close to the length-lexicographical normal form, but more symmetric. Both normal forms are geodesic. This means the normal form of an element u-1v yields the shortest path between u and v in the Cayley graph. For horocyclic elements the peak normal form and the length-lexicographical normal form coincide. The main result of this paper is that we can compute the peak normal form in polynomial time if p divides q. As consequence we can compute geodesic lengths in this case. In particular, this gives a partial answer to Question 1 in [4] For arbitrary p and q it is possible to compute the peak normal form (length-lexicographical normal form resp.) also the for elements in the horocyclic subgroup and, more generally, for elements which we call hills. This approach leads to a linear time reduction of the problem of computing geodesics to the problem of computing geodesics for Britton-reduced words where the t-sequence starts with t-1 and ends with t.

Two experiments investigated possible interactions between existing equivalence relations among stimuli and the acquisition of simple discriminations involving the same stimuli. During the first phase of Experiment 1, two adult humans learned two sets of six conditional relations among arbitrary visual stimuli designed to produce two independent sets of three, three-member equivalence classes (Set 1 and Set 2). Subjects whose performance met accuracy criteria on tests for equivalence relations proceeded to a second phase in which subjects were taught, via reinforcement, to make the same response to a cluster of three stimuli (e.g., response 1 in the presence of A1, B1, and C1). The clusters of three stimuli were initially consistent with documented equivalence relations for both Set 1 and Set 2 stimuli. In the second half of this phase, the contingencies were realigned such that the simple discriminations being established were consistent with documented equivalence relations for Set 1 stimuli but inconsistent for Set 2 stimuli. Results showed that the simple discriminations were acquired faster for Set 1 stimuli than for Set 2 stimuli. Experiment 2 introduced a third set of stimuli as a control set and systematically replicated Experiment 1 with four subjects. This study found that three of four subjects acquired simple discriminations faster when the cluster of stimuli were drawn from within equivalence classes rather than across equivalence classes. The development of equivalence relations with Set 3 stimuli constrained the utility of the control condition in this experiment. The results are best interpreted as suggesting that existing equivalence relations among stimuli can interact with the development of analytic units such as simple discriminations to either facilitate or retard their development.

On hyperbolic once-punctured-torus bundles IV: Automata for lightning curves

A method of representing knots, links, and singular knots and links by words in a finite alphabet and so-called d-diagrams is given. A representation of the chord diagram algebra by words in a finite alphabet is considered. This method, unlike coding by Gauss diagrams, allows one to consider knots and links simultaneously. An algorithm for recognition of diagrams of classical knots in terms of d-diagrams is given. Bibliography: 9 titles.

We consider several basic problems for texts and show that if the input texts are given by their Lempel-Ziv codes then the problems can be solved deterministically in polynomial time in the case when the original (uncompressed) texts are of exponential size. The growing importance of massively stored information requires new approaches to algorithms for compressed texts without decompressing. Denote by LZ(ω) the version of a string ω produced by Lempel-Ziv encoding algorithm. For given compressed strings LZ(T), LZ(P) we give the first known deterministic polynomial time algorithms to compute compressed representations of the set of all occurrences of the pattern P in T, all periods of T, all palindromes of T, and all squares of T. Then we consider several classical language recognition problems: We show that the first recognition problem has a polynomial time algorithm and the other two problems are NP-hard. We show also that the LZ encoding can be computed on-line in polynomial time delay and small space (i.e. proportional to the size of the compressed text). Also the compressed representation of a patternmatching automaton for the compressed pattern is computed in polynomial time.

A picture over a finite alphabet \(\varGamma \) is a matrix whose entries are drawn from \(\varGamma \). Let \(\pi :\varSigma \rightarrow \varGamma \) be a function between finite alphabets \(\varSigma \) and \(\varGamma \), and let \(V,H\subseteq \varSigma \times \varSigma \) be binary relations over \(\varSigma \). Given a picture \(N\) over \(\varGamma \), the picture satisfiability problem consists in determining whether there exists a picture M over \(\varSigma \) such that \(\pi (M_{ij})=N_{ij}\), and such that the constraints imposed by V and H on adjacent vertical and horizontal positions of M are respectively satisfied. This problem can be easily shown to be NP-complete. In this work we introduce the notion of s-smooth picture. Our main result states the satisfiability problem for s-smooth pictures can be solved in time polynomial on s and on the size of the input picture. With each picture \(N\), one can naturally associate a CNF formula \(F(N)\) which is satisfiable if and only if \(N\) is satisfiable. In our second result, we define an infinite family of unsatisfiable pictures which intuitively encodes the pigeonhole principle. We show that this family of pictures is polynomially smooth. In contrast we show that the formulas which naturally arise from these pictures are hard for bounded-depth Frege proof systems. This shows that there are families of pictures for which our algorithm for the satisfiability for smooth pictures performs exponentially better than certain classical variants of SAT solvers based on the technique of conflict-driven clause-learning (CDCL).

16 - Normal Forms: Near-Identity Transformations

Conditional "if-then" relations between drug (interoceptive) stimuli and visual (exteroceptive) stimuli were taught to 4 normal humans. Interoceptive stimuli were the effects produced by 0.32 mg/70 kg triazolam (a prototypical benzodiazepine) and placebo (lactose-filled capsules); exteroceptive stimuli were black symbols on white flash cards. Following the training of the prerequisite conditional relations, tests of emergent relations were conducted between exteroceptive stimuli and between interoceptive and exteroceptive stimuli. Equivalence relations emerged immediately without explicit training for all 4 subjects. Accuracy of responding during the interoceptive-exteroceptive equivalence tests and subjects' self-reports showed consistent discrimination between the drug effects of triazolam and placebo. Finally, a generalization test assessed whether a novel visual stimulus presented in the context of the placebo (i.e., no drug) would generalize to visual stimuli belonging to the placebo stimulus class. All 3 subjects who completed this test reliably chose the visual stimuli belonging to the placebo class and not the visual stimuli belonging to the triazolam stimulus class. The development of equivalence relations between interoceptive and exteroceptive stimuli demonstrates that private and public stimulus events can emerge as members of the same equivalence class. Theoretical and clinical implications are discussed.

For fixed integers r,l >= 0, a graph G is called an (r,l)-graph if the vertex set V(G) can be partitioned into r independent sets and l cliques. Such a graph is also said to have cochromatic number r+l. The class of (r,l) graphs generalizes r-colourable graphs (when l=0) and hence not surprisingly, determining whether a given graph is an (r,l)-graph is NP-hard even when r >= 3 or l >= 3 in general graphs. When r and ell are part of the input, then the recognition problem is NP-hard even if the input graph is a perfect graph (where the Chromatic Number problem is solvable in polynomial time). It is also known to be fixed-parameter tractable (FPT) on perfect graphs when parameterized by r and l. I.e. there is an f(r+l) n^O(1) algorithm on perfect graphs on n vertices where f is a function of r and l. Observe that such an algorithm is unlikely on general graphs as the problem is NP-hard even for constant r and l. In this paper, we consider the parameterized complexity of the following problem, which we call Vertex Partization. Given a perfect graph G and positive integers r,l,k decide whether there exists a set S subset or equal to V(G) of size at most k such that the deletion of S from G results in an (r,l)-graph. This problem generalizes well studied problems such as Vertex Cover (when r=1 and l=0), Odd Cycle Transversal (when r=2, l=0) and Split Vertex Deletion (when r=1=l). 1. Vertex Partization on perfect graphs is FPT when parameterized by k+r+l. 2. The problem, when parameterized by k+r+l, does not admit any polynomial sized kernel, under standard complexity theoretic assumptions. In other words, in polynomial time, the input graph cannot be compressed to an equivalent instance of size polynomial in k+r+l. In fact, our result holds even when k=0. 3. When r,ell are universal constants, then Vertex Partization on perfect graphs, parameterized by k, has a polynomial sized kernel.

The maximization of a full-rank quadratic form over a finite alphabet is NP-hard in both a worst-case sense and an average sense. Interestingly, if the rank of the form is not a function of the problem size, then it can be maximized in polynomial time. An algorithm for the efficient computation of the binary vector that maximizes a rank-deficient quadratic form is developed based on an analytic procedure. Auxiliary spherical coordinates are introduced and the multi-dimensional space is partitioned into a polynomial-size set of regions; each region corresponds to a distinct binary vector. The binary vector that maximizes the rank-deficient quadratic form is shown to belong to the polynomial-size set of candidate vectors. Thus, the size of the feasible set is efficiently reduced from exponential to polynomial.

The purpose of this study was to examine the transfer of consequential (reinforcement and punishment) functions through equivalence relations. In Experiment 1, 9 subjects acquired three three-member equivalence classes through matching-to-sample training using arbitrary visual forms. Comparison stimuli were then given conditioned reinforcement or punishment functions by pairing them with verbal feedback during a sorting task. For 8 of the 9 subjects, trained consequential functions transferred through their respective equivalence classes without additional training. In Experiment 2, transfer of function was initially tested before equivalence testing per se. Three of 4 subjects showed the transfer without a formal equivalence test. In Experiment 3, 3 subjects were given training that gave rise to six new three-member conditional equivalence classes. For 2 of the subjects, the same stimulus could have either a reinforcement or punishment function on the basis of contextual cues that defined its class membership. Experiment 4 assessed whether equivalence training had established general or specific consequential functions primarily by adding novel stimuli in the transfer test. Subjects treated even novel feedback stimuli in the transfer test as consequences, but the direction of consequential effects depended upon the transfer of specific consequential functions through equivalence relations.