Herbrand Model Research Articles

A Complete Diagrammatic Calculus for Boolean Satisfiability

We propose a calculus of string diagrams to reason about satisfiability of Boolean formulas, and prove it to be sound and complete. We then showcase our calculus in a few case studies. First, we consider SAT-solving. Second, we consider Horn clauses, which leads us to a new decision method for propositional logic programs equivalence under Herbrand model semantics.

Alpha-Beta Privacy

The formal specification of privacy goals in symbolic protocol models has proved to be not quite trivial so far. The most widely used approach in formal methods is based on the static equivalence of frames in the applied pi-calculus, basically asking whether or not the intruder is able to distinguish two given worlds. But then a subtle question emerges: How can we be sure that we have specified all pairs of worlds to properly reflect our intuitive privacy goal? To address this problem, we introduce in this article a novel and declarative way to specify privacy goals, called (α, β)-privacy. This new approach is based on specifying two formulae α and β in first-order logic with Herbrand universes, where α reflects the intentionally released information and β includes the actual cryptographic (“technical”) messages the intruder can see. Then (α, β)-privacy means that the intruder cannot derive any “nontechnical” statement from β that he cannot derive from α already. We describe by a variety of examples how this notion can be used in practice. Even though (α, β)-privacy does not directly contain a notion of distinguishing between worlds, there is a close relationship to static equivalence of frames that we investigate formally. This allows us to justify (and criticize) the specifications that are currently used in verification tools and obtain a decision procedure for a large fragment of (α, β)-privacy.

Thresholded semantic framework for a fully integrated fuzzy logic language

This work proposes a declarative semantics based on a fuzzy variant of the classical notion of least Herbrand model for the so-called FASILL programming language (acronym of “Fuzzy Aggregators and Similarity Into a Logic Language”) which has been recently designed and implemented for coping with implicit/explicit truth degree annotations, a great variety of connectives and unification by similarity. Also, we define an immediate consequence operator that allows us to give a fixpoint characterization of the least Herbrand model for a FASILL program. A remarkable aspect of our declarative and fixpoint semantics is the fact that both have been enriched with the use of thresholds thanks to the natural ability of the underlying fuzzy language for managing such constructs. Moreover, we have connected both semantics with an operational one, which also manages thresholds in an efficient way, by proving the correctness of the whole semantic framework.

Structural Resolution with Co-inductive Loop Detection

A way to combine co-SLD style loop detection with structural resolution was found and is introduced in this work, to extend structural resolution with co-induction. In particular, we present the operational semantics, called co-inductive structural resolution, of this novel combination and prove its soundness with respect to the greatest complete Herbrand model.

A linear algebraic approach to datalog evaluation

AbstractWe propose a fundamentally new approach to Datalog evaluation. Given a linear Datalog program DB written usingNconstants and binary predicates, we first translate if-and-only-if completions of clauses in DB into a setEq(DB) of matrix equations with a non-linear operation, where relations inMDB, the least Herbrand model of DB, are encoded as adjacency matrices. We then translateEq(DB) into another, but purely linear matrix equationsẼq(DB). It is proved that the least solution ofẼq(DB) in the sense of matrix ordering is converted to the least solution ofEq(DB) and the latter givesMDBas a set of adjacency matrices. Hence, computing the least solution ofẼq(DB) is equivalent to computingMDBspecified by DB. For a class of tail recursive programs and for some other types of programs, our approach achievesO(N3) time complexity irrespective of the number of variables in a clause since only matrix operations costingO(N3) or less are used. We conducted two experiments that compute the least Herbrand models of linear Datalog programs. The first experiment computes transitive closure of artificial data and real network data taken from the Koblenz Network Collection. The second one compared the proposed approach with the state-of-the-art symbolic systems including two Prolog systems and two ASP systems, in terms of computation time for a transitive closure program and the same generation program. In the experiment, it is observed that our linear algebraic approach runs 101~ 104times faster than the symbolic systems when data is not sparse. Our approach is inspired by the emergence of big knowledge graphs and expected to contribute to the realization of rich and scalable logical inference for knowledge graphs.

Scalable learning and inference in Markov logic networks

Markov logic networks (MLNs) have emerged as a powerful representation that incorporates first-order logic and probabilistic graphical models. They have shown very good results in many problem domains. However, current implementations of MLNs do not scale well due to the large search space and the intractable clause groundings, which is preventing their widespread adoption. In this paper, we propose a general framework named Ground Network Sampling (GNS) for scaling up MLN learning and inference. GNS offers a new instantiation perspective by encoding ground substitutions as simple paths in the Herbrand universe, which uses the interactions existing among the objects to constrain the search space. To further make this search tractable for large scale problems, GNS integrates random walks and subgraph pattern mining, gradually building up a representative subset of simple paths. When inference is concerned, a template network is introduced to quickly locate promising paths that can ground given logical statements. The resulting sampled paths are then transformed into ground clauses, which can be used for clause creation and probabilistic inference. The experiments on several real-world datasets demonstrate that our approach offers better scalability while maintaining comparable or better predictive performance compared to state-of-the-art MLN techniques.

On the existence of free models in fuzzy universal Horn classes

This paper is a contribution to the study of the universal Horn fragment of predicate fuzzy logics, focusing on some relevant notions in logic programming. We introduce the notion of term structure associated to a set of formulas in the fuzzy context and we show the existence of free models in fuzzy universal Horn classes. We prove that every equality-free consistent universal Horn fuzzy theory has a Herbrand model.

On definite program answers and least Herbrand models

AbstractA sufficient and necessary condition is given under which least Herbrand models exactly characterize the answers of definite clause programs.

Tabulation proof procedures for fuzzy linguistic logic programming

• The non-deterministic procedure is goal-oriented, sound, and complete and terminates. • The deterministic procedure is more efficient than the non-deterministic one. • The number and size of trees created by the deterministic procedure are less. • The deterministic procedure gives all and only the most general answers to a query. • The deterministic procedure enables threshold computation and top-k retrieval. Fuzzy linguistic logic programming is a logical system for representing and reasoning with linguistically-expressed human knowledge. In fuzzy linguistic logic programming, up until now, there have been only two methods to compute answers to a query w.r.t. a logic program: (i) by bottom-up iterating the immediate consequence operator T P ; or (ii) by using the procedural semantics. Nevertheless, the former is exhaustive and not goal-oriented. Indeed, it requires computation of the whole least Herbrand model despite the fact that not all the results are required to determine the answer of the query. On the other hand, although the latter is goal-oriented, it may lead to an infinite loop and may recompute atoms in rule bodies. Furthermore, it may not, in general, give a most general answer to a query. In this paper, we develop two query answering procedures which can overcome these problems for fuzzy linguistic logic programming. More precisely, the non-deterministic tabulation procedure is close to the procedural semantics and gives all answers to a query. The deterministic tabulation procedure is more efficient than the non-deterministic one and gives all and only the most general answers to a query. The deterministic procedure also enables threshold computation and top- k retrieval.

Reactive Computing as Model Generation

In this paper we propose a logic-based, framework inspired by artificial intelligence, but scaled down for practical database and programming applications. Computation in the framework is viewed as the task of generating a sequence of state transitions, with the purpose of making an agent’s goals all true. States are represented by sets of atomic sentences (or facts), representing the values of program variables, tuples in a coordination language, facts in relational databases, or Herbrand models. In the model-theoretic semantics, the entire sequence of states and events are combined into a single model-theoretic structure, by associating timestamps with facts and events. But in the operational semantics, facts are updated destructively, without timestamps. We show that the model generated by destructive updates is identical to the model generated by reasoning with facts containing timestamps. We also extend the model with intentional predicates and composite event predicates defined by logic programs containing conditions in first-order logic, which query the current state.

Concept Learning in the Description Logic ALCH(D) Based on Minimal Model Reasoning for RDF Data

In this paper, we propose an algorithm for ALCH(D) concept learning from RDF data using minimal model reasoning. This algorithm generates concept expressions in the Description Logic ALCH(D) by giving background knowledge and positive and negative examples in the RDF form. Our method can be widely applied to RDF data on the Web, as background knowledge. An advantage of the method for RDF data is that reasoning on RDF graphs is tractable compared to logical reasoning for OWL data. We solve the problem that RDF data cannot be directly applied to the concept learning due to its less expressive power, speci.cally, the lack of negative expressions. In order to construct expressive ALCH(D) concepts from less expressive RDF data in the concept learning, we introduce (nonmonotonic) inference rules based on minimal model reasoning which derive implicit subclass and subproperty relations from the background knowledge in the RDF form. We prove the soundness, completeness and decidability of the nonmonotonic RDF reasoning in the minimal Herbrand models for RDF graphs. The process of concept learning is divided in two parts: (i) concept generation and (ii) concept evaluation. In the concept generation, minimal model reasoning enables us to derive complex concepts consisting of negation, conjunction, disjunction and quanti.ers and to exclude inconsistent concepts. In the concept evaluation, we evaluate hypothesis concepts with class and property hierarchies where minimal model reasoning is used for expressing more speci.c concepts as the answer for learning. We implement a system that learns some ALCH(D) concepts describing the features of given examples.

Extensional Higher-Order Logic Programming

We propose a purely extensional semantics for higher-order logic programming. In this semantics program predicates denote sets of ordered tuples, and two predicates are equal iff they are equal as sets. Moreover, every program has a unique minimum Herbrand model which is the greatest lower bound of all Herbrand models of the program and the least fixed-point of an immediate consequence operator. We also propose an SLD-resolution proof system which is proven sound and complete with respect to the minimum Herbrand model semantics. In other words, we provide a purely extensional theoretical framework for higher-order logic programming which generalizes the familiar theory of classical (first-order) logic programming.

Constraint-based correctness proofs for logic program transformations

Abstract Many approaches proposed in the literature for proving the correctness of unfold/fold transformations of logic programs make use of measures associated with program clauses. When from a program P 1 we derive a program P 2 by applying a sequence of transformations, suitable conditions on the measures of the clauses in P 2 guarantee that the transformation of P 1 into P 2 is correct, that is, P 1 and P 2 have the same least Herbrand model. In the approaches proposed so far, clause measures are fixed in advance, independently of the transformations to be proved correct. In this paper we propose a method for the automatic generation of clause measures which, instead, takes into account the particular program transformation at hand. During the application of a sequence of transformations we construct a system of linear equalities and inequalities over nonnegative integers whose unknowns are the clause measures to be found, and the correctness of the transformation is guaranteed by the satisfiability of that system. Through some examples we show that our method is more powerful and practical than other methods proposed in the literature. In particular, we are able to establish in a fully automatic way the correctness of program transformations which, by using other methods, are proved correct at the expense of fixing in advance sophisticated clause measures.

A Tableau Calculus for Minimal Modal Model Generation

Model generation and minimal model generation is useful for fault analysis, verification of systems and validation of data models. Whereas for classical propositional and first-order logic several model minimization approaches have been developed and studied, for non-classical logic the topic has been much less studied. In this paper we introduce a minimal model generation calculus for multi-modal logic K(m) and extensions of K(m) with the axioms T and B. The calculus provides a method to generate all and only minimal modal Herbrand models, and each model is generated exactly once. A novelty of the calculus is a non-standard complement splitting rule designed for minimal model generation. Experiments show the rule has the added benefit of reducing the search space.

Idealized coinductive type systems for imperative object-oriented programs

In recent work we have proposed a novel approach to define idealized type systems for object-oriented languages, based on abstract compilation of programs into Horn formulas which are interpreted w.r.t. the coinductive (that is, the greatest) Herbrand model. In this paper we investigate how this approach can be applied also in the presence of imperative features. This is made possible by considering a natural translation of Static Single Assignment intermediate form programs into Horn formulas, where φ functions correspond to union types.

GOPT Resolution

Some resolution strategies, such as SLD-resolution, are such that a derivation may be infinite even on a logic program that has a finite Herbrand universe. This paper introduces GOPT-resolution, a new deduction strategy for deriving solutions from a set of rules that improves on previous methods by preventing derivations that have infinite recursion. The paper outlines the process behind the development of GOPT-resolution based on PT-resolution. GOPT-resolution is then developed by distinguishing between goal-relevant and goal-irrelevant P-domains.

Integrating Positional and Slotted Knowledge on the Semantic Web

POSL is a Semantic Web language for knowledge interchange, reconciling Horn logic's positional and F-logic's slotted formulas for representing facts and rules on the Web, optionally referring to RDFS or OWL classes for order-sorted typing. The POSL semantics directly enhances Herbrand models for n-ary relations by accommodating slotted clause instantiation and ground equality, further restricted through signatures and types. Webizing uses IRIs in the IETF form of N3 for individuals, relations, slots, and types. Webized atoms further permit the representation of F-logic objects and RDF descriptions as anchored slotted facts enhanced by rules. All POSL notions are exemplified using an e-Business use case in logistics. The online trans- lator from POSL to OO RuleML and POSL engine OO jDREW have enabled Semantic Web applications in business information integration, touristic planning, and distributed expert/symposium/wellness profile querying.

Coinductive subtyping for abstract compilation of object-oriented languages into Horn formulas

In recent work we have shown how it is possible to define very precise type systems for object-oriented languages by abstractly compiling a program into a Horn formula f. Then type inference amounts to resolving a certain goal w.r.t. the coinductive (that is, the greatest) Herbrand model of f. Type systems defined in this way are idealized, since in the most interesting instantiations both the terms of the coinductive Herbrand universe and goal derivations cannot be finitely represented. However, sound and quite expressive approximations can be implemented by considering only regular terms and derivations. In doing so, it is essential to introduce a proper subtyping relation formalizing the notion of approximation between types. In this paper we study a subtyping relation on coinductive terms built on union and object type constructors. We define an interpretation of types as set of values induced by a quite intuitive relation of membership of values to types, and prove that the definition of subtyping is sound w.r.t. subset inclusion between type interpretations. The proof of soundness has allowed us to simplify the notion of contractive derivation and to discover that the previously given definition of subtyping did not cover all possible representations of the empty type.

Totally correct logic program transformations via well-founded annotations

We address the problem of proving the total correctness of transformations of definite logic programs. We consider a general transformation rule, called clause replacement, which consists in transforming a program P into a new program Q by replacing a set Γ 1 of clauses occurring in P by a new set Γ 2 of clauses, provided that Γ 1 and Γ 2 are equivalent in the least Herbrand model M(P) of the program P. We propose a general method for proving that transformations based on clause replacement are totally correct, that is, M(P)=M(Q). Our method consists in showing that the transformation of P into Q can be performed by: (i) adding extra arguments to predicates, thereby deriving from the given program P an annotated program $\overline{P}$ , (ii) applying a variant of the clause replacement rule and transforming the annotated program $\overline{P}$ into a terminating annotated program $\overline{Q}$ , and (iii) erasing the annotations from $\overline{Q}$ , thereby getting Q. Our method does not require that either P or Q are terminating and it is parametric with respect to the annotations. By providing different annotations we can easily prove the total correctness of program transformations based on various versions of the popular unfolding, folding, and goal replacement rules, which can all be viewed as particular cases of our clause replacement rule.

Combining answer set programming with description logics for the Semantic Web

We propose a combination of logic programming under the answer set semantics with the description logics SHIF ( D ) and SHOIN ( D ) , which underly the Web ontology languages OWL Lite and OWL DL, respectively. To this end, we introduce description logic programs (or dl-programs), which consist of a description logic knowledge base L and a finite set P of description logic rules (or dl-rules). Such rules are similar to usual rules in nonmonotonic logic programs, but they may also contain queries to L, possibly under default negation, in their bodies. They allow for building rules on top of ontologies but also, to a limited extent, building ontologies on top of rules. We define a suite of semantics for various classes of dl-programs, which conservatively extend the standard semantics of the respective classes and coincide with it in absence of a description logic knowledge base. More concretely, we generalize positive, stratified, and arbitrary normal logic programs to dl-programs, and define a Herbrand model semantics for them. We show that they have similar properties as ordinary logic programs, and also provide fixpoint characterizations in terms of (iterated) consequence operators. For arbitrary dl-programs, we define answer sets by generalizing Gelfond and Lifschitz's notion of a transform, leading to a strong and a weak answer set semantics, which are based on reductions to the semantics of positive dl-programs and ordinary positive logic programs, respectively. We also show how the weak answer sets can be computed utilizing answer sets of ordinary normal logic programs. Furthermore, we show how some advanced reasoning tasks for the Semantic Web, including different forms of closed-world reasoning and default reasoning, as well as DL-safe rules, can be realized on top of dl-programs. Finally, we give a precise picture of the computational complexity of dl-programs, and we describe efficient algorithms and a prototype implementation of dl-programs which is available on the Web.