Specifying Well-Formed Part-Whole Relations in Coq

Generally, part-whole relations are modeled using fragments of first-order logic(FOL) and difficulties arise when meta-reasoning is done over their properties, leading to reason outside the logic. Alternatively, classical languages for ontological reasoning such as Description Logics + Logic Programming lack of expressive formal foundations resulting in ambiguous interpretations of relations. Moreover, they show some difficulties to prove that a given meta property is logically correct. In order to address these problems, we suggest a formal framework using a dependent (higher-order) type theory such as those used in program checking and theorem provers (e.g., Coq). All properties of part-whole relations are formalized through abstract constructs called parameterized specifications (p-specifications). We detail their content and explain how they are suitable to build an ontology of formal properties that can be further used for reasoning about higher-order properties.

Conceptual modeling is an important step for successful system development. It helps system designers and business practitioners share the same view on domain knowledge. If the work is successful, a result of conceptual modeling can be beneficial in increasing productivity and reducing failures. However, the value of conceptual modeling is unlikely to be evaluated uniformly because we are lack of agreement on how to elicit concepts and how to represent those with conceptual modeling constructs. Especially, designing relationships between components, also known as part-whole relationships, have been regarded as complicated work. The recent study, Representing Part-Whole Relations in Conceptual Modeling : An Empirical Evaluation (Shanks et al., 2008), published in MIS Quarterly, can be regarded as one of positive efforts. Not only the study is one of few attempts of trying to clarify how to select modeling alternatives in part-whole design, but also it shows results based on an empirical experiment. Shanks et al. argue that there are two modeling alternatives to represent part-whole relationships : an implicit representation and an explicit one. By conducting an experiment, they insist that the explicit representation increases the value of a conceptual model. Moreover, Shanks et al. justify their findings by citing the BWW ontology. Recently, the study from Shanks et al. faces criticism. Allen and March (2012) argue that Shanks et al.'s experiment is lack of validity and reliability since the experimental setting suffers from error-prone and self-defensive design. They point out that the experiment is intentionally fabricated to support the idea, as such that using concrete UML concepts results in positive results in understanding models. Additionally, Allen and March add that the experiment failed to consider boundary conditions; thus reducing credibility. Shanks and Weber (2012) contradict flatly the argument suggested by Allen and March (2012). To defend, they posit the BWW ontology is righteously applied in supporting the research. Moreover, the experiment, they insist, can be fairly acceptable. Therefore, Shanks and Weber argue that Allen and March distort the true value of Shanks et al. by pointing out minor limitations. In this study, we try to investigate the dispute around Shanks et al. in order to answer to the following question : What is the proper value of the study conducted by Shanks et al.? More profoundly, we question whether or not using the BWW ontology can be the only viable option of exploring better conceptual modeling methods and procedures. To understand key issues around the dispute, first we reviewed previous studies relating to the BWW ontology. We critically reviewed both of Shanks and Weber and Allen and March. With those findings, we further discuss theories on part-whole (or part-of) relationships that are rarely treated in the dispute. As a result, we found three additional evidences that are not sufficiently covered by the dispute. The main focus of the dispute is on the errors of experimental methods: Shanks et al. did not use Bunge's Ontology properly; the refutation of a paradigm shift is lack of concrete, logical rationale; the conceptualization on part-whole relations should be reformed. Conclusively, Allen and March indicate properly issues that weaken the value of Shanks et al. In general, their criticism is reasonable; however, they do not provide sufficient answers how to anchor future studies on part-whole relationships. We argue that the use of the BWW ontology should be rigorously evaluated by its original philosophical rationales surrounding part-whole existence. Moreover, conceptual modeling on the part-whole phenomena should be investigated with more plentiful lens of alternative theories. The criticism on Shanks et al. should not be regarded as a contradiction on evaluating modeling methods of alternative part-whole representations. To the contrary, it should be viewed as a call for research on usable and useful approaches to increase value of conceptual modeling.

On Expressive Completeness of Duration and Mean Value Calculi (Extended Abstract)

The role of games in computing science has been given some consideration in the literature. We aim to design games that can be used for teaching logic. Specifically, we build on the Minefield game used in the Stanford Intro to Logic course, and present a version, which we call MineFOL, that incorporates time constraints. We also move beyond the Stanford examples in that we provide a formalisation of the game, with a view to eventually generating game instances automatically as well as from user input, and show how MineFOL can be used as a learning tool. We also developed an online platform supporting learners in at least three ways. First, the learners can practice reasoning in First Order Logic (FOL) and proving strategies such as resolution through various MineFOL games. Second, the learners can practice formalisation in FOL, by allowing learners to build their own games. Third, the learners become aware of the need of interleaving various technologies to solve a logicbased task. In this line, MineFOL is formalised in such way to allow augmenting reasoning in FOL with search strategies.

In this paper, we present a system that translates first order logic (FOL) formulas into natural language (NL) sentences. The motivation comes from an intelligent tutoring system teaching logic as a knowledge representation language, where it is used as a means for feedback to the users. FOL to NL conversion is achieved by using a rule-based approach, where we exploit the pattern matching capabilities of rules. So, the system consists of a rule-based component and a lexicon. The rule-based unit implements the conversion process, which is based on a linguistic analysis of a FOL sentence, and the lexicon provides lexical and grammatical information that helps in producing the NL sentences. The whole system is implemented in Jess, a java-based expert system shell. The conversion process currently covers a restricted set of FOL formulas.

Approximate pattern matching and transitive closure logics

Intuitionistic completeness of first-order logic

The paper investigates interpretations of propositional and first-order logic in which validity is defined in terms of partial indices; sometimes called possibilities but here understood as non-empty subsets of a set W of possible worlds. Truth at a set of worlds is understood to be truth at every world in the set. If all subsets of W are permitted the logic so determined is classical first-order predicate logic. Restricting allowable subsets and then imposing certain closure conditions provides a modelling for intuitionistic predicate logic. The same semantic interpretation rules are used in both logics for all the operators.

A Lindström theorem for intuitionistic first-order logic

We study Sowa’s conceptual graphs (CGs) with both existential and universal quantifiers. We explore in detail the existential fragment. We extend and modify Sowa’s original graph derivation system with new rules and prove the soundness and completeness theorem with respect to Sowa’s standard interpretation of CGs into first order logic (FOL). The proof is obtained by reducing the graph derivation to a question-answering problem. The graph derivation can be equivalently obtained by querying a Definite Horn Clauses program by a conjunction of positive atoms. Moreover, the proof provides an algorithm for graph derivation in a pure proof-theoretic fashion, namely by means of a slight enhancement of the standard PROLOG interpreter. The graph derivation can be rebuilt step-by-step and constructively from the resolution-based proof. We provide a notion of CGs in normal form (the table of the conceptual graph) and show that the PROLOG interpreter also gives a projection algorithm between normal CGs. The normal forms are obtained by extending the FOL language by witnesses (new constants) and extending the graph derivation system. By applying iteratively a set of rules the reduction process terminates with the normal form of a conceptual graph. We also show that graph derivation can be reduced to a question-answering problem in propositional datalog for a subclass of simple CGs. The embedding into propositional datalog makes the complexity of the derivation polynomial.

The expressiveness of First Order Logic (FOL) languages is severely counterbalanced by the complexity of matching formulas on a universe. Matching is an instance of the class of Constraint Satisfaction Problems (CSP), which have shown to undergo a phase transition with respect to two order parameters: constraint density and constraint tightness. This paper analyzes the problem of satisfying FOL Horn clauses in the light of these recent results. By means of an extensive experimental analysis, we show how Horn clause verification exhibits a typical phase transition with respect to the number of binary (or greater arity) predicates, and with respect to the ratio between the number of constants in the universe and the cardinality of the basic predicates extension.

Software Product Line (SPL) is a new methodology that develops software products by configuring a software product from artifacts repository. Variability is one of the important issues in designing SPL. It reflects the diversity and commonality of the artifacts in a product line. The success of SPL is basically dependent on model's variability. SPL contains three main issues: variability modeling, configuration of new software and the analysis of SPL. Validation is a vital operation among all these issues. In the literature, many methods of modeling variability are proposed but none of them focuses on the validation. In this paper, an intelligent method for validating SPL is introduced. The proposed method is based on two layers. The higher layer is a graphical representation (satisfying visualization condition), and the lower layer is logical representation of the variability using first order logic (FOL). The way that the proposed method can be used to model variability, and support the analysis and validation of SPL is described. Later a new operation in SPL validation issue is presented. Finally, the implementation of the two basic operations in the analysis of SPL is illustrated.

The question, whether second order logic is a better foundation for mathematics than set theory, is addressed. The main difference between second order logic and set theory is that set theory builds up a transfinite cumulative hierarchy while second order logic stays within one application of the power sets. It is argued that in many ways this difference is illusory. More importantly, it is argued that the often stated difference, that second order logic has categorical characterizations of relevant mathematical structures, while set theory has non-standard models, amounts to no difference at all. Second order logic and set theory permit quite similar categoricity results on one hand, and similar non-standard models on the other hand.

Software product Line (SPL) is an emerging methodology for developing software products. A successful software product is highly dependent on the validity of a SPL. Therefore, validation is a significant process within SPL. In this paper, inconsistency detection is investigated as operation for validating SPL. Intelligent rules are formulated detecting inconsistency based on deducing the results from predefined cases. First, variability is modeled using First Order Logic (FOL) predicates as a prerequisite for inconsistency detection. Later, inconsistency is categorized in three groups. For each group a general form is formulated that can coffer all possible cases. Finally, an intelligent rule (based on FOL) is illustrated for implementing each possibility. As results, all cases of inconsistency in the domain-engineering process are defined.

In this paper we will present a graph-transformation based method for the verification of heterogeneous first order logic (FOL) and Euler/Venn proofs. In previous work, it has been shown that a special collection of directed acyclic graphs (DAGs) can be used interchangeably with Euler/Venn diagrams in reasoning processes. Thus, proofs which include Euler/Venn diagrams can be thought of as proofs with DAGs where steps involving only Euler/Venn diagrams can be treated as particular DAG transformations. Here we will show how the characterization of these manipulations can be used to verify Euler/Venn proofs. Also, a method for verifying the use of heterogeneous Euler/Venn and FOL reasoning rules will be presented that is also based upon DAG transformations .