Expressive power and complexity of disjunctive datalog under the stable model semantics

1994 ◽  
pp. 83-103 ◽  
Author(s):  
Thomas Eiter ◽  
Georg Gottlob ◽  
Heikki Mannila
Keyword(s):  
Expressive Power ◽  
Stable Model

On the Implementation of a Logic Language for NP Search and Optimization Problems

2009 ◽  
pp. 798-804
Author(s):  
Sergio Greco ◽  
Cristian Molinaro ◽  
Irina Trubitsyna ◽  
Ester Zumpano
Keyword(s):  
Optimization Problems ◽  
Vertex Cover ◽  
Expressive Power ◽  
Stable Model ◽  
Restricted Version ◽  
Vertex Cover Problem ◽  
Cover Problem ◽  
Exclusive Disjunction

It is well known that NP search and optimization problems can be formulated as DATALOG¬ (datalog with unstratified negation; Abiteboul, Hull, & Vianu, 1994) queries under nondeterministic stable-model semantics so that each stable model corresponds to a possible solution (Gelfond & Lifschitz, 1988; Greco & Saccà, 1997; Kolaitis & Thakur, 1994). Although the use of (declarative) logic languages facilitates the process of writing complex applications, the use of unstratified negation allows programs to be written that in some cases are neither intuitive nor efficiently valuable. This article presents the logic language NP Datalog, a restricted version of DATALOG¬ that admits only controlled forms of negation, such as stratified negation, exclusive disjunction, and constraints. NP Datalog has the same expressive power as DATALOG¬, enables a simpler and intuitive formulation for search and optimization problems, and can be easily translated into other formalisms. The example below shows how the vertex cover problem can be expressed in NP Datalog.

The expressive power of unique total stable model semantics

1997 ◽  
pp. 849-859 ◽  
Author(s):  
Francesco Buccafurri ◽  
Sergio Greco ◽  
Domenico Saccà
Keyword(s):  
Expressive Power ◽  
Stable Model

scholarly journals Complexity and Expressive Power of Disjunction and Negation in Limit Datalog

2020 ◽  
Vol 34 (03) ◽  
pp. 2862-2869 ◽  
Author(s):  
Mark Kaminski ◽  
Bernardo Cuenca Grau ◽  
Egor V. Kostylev ◽  
Ian Horrocks
Keyword(s):  
Data Analysis ◽  
Expressive Power ◽  
Arithmetic Functions ◽  
Stable Model ◽  
Declarative Language ◽  
Ordered Structures ◽  
Disjunctive Programs ◽  
Linearity Condition

Limit Datalog is a fragment of Datalogℤ—the extension of Datalog with arithmetic functions over the integers—which has been proposed as a declarative language suitable for capturing data analysis tasks. In limit Datalog programs, all intensional predicates with a numeric argument are limit predicates that keep maximal (or minimal) bounds on numeric values. Furthermore, to ensure decidability of reasoning, limit Datalog imposes a linearity condition restricting the use of multiplication in rules. In this paper, we study the complexity and expressive power of limit Datalog programs extended with disjunction in the heads of rules and non-monotonic negation under the stable model semantics. We show that allowing for unrestricted use of negation leads to undecidability of reasoning. Decidability can be restored by stratifying the use of negation over predicates carrying numeric values. We show that the resulting language is Π2EXP -complete in combined complexity and that it captures Π2P over ordered structures in the sense of descriptive complexity.We also provide a study of several fragments of this language: we show that the complexity and expressive power of the full language are already reached for disjunction-free programs; furthermore, we show that semi-positive disjunctive programs are coNEXPcomplete and that they capture coNP.

A Study of the Planning of Water Pollution Control for Second Songhua River in China

1991 ◽  
Vol 23 (1-3) ◽  
pp. 41-48
Author(s):  
Yin Jun
Keyword(s):  
Water Pollution ◽  
Mathematical Model ◽  
Pollution Control ◽  
Sewage Treatment Plant ◽  
Treatment Plant ◽  
Initial Structure ◽  
Stable Model ◽  
Control Plan ◽  
Songhua River ◽  
The Second Songhua River

The paper takes the Second Songhua River as an object for research and selects Thomas's BOD-DO stable model as the initial structure on the basis of overall investigations and analyses on water pollution in every reach. In view of the characteristics of the river being located at the north, values k’1, k’2 and k’3 in dry season of winter were determined and calculated, and a series analyses have been made. The self-purification ability of the river and the total elimination amount of the main pollutants BOD5 were also calculated. In order to minimize the required cost, we distributed the cost to the main pollution sources, which are to be controlled. We firstly set a cost function of sewage treatment plant by series design and calculated the related cost parameters, then calculated two kinds of optimal distributing models of BOD5 elimination, which were a mathematical model of extreme value of conditions and a matrix mathematical model. Now they have been applied to the practical pollution control plan for the Second Songhua River.

Recursive generation in language

2017 ◽  
Author(s):  
David J. Lobina
Keyword(s):  
Mathematical Logic ◽  
Production System ◽  
General Property ◽  
Recent Literature ◽  
Expressive Power ◽  
The Self ◽  
Recursive Structure ◽  
Embedded Structures ◽  
Mechanical Procedure ◽  
The 1950S

The introduction of recursion into linguistics was the result of applying some of the results of mathematical logic to the study of language. In particular, recursion was introduced in the 1950s as a general property of the mechanical procedure underlying the grammar, in order to account for language’s discrete infinity and expressive power—in the 1950s, this mechanical procedure was a production system, whereas more recently, of course, it is the set-operator merge. Unfortunately, the recent literature has confused the general recursive property of a grammar with specific instances of (recursive) rules/operations within a grammar; more worryingly still, there has been a general conflation of these recursive rules with some of the self-embedded structures these rules can generate, adding to the confusion. The conflation is manifold but always fallacious. Moreover, language manifests a much more generally recursive structure than is usually recognized: bundles of the universal (Specifier)-Head-Complement(s) geometry.

scholarly journals Probability logic: A model-theoretic perspective

2020 ◽  
Author(s):  
M Pourmahdian ◽  
R Zoghifard
Keyword(s):  
Characterization Theorem ◽  
Expressive Power ◽  
Probability Logic ◽  
Theoretic Analysis ◽  
Compactness Property ◽  
Probability Models ◽  
Finitely Additive Probability ◽  
Additive Probability ◽  
Basic Probability ◽  
Abstract Logics

Abstract This paper provides some model-theoretic analysis for probability (modal) logic ($PL$). It is known that this logic does not enjoy the compactness property. However, by passing into the sublogic of $PL$, namely basic probability logic ($BPL$), it is shown that this logic satisfies the compactness property. Furthermore, by drawing some special attention to some essential model-theoretic properties of $PL$, a version of Lindström characterization theorem is investigated. In fact, it is verified that probability logic has the maximal expressive power among those abstract logics extending $PL$ and satisfying both the filtration and disjoint unions properties. Finally, by alternating the semantics to the finitely additive probability models ($\mathcal{F}\mathcal{P}\mathcal{M}$) and introducing positive sublogic of $PL$ including $BPL$, it is proved that this sublogic possesses the compactness property with respect to $\mathcal{F}\mathcal{P}\mathcal{M}$.

The Complexity of Model-Checking Tail-Recursive Higher-Order Fixpoint Logic

2021 ◽  
Vol 178 (1-2) ◽  
pp. 1-30
Author(s):  
Florian Bruse ◽  
Martin Lange ◽  
Etienne Lozes
Keyword(s):  
Model Checking ◽  
Lower Bounds ◽  
Expressive Power ◽  
Higher Order ◽  
Order Logic ◽  
High Complexity ◽  
Transition Systems ◽  
Recursive Formulas ◽  
Second Order Logic ◽  
Monadic Second Order Logic

Higher-Order Fixpoint Logic (HFL) is a modal specification language whose expressive power reaches far beyond that of Monadic Second-Order Logic, achieved through an incorporation of a typed λ-calculus into the modal μ-calculus. Its model checking problem on finite transition systems is decidable, albeit of high complexity, namely k-EXPTIME-complete for formulas that use functions of type order at most k < 0. In this paper we present a fragment with a presumably easier model checking problem. We show that so-called tail-recursive formulas of type order k can be model checked in (k − 1)-EXPSPACE, and also give matching lower bounds. This yields generic results for the complexity of bisimulation-invariant non-regular properties, as these can typically be defined in HFL.

scholarly journals Audio-to-score singing transcription based on a CRNN-HSMM hybrid model

2021 ◽  
Vol 10 ◽  
Author(s):  
Ryo Nishikimi ◽  
Eita Nakamura ◽  
Masataka Goto ◽  
Kazuyoshi Yoshii
Keyword(s):  
Hybrid Model ◽  
Viterbi Algorithm ◽  
Language Model ◽  
Expressive Power ◽  
Acoustic Model ◽  
Musical Language ◽  
Musical Score ◽  
Singing Voice ◽  
Generative Process ◽  
Music Signal

This paper describes an automatic singing transcription (AST) method that estimates a human-readable musical score of a sung melody from an input music signal. Because of the considerable pitch and temporal variation of a singing voice, a naive cascading approach that estimates an F0 contour and quantizes it with estimated tatum times cannot avoid many pitch and rhythm errors. To solve this problem, we formulate a unified generative model of a music signal that consists of a semi-Markov language model representing the generative process of latent musical notes conditioned on musical keys and an acoustic model based on a convolutional recurrent neural network (CRNN) representing the generative process of an observed music signal from the notes. The resulting CRNN-HSMM hybrid model enables us to estimate the most-likely musical notes from a music signal with the Viterbi algorithm, while leveraging both the grammatical knowledge about musical notes and the expressive power of the CRNN. The experimental results showed that the proposed method outperformed the conventional state-of-the-art method and the integration of the musical language model with the acoustic model has a positive effect on the AST performance.

scholarly journals Categorification of the Müller-Wichards System Performance Estimation Model: Model Symmetries, Invariants, and Closed Forms

Systems ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 6
Author(s):  
Allen D. Parks ◽  
David J. Marchette
Keyword(s):  
Closed Form ◽  
System Performance ◽  
Algebraic Method ◽  
Expressive Power ◽  
Computer Applications ◽  
Performance Estimation ◽  
Parallel Computer ◽  
Estimation Model ◽  
Data Movement ◽  
Fundamental Symmetry

The Müller-Wichards model (MW) is an algebraic method that quantitatively estimates the performance of sequential and/or parallel computer applications. Because of category theory’s expressive power and mathematical precision, a category theoretic reformulation of MW, i.e., CMW, is presented in this paper. The CMW is effectively numerically equivalent to MW and can be used to estimate the performance of any system that can be represented as numerical sequences of arithmetic, data movement, and delay processes. The CMW fundamental symmetry group is introduced and CMW’s category theoretic formalism is used to facilitate the identification of associated model invariants. The formalism also yields a natural approach to dividing systems into subsystems in a manner that preserves performance. Closed form models are developed and studied statistically, and special case closed form models are used to abstractly quantify the effect of parallelization upon processing time vs. loading, as well as to establish a system performance stationary action principle.

Unsatisfiable Core Analysis and Aggregates for Optimum Stable Model Search

2020 ◽  
Vol 176 (3-4) ◽  
pp. 271-297
Author(s):  
Mario Alviano ◽  
Carmine Dodaro
Keyword(s):  
State Of The Art ◽  
Answer Set Programming ◽  
Efficient Algorithms ◽  
Satisfiability Problem ◽  
Core Analysis ◽  
Stable Model ◽  
Maximum Satisfiability ◽  
Model Search ◽  
Unsatisfiable Core ◽  
Stable Models

Many efficient algorithms for the computation of optimum stable models in the context of Answer Set Programming (ASP) are based on unsatisfiable core analysis. Among them, algorithm OLL was the first introduced in the context of ASP, whereas algorithms ONE and PMRES were first introduced for solving the Maximum Satisfiability problem (MaxSAT) and later on adapted to ASP. In this paper, we present the porting to ASP of another state-of-the-art algorithm introduced for MaxSAT, namely K, which generalizes ONE and PMRES. Moreover, we present a new algorithm called OLL-IN-ONE that compactly encodes all aggregates of OLL by taking advantage of shared aggregate sets propagators. The performance of the algorithms have been empirically compared on instances taken from the latest ASP Competition.

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