Clause Density Research Articles

Solution of SAT problems with the adaptive-bias quantum approximate optimization algorithm

The quantum approximate optimization algorithm (QAOA) is a promising method for solving certain classical combinatorial optimization problems on near-term quantum devices. When employing the QAOA to 3-SAT and Max-3-SAT problems, the quantum cost exhibits an easy-hard-easy or easy-hard pattern respectively as the clause density is changed. The quantum resources needed in the hard-region problems are out of reach for current NISQ devices. We show by numerical simulations with up to 14 variables and analytical arguments that the adaptive-bias QAOA (ab-QAOA) greatly improves performance in the hard region of the 3-SAT problems and hard region of the Max-3-SAT problems. For similar accuracy, on average, ab-QAOA needs 3 levels for 10-variable 3-SAT problems as compared to 22 for QAOA. For 10-variable Max-3-SAT problems, the numbers are 7 levels and 62 levels. The improvement comes from a more targeted and more limited generation of entanglement during the evolution. We demonstrate that classical optimization is not strictly necessary in the ab-QAOA since local fields are used to guide the evolution. This leads us to propose an optimization-free ab-QAOA that can solve the hard-region 3-SAT and Max-3-SAT problems effectively with significantly fewer quantum gates as compared to the original ab-QAOA. Our work paves the way for realizing quantum advantages for optimization problems on NISQ devices.

Writing Development in DHH Students: A Bimodal Bilingual Approach.

This article describes the lexical and syntactic development of written narratives in 24 deaf and hard-of-hearing (DHH) students aged between 8 and 18 and takes into account how their varying linguistic backgrounds may predict their written performance. All participants use spoken Swedish, but the study also considers their proficiency in Swedish Sign language, which ranged from zero to fluent. Their narrative texts were analyzed in regard to syntax and lexicon, which demonstrated a strong developmental trend in which increased age predicted more complex texts. Age of acquisition did not predict any writing outcome, which is suggested to occur because all participants were exposed to language early, that is, within the critical time window for language acquisition. Sign language proficiency showed a tendency to predict adjective density and number of clauses, which encourages future research in this area, especially since this connection argues for the benefits of early access to a language and the positive relationship between sign language proficiency and writing.

Proof of the satisfiability conjecture for large $k$

We establish the satisfiability threshold for random $k$-SAT for all $k\ge k_0$, with $k_0$ an absolute constant. That is, there exists a limiting density $\alpha_{\rm sat}(k)$ such that a random $k$-SAT formula of clause density $\alpha$ is with high probability satisfiable for $\alpha\lt \alpha_{\rm sat}$, and unsatisfiable for $\alpha>\alpha_{\rm sat}$. We show that the threshold $\alpha_{\rm sat}(k)$ is given explicitly by the one-step replica symmetry breaking prediction from statistical physics. The proof develops a new analytic method for moment calculations on random graphs, mapping a high-dimensional optimization problem to a more tractable problem of analyzing tree recursions. We believe that our method may apply to a range of random CSPs in the $1$-RSB universality class.

Information Density and the Extraposition of German Relative Clauses.

This paper aims to find a correlation between Information Density (ID) and extraposition of Relative Clauses (RC) in Early New High German. Since surprisal is connected to perceiving difficulties, the impact on the working memory is lower for frequent combinations with low surprisal-values than it is for rare combinations with higher surprisal-values. To improve text comprehension, producers therefore distribute information as evenly as possible across a discourse. Extraposed RC are expected to have a higher surprisal-value than embedded RC. We intend to find evidence for this idea in RC taken from scientific texts from the 17th to 19th century. We built a corpus of tokenized, lemmatized and normalized papers about medicine from the 17th and 19th century, manually determined the RC-variants and calculated a skipgram-Language Model to compute the 2-Skip-bigram surprisal of every word of the relevant sentences. A logistic regression over the summed up surprisal values shows a significant result, which indicates a correlation between surprisal values and extraposition. So, for these periods it can be said that RC are more likely to be extraposed when they have a high total surprisal value. The influence of surprisal values also seems to be stable across time. The comparison of the analyzed language periods shows no significant change.

Comparing Syntactic Complexity: Research Papers Written by Undergraduate Researchers and Published Research Papers

This study compared syntactic complexity between undergraduate students’ research papers and published research papers. A 2 (Groups: experts, students) × 4 (Sections: introduction, methods, results, discussions) mixed factorial analysis of variance (ANOVA) was conducted on each of the following variables: T-unit length, clause density, the frequencies of adverbial clauses, relative clauses, and nominal clauses. The analyses showed there were longer T-units and fewer adverbial clauses in the published papers than in the student papers. Section effects were robust. Qualitative analyses were further conducted to reveal how the student research paper is different from the published papers syntactically. Implications for education were discussed.

Language Sample Analysis of Writing in Children and Adolescents

Competent writing eludes many school children and adolescents, especially those with developmental language and reading disorders. Language sample analysis (LSA) of writing is a powerful but underused assessment tool that can also inform instruction. Similar to LSA of oral language, writing has been analyzed at word, sentence, and text levels using measures that include lexical diversity, sentence length, clause density, grammaticality, productivity, and text organization and content. Unique observations of writing include spelling and literate semantic and syntactic features. This article offers a review of analytic writing measures in the LSA literature from perspectives that include developmental change, language ability differences, relation to quality ratings, practical utility, and effects of genre and task. Writing samples from two 12-year-old students, with and without a language disorder, are used to illustrate application of these measures and suggest potential instruction targets.

Sentence Pattern and Usage in Nigeria’s 2015 Presidential Debate

This study of sentence pattern and usage in Nigeria’s 2015 Presidential Debate identifies and accounts for all occurring major and minor sentences, classifies the major sentences into simple, compound and complex subtypes, determines their typological and thematic distribution, and demonstrates how they were strategically employed to articulate each debater’s points. The data comprises 1876 sentences and was analysed using an improved version of the systemic grammatical model. Results show that major and minor sentences represent 92 and 8 per cent respectively and that while the simple sentence accounted for 77 per cent, compound and complex sentences make up 10 and 13 per cent respectively. Mean length of sentence was 13.6 words and clause density in respect of compound and complex sentences was 1.6 and 2.22 per sentence. Of the seventeen variants of the simple sentence isolated 21 per cent had their elements of structure realised by rankshifted clauses while 20 per cent were affected by multiplicity, mobility and inversion. Though the rest 59 per cent were kernel sentences of the basic SPCA structural pattern, it was not uncommon to find structurally complex groups as elements of clause structure. What determined which of alpha or beta was clause-initially was the focus of the message conveyed. So thematic fronting is not limited to the single clause sentence. Sentence length and type, positioning of clauses or parts thereof, and decisions on conjunctions and finite or non-finite clauses, were greatly governed by theme and the speaker’s mediate goals and grammatical sophistication. These are proofs of the strategic use of the sentence by politicians.

Classifying Cases in Federal Studies. An Illustration of why Political Scientists should do more Cluster Analysis

Typologies are widely used in research on federalism, e.g. to distinguish dual from cooperative or coming-together from holding-together federations. More general, ideal types, archetypes and categories are frequently used in political science research to define concepts and classify cases. As recently as in 2014, Filho et al. pointed out that Cluster Analysis is still hardly used when it comes to developing typologies in political science. Rather, political scientists rely on more intuitive methods or factor analysis. Our paper argues that Cluster Analysis is of great usefulness because it a) focuses on the relationship between cases and not variables and b) draws on empirical data when identifying the clusters. This paper proposes to apply this fruitful approach to the field of federalism to exemplify its major heuristic potential. Furthermore, we emphasize that testing the secondary validity is a crucial step. Our paper provides two original examples from comparative federal politics and public management that illustrate the strength of Cluster Analysis both in testing and generating hypotheses through the establishment of typologies. For both examples, the validity of the Cluster Analysis is tested by checking for correlations between the clusters and the distribution of power. Hence, the typologies established through Cluster Analysis not only define our respective dependent variables related to aspects of intergovernmental coordination within federations and the normative density of evaluation clauses in the Swiss federation, but also offer strong insights in issues of regional autonomy.

Syntactic development in Chilean students’ narrative and expository discourse throughout schooling: different measures of syntactic complexity / Desarrollo sintáctico en narraciones y exposiciones orales de estudiantes chilenos a través de la escolaridad: diferentes mediciones de complejidad sintáctica

While there are studies that describe early syntactic development, less research has focused on oral syntactic development in Spanish throughout schooling according to text structure. This article describes syntactic patterns in Chilean school age students and compares the differences between narrative and expository discourse. The oral production of 462 students aged 5–18 years old from eight schools in a central Chilean region was analysed. Their oral productions were transcribed and segmented at clause and clause package levels; clause package length, clause package density and the clause density were calculated. Results show that students are more productive and construct more complex syntactical discourse as they advance through their education. The results also suggest that syntactic development patterns function differently in narrative and expository discourses. This indicates that students firstly master their syntactic resources before producing narratives. In addition, the implications of using different syntactic measures to describe syntactic development by text structure are explored.

Probing for quantum speedup in spin-glass problems with planted solutions

The availability of quantum annealing devices with hundreds of qubits has made the experimental demonstration of a quantum speedup for optimization problems a coveted, albeit elusive goal. Going beyond earlier studies of random Ising problems, here we introduce a method to construct a set of frustrated Ising-model optimization problems with tunable hardness. We study the performance of a D-Wave Two device (DW2) with up to 503 qubits on these problems and compare it to a suite of classical algorithms, including a highly optimized algorithm designed to compete directly with the DW2. The problems are generated around predetermined ground-state configurations, called planted solutions, which makes them particularly suitable for benchmarking purposes. The problem set exhibits properties familiar from constraint satisfaction (SAT) problems, such as a peak in the typical hardness of the problems, determined by a tunable clause density parameter. We bound the hardness regime where the DW2 device either does not or might exhibit a quantum speedup for our problem set. While we do not find evidence for a speedup for the hardest and most frustrated problems in our problem set, we cannot rule out that a speedup might exist for some of the easier, less frustrated problems. Our empirical findings pertain to the specific D-Wave processor and problem set we studied and leave open the possibility that future processors might exhibit a quantum speedup on the same problem set.

Comparison of measures of morphosyntactic complexity in French-speaking school-aged children

This study examined the validity and reliability of different measures of morphosyntactic complexity, including the Morphosyntactic Complexity Scale (MSCS), a novel adaptation of the Developmental Sentence Scoring, in French-speaking school-aged children. Seventy-three Quebec children from kindergarten to Grade 3 completed a definition task and a narration task. Mean length of utterance (MLU), clause density and MSCS global score, average frequency scores and average complexity scores were calculated from the transcripts of the two contexts. MLU, clause density and MSCS global score were correlated with vocabulary knowledge and narrative skills, and they increased as a function of school level, suggesting that they are valid measures of morphosyntactic complexity. Moreover, the three scores were correlated across contexts, suggesting that they are also reliable measures. However, no MSCS average frequency or average complexity score was found to be both valid and reliable. These findings will guide researchers and practitioners who desire to assess the language skills of French-speaking school-aged children.

Narrative Discourse Impairments in Persian-Speaking Persons with Traumatic Brain Injury: A Pilot Study

Objective: Studies have shown the presence of narrative discourse difficulties in persons with traumatic brain injury (TBI), even in those who do not suffer from aphasia. Yet, there still exist inconsistencies between the results of different studies, in particular at the microlinguistic level. Moreover, a limited number of languages have been studied in this regard. Therefore, this study aimed at examining these skills in Persian-speaking individuals with TBI. The purpose of this pilot study was to analyse the microlinguistic and macrolinguistic skills of these individuals to determine impaired linguistic measures at different levels of narrative discourse. Participants and Methods: Fourteen non-aphasic Persian-speaking persons with TBI (9 with severe TBI and 5 with moderate TBI), aged 19-40 years (mean = 25.84, SD = 5.69), and 61 age-matched healthy adults completed a narrative task. Measures of language productivity, clause density, verbal error ratio, and cohesion ratio were calculated. Also, test-retest and inter-rater reliability coefficients were analysed. Results: The TBI group was impaired in some microlinguistic and all macrolinguistic measures compared to their control peers. Conclusion: The results of this study suggest that multi-level narrative discourse analyses of Persian-speaking individuals with TBI may be useful for speech/language pathologists wishing to evaluate communication disorders in persons with TBI.

An analysis of the relationship between cohesion and clause combination in English discourse employing NLP and data mining approaches

This study examines the relationship between the frequencies of clause combination and the distribution of discourse-pragmatic markers of cohesion in a subsample of the Susanne corpus. It addresses the theory that clause grammar constitutes a form of grammar-cued discourse coherence which functions as an integrated system with other methods of managing coherence in language. Evidence is sought for whether increased clause density in a corpus correlates with a reduction in explicit cohesive devices. To address this, a computational approach is outlined for the coding of cohesion in a corpus, using a semi-automated data mining procedure. To validate this approach, it is compared with cohesion measures on the same data using the NLP tool Coh-Metrix 3.0. The two approaches are shown to positively correlate on a series of measures, suggesting they significantly overlap in quantifying the cohesion construct. The final analysis of the tagged corpus indicates that as frequencies of clause combination increase in a text, the use of explicit lexical cohesive devices decrease. Also, higher frequencies of clause combination positively correlate with an increased use of grammatical cohesive devices. Findings are interpreted as generally aligning with the expectations of the theoretical framework known as the Adaptive Approach to Grammar.

Max 2-SAT with up to 108 qubits

We experimentally study the performance of a programmable quantum annealing processor, the D-Wave One (DW1) with up to 108 qubits, on maximum SAT problem with 2 variables per clause (MAX 2-SAT) problems. We consider ensembles of random problems characterized by a fixed clause density, an external parameter which we tune through its critical value in our experiments. We demonstrate that the DW1 is sensitive to the critical value of the clause density. The DW1 results are verified and compared with akmaxsat, an exact, state-of-the-art algorithm. We study the relative performance of the two solvers and how they correlate in terms of problem hardness. We find that the DW1 performance scales more favorably with problem size and that problem hardness correlation is essentially non-existent. We discuss the relevance and limitations of such a comparison.

Satisfiability-unsatisfiability transition in the adversarial satisfiability problem.

Adversarial satisfiability (AdSAT) is a generalization of the satisfiability (SAT) problem in which two players try to make a Boolean formula true (resp. false) by controlling their respective sets of variables. AdSAT belongs to a higher complexity class in the polynomial hierarchy than SAT, and therefore the nature of the critical region and the transition are not easily parallel to those of SAT and worthy of independent study. AdSAT also provides an upper bound for the transition threshold of the quantum satisfiability problem (QSAT). We present acompletealgorithm for AdSAT, show that 2-AdSAT is in P, and then study two stochastic algorithms (simulated annealing and its improved variant) and compare their performances in detail for 3-AdSAT. Varying the density of clauses α we claim that there is a sharp SAT-UNSAT transition at a critical value whose upper bound is αc≲1.5, suggesting a much stricter upper bound for the QSAT transition than those previously found.

Solution Space Coupling in the Random K-Satisfiability Problem

The random K-satisfiability (K-SAT) problem is very difficult when the clause density is close to the satisfiability threshold. In this paper we study this problem from the perspective of solution space coupling. We divide a given difficult random K-SAT formula into two easy sub-formulas and let the two corresponding solution spaces to interact with each other through a coupling field x. We investigate the statistical mechanical property of this coupled system by mean field theory and computer simulations. The coupled system has an ergodicity-breaking (clustering) transition at certain critical value xd of the coupling field. At this transition point, the mean overlap value between the solutions of the two solution spaces is very close to 1. The mean energy density of the coupled system at its clustering transition point is less than the mean energy density of the original K-SAT problem at the temperature-induced clustering transition point. The implications of this work for designing new heuristic K-SAT solvers are discussed.

Witness of unsatisfiability for a random 3-satisfiability formula

The random 3-satisfiability (3-SAT) problem is in the unsatisfiable (UNSAT) phase when the clause density α exceeds a critical value α(s)≈4.267. Rigorously proving the unsatisfiability of a given large 3-SAT instance is, however, extremely difficult. In this paper we apply the mean-field theory of statistical physics to the unsatisfiability problem, and show that a reduction to 3-XORSAT, which permits the construction of a specific type of UNSAT witnesses (Feige-Kim-Ofek witnesses), is possible when the clause density α>19. We then construct Feige-Kim-Ofek witnesses for single 3-SAT instances through a simple random sampling algorithm and a focused local search algorithm. The random sampling algorithm works only when α scales at least linearly with the variable number N, but the focused local search algorithm works for clause density α>cN(b) with b≈0.59 and prefactor c≈8. The exponent b can be further decreased by enlarging the single parameter S of the focused local search algorithm.

On the concentration of the number of solutions of random satisfiability formulas

ABSTRACTLet Z(F) be the number of solutions of a random k‐satisfiability formula F with n variables and clause density α. Assume that the probability that F is unsatisfiable is for some . We show that (possibly excluding a countable set of “exceptional” α's) the number of solutions concentrates, i.e., there exists a non‐random function such that, for any , we have with high probability. In particular, the assumption holds for all , which proves the above concentration claim in the whole satisfiability regime of random 2‐SAT. We also extend these results to a broad class of constraint satisfaction problems. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 362–382, 2014

Exponential bounds for the random walk algorithm on random planted 3-SAT

Random walk algorithm (RWalkSAT) is one of the simplest and oldest heuristics for satisfiability problems. In contrast to many experimental results, relatively few rigorous analyses of RWalkSAT are available. Up to now, runtime results of small density random 3-SAT have been achieved showing RWalkSAT terminates successfully in linear time up to clause density 1.63. This paper presents a rigorous runtime analysis of RWalkSAT for 3-CNF formulas generated under the planted assignment distribution. It proves that with overwhelming probability, when starting from any initial assignment | x |< 0.9999n, the expected number of steps until RWalkSAT on random planted formulas with a large constant density finds the planted assignment (1, ..., 1) is at least Ω(1.1514 n ) and at most O(n 21.1517 n ).

Exact thresholds for DPLL on random XOR-SAT and NP-complete extensions of XOR-SAT

This paper discusses a model of constraint satisfaction problems known as uniquely extendible constraint satisfaction problems. This model includes and generalizes XOR-SAT, and the model includes an NP-complete problem that appears to share many of the threshold characteristics of random SAT. In this paper we find an exact threshold in the behavior of two versions of DPLL on random instances of this problem. One version uses the unit clause heuristic, and the other uses the generalized unit clause heuristic. Specifically, for DPLL with the unit clause heuristic, we prove that there is a clause density c, smaller than the satisfiability threshold, such that for random instances with density smaller than this threshold, DPLL with unit clause will find a satisfying assignment in linear time, with uniformly positive probability. However, for random instances with density larger than this threshold, DPLL with unit clause will require exponential time, with uniformly positive probability, to find a satisfying assignment. We then find the equivalent threshold density for DPLL with the generalized unit clause heuristic. We also prove the analog of the (2+p)-SAT Conjecture for this class of problems.