Set Unification

The unification problem in algebras capable of describing sets has been tackled, directly or indirectly, by many researchers and it finds important applications in various research areas--e.g., deductive databases, theorem proving, static analysis, rapid software prototyping. The various solutions proposed are spread across a large literature. In this paper we provide a uniform presentation of unification of sets, formalizing it at the level of set theory. We address the problem of deciding existence of solutions at an abstract level. This provides also the ability to classify different types of set unification problems. Unification algorithms are uniformly proposed to solve the unification problem in each of such classes. The algorithms presented are partly drawn from the literature--and properly revisited and analyzed--and partly novel proposals. In particular, we present a new goal-driven algorithm for general ACI1 unification and a new simpler algorithm for general (Ab)(Cl) unification.Comment: 58 pages, 9 figures, 1 table. To appear in Theory and Practice of Logic Programming (TPLP

A Formalization of the Theorem of Existence of First-Order Most General Unifiers

This work presents a formalization of the theorem of existence of most general unifiers in first-order signatures in the higher-order proof assistant PVS. The distinguishing feature of this formalization is that it remains close to the textbook proofs that are based on proving the correctness of the well-known Robinson's first-order unification algorithm. The formalization was applied inside a PVS development for term rewriting systems that provides a complete formalization of the Knuth-Bendix Critical Pair theorem, among other relevant theorems of the theory of rewriting. In addition, the formalization methodology has been proved of practical use in order to verify the correctness of unification algorithms in the style of the original Robinson's unification algorithm.Comment: In Proceedings LSFA 2011, arXiv:1203.542

Logic Programming and Logarithmic Space

We present an algebraic view on logic programming, related to proof theory and more specifically linear logic and geometry of interaction. Within this construction, a characterization of logspace (deterministic and non-deterministic) computation is given via a synctactic restriction, using an encoding of words that derives from proof theory. We show that the acceptance of a word by an observation (the counterpart of a program in the encoding) can be decided within logarithmic space, by reducing this problem to the acyclicity of a graph. We show moreover that observations are as expressive as two-ways multi-heads finite automata, a kind of pointer machines that is a standard model of logarithmic space computation

Handling Network Partitions and Mergers in Structured Overlay Networks

Structured overlay networks form a major class of peer-to-peer systems, which are touted for their abilities to scale, tolerate failures, and self-manage. Any long-lived Internet-scale distributed system is destined to face network partitions. Although the problem of network partitions and mergers is highly related to fault-tolerance and self-management in large-scale systems, it has hardly been studied in the context of structured peer-to-peer systems. These systems have mainly been studied under churn (frequent joins/failures), which as a side effect solves the problem of network partitions, as it is similar to massive node failures. Yet, the crucial aspect of network mergers has been ignored. In fact, it has been claimed that ring-based structured overlay networks, which constitute the majority of the structured overlays, are intrinsically ill-suited for merging rings. In this paper, we present an algorithm for merging multiple similar ring-based overlays when the underlying network merges. We examine the solution in dynamic conditions, showing how our solution is resilient to churn during the merger, something widely believed to be difficult or impossible. We evaluate the algorithm for various scenarios and show that even when falsely detecting a merger, the algorithm quickly terminates and does not clutter the network with many messages. The algorithm is flexible as the tradeoff between message complexity and time complexity can be adjusted by a parameter

Performance-based control system design automation via evolutionary computing

This paper develops an evolutionary algorithm (EA) based methodology for computer-aided control system design (CACSD) automation in both the time and frequency domains under performance satisfactions. The approach is automated by efficient evolution from plant step response data, bypassing the system identification or linearization stage as required by conventional designs. Intelligently guided by the evolutionary optimization, control engineers are able to obtain a near-optimal ‘‘off-thecomputer’’ controller by feeding the developed CACSD system with plant I/O data and customer specifications without the need of a differentiable performance index. A speedup of near-linear pipelineability is also observed for the EA parallelism implemented on a network of transputers of Parsytec SuperCluster. Validation results against linear and nonlinear physical plants are convincing, with good closed-loop performance and robustness in the presence of practical constraints and perturbations

Linear Compressed Pattern Matching for Polynomial Rewriting (Extended Abstract)

This paper is an extended abstract of an analysis of term rewriting where the terms in the rewrite rules as well as the term to be rewritten are compressed by a singleton tree grammar (STG). This form of compression is more general than node sharing or representing terms as dags since also partial trees (contexts) can be shared in the compression. In the first part efficient but complex algorithms for detecting applicability of a rewrite rule under STG-compression are constructed and analyzed. The second part applies these results to term rewriting sequences. The main result for submatching is that finding a redex of a left-linear rule can be performed in polynomial time under STG-compression. The main implications for rewriting and (single-position or parallel) rewriting steps are: (i) under STG-compression, n rewriting steps can be performed in nondeterministic polynomial time. (ii) under STG-compression and for left-linear rewrite rules a sequence of n rewriting steps can be performed in polynomial time, and (iii) for compressed rewrite rules where the left hand sides are either DAG-compressed or ground and STG-compressed, and an STG-compressed target term, n rewriting steps can be performed in polynomial time.Comment: In Proceedings TERMGRAPH 2013, arXiv:1302.599

A Process Modelling Framework Based on Point Interval Temporal Logic with an Application to Modelling Patient Flows

This thesis considers an application of a temporal theory to describe and model the patient journey in the hospital accident and emergency (A&E) department. The aim is to introduce a generic but dynamic method applied to any setting, including healthcare. Constructing a consistent process model can be instrumental in streamlining healthcare issues. Current process modelling techniques used in healthcare such as flowcharts, unified modelling language activity diagram (UML AD), and business process modelling notation (BPMN) are intuitive and imprecise. They cannot fully capture the complexities of the types of activities and the full extent of temporal constraints to an extent where one could reason about the flows. Formal approaches such as Petri have also been reviewed to investigate their applicability to the healthcare domain to model processes. Additionally, to schedule patient flows, current modelling standards do not offer any formal mechanism, so healthcare relies on critical path method (CPM) and program evaluation review technique (PERT), that also have limitations, i.e. finish-start barrier. It is imperative to specify the temporal constraints between the start and/or end of a process, e.g., the beginning of a process A precedes the start (or end) of a process B. However, these approaches failed to provide us with a mechanism for handling these temporal situations. If provided, a formal representation can assist in effective knowledge representation and quality enhancement concerning a process. Also, it would help in uncovering complexities of a system and assist in modelling it in a consistent way which is not possible with the existing modelling techniques. The above issues are addressed in this thesis by proposing a framework that would provide a knowledge base to model patient flows for accurate representation based on point interval temporal logic (PITL) that treats point and interval as primitives. These objects would constitute the knowledge base for the formal description of a system. With the aid of the inference mechanism of the temporal theory presented here, exhaustive temporal constraints derived from the proposed axiomatic system’ components serves as a knowledge base. The proposed methodological framework would adopt a model-theoretic approach in which a theory is developed and considered as a model while the corresponding instance is considered as its application. Using this approach would assist in identifying core components of the system and their precise operation representing a real-life domain deemed suitable to the process modelling issues specified in this thesis. Thus, I have evaluated the modelling standards for their most-used terminologies and constructs to identify their key components. It will also assist in the generalisation of the critical terms (of process modelling standards) based on their ontology. A set of generalised terms proposed would serve as an enumeration of the theory and subsume the core modelling elements of the process modelling standards. The catalogue presents a knowledge base for the business and healthcare domains, and its components are formally defined (semantics). Furthermore, a resolution theorem-proof is used to show the structural features of the theory (model) to establish it is sound and complete. After establishing that the theory is sound and complete, the next step is to provide the instantiation of the theory. This is achieved by mapping the core components of the theory to their corresponding instances. Additionally, a formal graphical tool termed as point graph (PG) is used to visualise the cases of the proposed axiomatic system. PG facilitates in modelling, and scheduling patient flows and enables analysing existing models for possible inaccuracies and inconsistencies supported by a reasoning mechanism based on PITL. Following that, a transformation is developed to map the core modelling components of the standards into the extended PG (PG*) based on the semantics presented by the axiomatic system. A real-life case (from the King’s College hospital accident and emergency (A&E) department’s trauma patient pathway) is considered to validate the framework. It is divided into three patient flows to depict the journey of a patient with significant trauma, arriving at A&E, undergoing a procedure and subsequently discharged. Their staff relied upon the UML-AD and BPMN to model the patient flows. An evaluation of their representation is presented to show the shortfalls of the modelling standards to model patient flows. The last step is to model these patient flows using the developed approach, which is supported by enhanced reasoning and scheduling

Minimizing finite automata is computationally hard

It is known that deterministic finite automata (DFAs) can be algorithmically minimized, i.e., a DFA M can be converted to an equivalent DFA M' which has a minimal number of states. The minimization can be done efficiently [6]. On the other hand, it is known that unambiguous finite automata (UFAs) and nondeterministic finite automata (NFAs) can be algorithmically minimized too, but their minimization problems turn out to be NP-complete and PSPACE-complete [8]. In this paper, the time complexity of the minimization problem for two restricted types of finite automata is investigated. These automata are nearly deterministic, since they only allow a small amount of non determinism to be used. On the one hand, NFAs with a fixed finite branching are studied, i.e., the number of nondeterministic moves within every accepting computation is bounded by a fixed finite number. On the other hand, finite automata are investigated which are essentially deterministic except that there is a fixed number of different initial states which can be chosen nondeterministically. The main result is that the minimization problems for these models are computationally hard, namely NP-complete. Hence, even the slightest extension of the deterministic model towards a nondeterministic one, e.g., allowing at most one nondeterministic move in every accepting computation or allowing two initial states instead of one, results in computationally intractable minimization problems

Towards Correctness of Program Transformations Through Unification and Critical Pair Computation

Correctness of program transformations in extended lambda calculi with a contextual semantics is usually based on reasoning about the operational semantics which is a rewrite semantics. A successful approach to proving correctness is the combination of a context lemma with the computation of overlaps between program transformations and the reduction rules, and then of so-called complete sets of diagrams. The method is similar to the computation of critical pairs for the completion of term rewriting systems. We explore cases where the computation of these overlaps can be done in a first order way by variants of critical pair computation that use unification algorithms. As a case study we apply the method to a lambda calculus with recursive let-expressions and describe an effective unification algorithm to determine all overlaps of a set of transformations with all reduction rules. The unification algorithm employs many-sorted terms, the equational theory of left-commutativity modelling multi-sets, context variables of different kinds and a mechanism for compactly representing binding chains in recursive let-expressions.Comment: In Proceedings UNIF 2010, arXiv:1012.455