Computational complexity of linear constraints over the integers

A causal and temporal reasoning model and its use in drug therapy applications

In recent years, several constraintdbased temporal reasoning frameworks have been proposed. They consider temporal points or intervals as domain elements linked by temporal constraints. Temporal reasoning in these systems is based on constraint propagation. In this paper, we argue that a language based on constraint propagation can be a suitable tool for expressing and reasoning about temporal problems. We concentrate on Constraint Logic Programming (CLP) which is a powerful programming paradigm combining the advantages of Logic Programming and the efficiency of constraint solving. However, CLP presents some limitations in dealing with temporal reasoning. First, it uses an “arc consistency” propagation algorithm which is embedded in the inference engine, cannot be changed by the user, and is too weak in many temporal frameworks. Second, CLP is not able to deal with qualitative temporal constraints. We present a general meta CLP architecture which maintains the advantages of CLP, but overcomes these two main limitations. Each architectural level is a finite domain constraint solver (CLP(FD)) that reasons about constraints of the underlying level. Based on this conceptual architecture, we extend the CLP(FD) language and we specialize the extension proposed on Vilain and Kautz’s Point Algebra, on Allen’s Interval Algebra and on the STP framework by Dechter, Meiri and Pearl. In particular, we show that we can cope effectively with disjunctive constraints even in an intervaldbased framework.

Many different applications in different areas need to deal with both: databases, in order to take into account large amounts of structured data; and quantitative and qualitative temporal constraints about such data. We propose an approach that extends: temporal databases and artificial intelligence temporal reasoning techniques and integrate them in order to face such a need. Regarding temporal reasoning, we consider some results that we proved recently about efficient query answering in the Simple Temporal Problem framework and we extend them in order to deal with partitioned sets of constraints and to support relational database operations. Regarding databases, we extend the relational model in order to consider also qualitative and quantitative temporal constraints both in the data (data expressiveness) and in the queries (query expressiveness). We then propose a modular architecture integrating a relational database with a temporal reasoner. We also consider classes of applications that fit into our approach and consider patient management in a hospital as an example.

Temporal reasoning is central to Artificial Intelligence (AI) and many of its applications. However, the existing algorithmic frameworks for temporal reasoning are not expressive enough to be applicable to robots with complex kinodynamic constraints typically described using differential equations. For example, while minimum and maximum velocity constraints can be encoded in Simple Temporal Networks (STNs), higher-order kinodynamic constraints cannot be represented in existing frameworks. In this paper, we present a novel framework for temporal reasoning called Kinodynamic Networks (KDNs). KDNs combine elements of existing temporal reasoning frameworks with the idea of Bernstein polynomials. The velocity profiles of robots are represented using Bernstein polynomials; and dynamic constraints on these velocity profiles can be converted to linear constraints on the to-be-determined coefficients of their Bernstein polynomials. We study KDNs for their attractive theoretical properties and apply them to the Multi-Agent Path Finding (MAPF) problem with higher-order kinodynamic constraints. We show that our approach is not only scalable but also yields smooth velocity profiles for all robots that can be executed by their controllers.

Scheduling in many practical industrial domains is complicated by the need to account for diverse and idiosyncratic constraints. Research in temporal reasoning and constraint satisfaction has produced problem solving models that operate with respect to general representational assumptions. These frameworks offer possibilities for developing scheduling technologies that naturally extend to accommodate the peculiarities of various application domains. One critical issue is whether such generality can be obtained without sacrificing scheduling performance. We investigate this issue through application of a previously developed constraint satisfaction problem solving (CSP) model for deadline scheduling to a complicated, multiproduct hoist scheduling problem encountered in printed circuit board (PCB) manufacturing. The goal is to maximize throughput of an automated PCB electroplating facility while ensuring feasibility with respect to process, capacity and material movement constraints. Building from a heuristic procedure generically referred to as PCP (precedence constraint posting), which relies on a temporal constraint graph representation of the problem, we straightforwardly define an extended solution procedure for minimizing makespan. In a series of comparative experiments, our procedure is found to significantly outperform previously published procedures for solving this hoist problem across a broad range of input assumptions.

Temporal constraint networks are a well-known powerful formalism for encoding temporal knowledge based on the CSP techniques. A redefinition of metric temporal constraints to cope with vagueness of temporal relations based on fuzzy sets, called fuzzy temporal constraint networks has been proposed previously. In this paper the authors identify those queries on a fuzzy temporal constraint network relevant to a knowledge-based reasoning system. The authors provide definitions for their answering and explore the functions on which they are based by discussing the different choices for them. These results can be useful to those interested in defining a system for temporal reasoning under uncertainty. For instance, the authors satisfactorily applied them to the definition of a possibilistic temporal reasoning system.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Establishing local consistency is one of the main algorithmic techniques in temporal and spatial reasoning. In this area, one of the central questions for the various proposed temporal and spatial constraint languages is whether local consistency implies global consistency. Showing that a constraint language Γ has this “local-to-global” property implies polynomial-time tractability of the constraint language, and has further pleasant algorithmic consequences.In the present paper, we study the “local-to-global” property by making use of a recently established connection of this property with universal algebra. Specifically, the connection shows that this property is equivalent to the presence of a so-called quasi near-unanimity polymorphism of the constraint language. We obtain new algorithmic results and give very concise proofs of previously known theorems. Our results concern well-known and heavily studied formalisms such as the point algebra and its extensions, Allen’s interval algebra, and the spatial reasoning language RCC-5.KeywordsConstraint SatisfactionConstraint Satisfaction ProblemRelation AlgebraSpatial ReasoningRelation SymbolThese 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.

We study the computational complexity of the qualitative algebra which is a temporal constraint formalism that combines the point algebra, the point-interval algebra and Allen's interval algebra. We identify all tractable fragments and show that every other fragment is NP-complete.

We consider qualitative temporal or spatial constraint networks whose constraints evolve over time in an ultimately periodic fashion: after an initial stretch of time, a fixed pattern of constraints (over an interval) is reproduced indefinitely. We propose a local propagation algorithm which is polynomial, and we show that it decides the consistency problem in some particular cases. We also show that the general problem of consistency for such networks is in PSPACE

With the rapid development of MLLMs, evaluating their visual capabilities has become increasingly crucial. Current benchmarks primarily fall into two main types: basic perception benchmarks,which focus on local details but lack deep reasoning (e.g., ''what is in the image?''), and mainstream reasoning benchmarks, which concentrate on prominent image elements but may fail to assess subtle clues requiring intricate analysis. However, profound visual understanding and complex reasoning depend more on interpreting subtle, inconspicuous local details than on perceiving salient, macro-level objects. These details, though occupying minimal image area, often contain richer, more critical information for robust analysis. To bridge this gap, we introduce the VER-Bench, a novel framework to evaluate MLLMs' ability to: 1) identify fine-grained visual clues, often occupying, on average, just 0.25% of the image area; 2) integrate these clues with world knowledge for complex reasoning. Comprising 374 carefully designed questions across Geospatial, Temporal, Situational, Intent, System State, and Symbolic reasoning, each question in VER-Bench is accompanied by structured evidence: visual clues and question-related reasoning derived from them. VER-Bench reveals current models' limitations in extracting subtle visual evidence and constructing evidence-based reasoning chains, highlighting the need to enhance models' capabilities in fine-grained visual evidence extraction, integration, and reasoning for genuine visual understanding and human-like analysis. The dataset is available at https://github.com/verbta/ACMMM-25-Materials.

Constraint-based reasoning is an important area of automated reasoning in artificial intelligence, with many applications. These include configuration and design problems, planning and scheduling, temporal and spatial reasoning, defeasible and causal reasoning, machine vision and language understanding, qualitative and diagnostic reasoning, and expert systems. Constraint-Based Reasoning presents current work in the field at several levels: theory, algorithms, languages, applications, and hardware. Constraint-based reasoning has connections to a wide variety of fields, including formal logic, graph theory, relational databases, combinatorial algorithms, operations research, neural networks, truth maintenance, and logic programming. The ideal of describing a problem domain in natural, declarative terms and then letting general deductive mechanisms synthesize individual solutions has to some extent been realized, and even embodied, in programming languages. Contents Introduction, E. C. Freuder, A. K. Mackworth • The Logic of Constraint Satisfaction, A. K. Mackworth • Partial Constraint Satisfaction, E. C. Freuder, R. J. Wallace • Constraint Reasoning Based on Interval Arithmetic: The Tolerance Propagation Approach, E. Hyvonen • Constraint Satisfaction Using Constraint Logic Programming, P. Van Hentenryck, H. Simonis, M. Dincbas • Minimizing Conflicts: A Heuristic Repair Method for Constraint Satisfaction and Scheduling Problems, S. Minton, M. D. Johnston, A. B. Philips, and P. Laird • Arc Consistency: Parallelism and Domain Dependence, P. R. Cooper, M. J. Swain • Structure Identification in Relational Data, R. Dechter, J. Pearl • Learning to Improve Constraint-Based Scheduling, M. Zweben, E. Davis, B. Daun, E. Drascher, M. Deale, M. Eskey • Reasoning about Qualitative Temporal Information, P. van Beek • A Geometric Constraint Engine, G. A. Kramer • A Theory of Conflict Resolution in Planning, Q. Yang Bradford Books imprint

The logic program formalism is commonly viewed as a modal or default logic. In this paper, we propose an alternative interpretation of the formalism as a terminological logic. A terminological logic is designed to represent two different forms of knowledge. A TBox represents definitions for a set of concepts. An ABox represents the assertional knowledge of the expert. In our interpretation, a logic program is a TBox providing definitions for all predicates; this interpretation is present already in Clark's completion semantics. We extend the logic program formalism such that some predicates can be left undefined and use classical logic as the language for the ABox. The resulting logic can be seen as an alternative interpretation of abductive logic program formalism. We study the expressivity of the formalism for representing uncertainty by proposing solutions for problems in temporal reasoning, with null values and open domain knowledge.

The anomalous extension problem in default reasoning

Since Simple Temporal Networks (STNs) were first introduced in 1991, there have been numerous theoretic and algorithmic advances that have made them practical for a wide variety of applications. However, the presentation of most of the important advances have been scattered across numerous conference papers and journal articles. As a result, it is too easy for even experienced researchers to be unaware of results that could positively impact their work. In this talk we review the most important results about STNs for researchers in Artificial Intelligence who are interested in incorporating the management of time and temporal constraints into their projects.

The theory of computational complexity is based on the tradeoff between two computational resources, memory space and computer time. This paper investigates the physical counterparts of these resources. Memory space is the number of bits of information required by a computer over the course of a computation. Physically, information is intimately connected with entropy: entropy can be thought of as random bits of information. The physical analogue of the amount of memory space required for a computation, Ф- SPACE, is the difference between the maximum entropy Smax of the computing system and its actual entropy, S.Ф-SPACE = S - Smax is the thermodynamic depth of the computation. Ф-SPACE is the maximum number of bits of memory space available for a physical system to perform computation. In conventional computational complexity theory, computer time is identified as the number of elementary logical operations (AND, OR, NOT, COPY) required to perform the computation. The physical analogue of time computational complexity is simply the time required for the physical computing system to perform the computation. In order to obtain a number that measures the number of physical operations that can be performed by the system over time t, we define Ф-TIME to be proportional to the computation time multiplied by the energy E of the computing system over the course of the computation: Ф-TIME = 2Et/πħ. Here, the proportionality factor 2/πħ is chosen to match the time it takes a computing system with energy E to flip a bit in time t. Ф-TIME is thus the number of bit flips that can take place in an information processing system. In addition, we propose a novel measure of physical complexity that combines these measures. The free energy F = E - TS of a system is the amount of energy that is available to do useful work - as opposed to the energy bound up in the thermal fluctuations of heat. We define the free complexity to be the free energy × time product, 2Ft/πħ required by a physical system to perform a computation. Physical spatial and temporal complexity provide upper bounds on the amount of computation that can be performed by any physical system. We discuss the implications of physical complexity for the computational complexity hierarchy, in particular, we show that in the presence of errors in the computation, Ф-SPACE is proportional Ф-TIME. We apply the resulting theory of physical computational complexity to analyze the computational power of black holes: exactly half of the energy of a black hole is available for physical computation.