Clause Logic Research Articles

Clausal Discovery

The clausal discovery engine claudien is presented. CLAUDIEN is an inductive logic programming engine that fits in the descriptive data mining paradigm. CLAUDIEN addresses characteristic induction from interpretations, a task which is related to existing formalisations of induction in logic. In characteristic induction from interpretations, the regularities are represented by clausal theories, and the data using Herbrand interpretations. Because CLAUDIEN uses clausal logic to represent hypotheses, the regularities induced typically involve multiple relations or predicates. CLAUDIEN also employs a novel declarative bias mechanism to define the set of clauses that may appear in a hypothesis.

A Disjunctive Positive Refinement of Model Elimination and its Application to Subsumption Deletion

The Model Elimination (ME) calculus is a refutationally complete, goal-oriented calculus for first-order clause logic. In this article, we introduce a new variant called disjunctive positive ME (DPME); it improves on Plaisted’s positive refinement of ME in that reduction steps are allowed only with positive literals stemming from clauses having at least two positive literals (so-called disjunctive clauses). DPME is motivated by its application to various kinds of subsumption deletion: in order to apply subsumption deletion in ME equally successful as in resolution, it is crucial to employ a version of ME that minimizes ancestor context (i.e., the necessary A-literals to find a refutation). DPME meets this demand. We describe several variants of ME with subsumption, the most important ones being ME with backward and forward subsumption and the Ta-Context Check. We compare their pruning power, also taking into consideration the well-known regularity restriction. All proofs are supplied. The practicability of our approach is demonstrated with experiments.

LINUS - A Link Instantion Prover with Unit Support

LINUS is a theorem prover for clause logic based on hyperlinking. The main new feature of the system is the emphasis on the preference for unit clauses, which are generated as a by-product of the hyperlinking method. The mechanism of unit support also motivates a slightly different interpretation of hyperlinking, namely, as mate saturation. Mate saturation can be viewed as a generalization of unit-resulting resolution that is compatible with any set-of-support strategy. We give a survey on the theoretical background and the architecture of the system, and decribe its performance.

Category-based modularisation for equational logic programming

Although modularisation is basic to modern computing, it has been little studied for logic-based programming. We treat modularisation for equational logic programming using the institution of categorybased equational logic in three different ways: (1) to provide a generic satisfaction condition for equational logics; (2) to give a category-based semantics for queries and their solutions; and (3) as an abstract definition of compilation from one (equational) logic programming language to another. Regarding (2), we study soundness and completeness for equational logic programming queries and their solutions. This can be understood as ordinary soundness and completeness in a suitable "non-logical" institution. Soundness holds for all module imports, but completeness only holds for conservative module imports. Category-based equational signatures are seen as modules, and morphisms of such signatures as module imports. Regarding (3), completeness corresponds to compiler correctness. The results of this research applies to languages based on a wide class of equational logic systems, including Horn clause logic, with or without equality; all variants of order and many sorted equational logic, including working modulo a set of axioms; constraint logic programming over arbitrary user-defined data types; and any combination of the above. Most importantly, due to the abstraction level, this research gives the possibility to have semantics and to study modularisation for equational logic programming developed over non-conventional structures.

On the minimum number of logical clauses inferred from examples

Given two sets of positive and negative examples, the inductive inference problem is to infer a small set of logical clauses which appropriately classify the examples. A graph theoretic approach is used to establish a lower limit on the minimum number of required clauses. Furthermore, the findings of this paper reveal methods for partitioning the original data and thus solving efficiently large scale problems.

Combining Algebraic and Set-Theoretic Specifications (Extended Version)

Specification frameworks such as B and Z provide power sets and cartesian<br />products as built-in type constructors, and employ a rich notation for<br />defining (among other things) abstract data types using formulae of predicate<br />logic and lambda-notation. In contrast, the so-called algebraic specification <br />frameworks often limit the type structure to sort constants and<br />first-order functionalities, and restrict formulae to (conditional) equations.<br />Here, we propose an intermediate framework where algebraic specifications<br />are enriched with a set-theoretic type structure, but formulae remain in the<br />logic of equational Horn clauses. This combines an expressive yet modest<br />specification notation with simple semantics and tractable proof theory.

Linear and unit-resulting refutations for Horn theories

We present a new transformation method by which a given Horn theory is transformed in such a way that resolution derivations can be carried out which are bot h linear (in the sense of Prologs SLD-resolution) and unit-resulting (i.e. the resolvents are unit clauses). Thi s is not trivial since although both strategies alone are complete, their nacombination is not. Completeness is recovered by our method through a completion procedure in the spirit of Knuth-Bendix completion, however with different ordering criteria. A powerful redundancy criterion helps to find a finite system quite often . The transformed theory can be used in combination with linear calculi such as e.g. (theory) model elimination to yield sound, complete and efficient calculi for full first o rder clause logic over the given Horn theory. As an example application, our method discovers a generalization of the well-known linear paramodulation calculus for the combined theory of equality and strict orderings. The method has been implemented and has been tested in conjunction with a model elimination theorem prover.

Is GUI programming a database research problem?

Programming nontrivial GUI applications is currently an arduous task. Just as the use of a declarative language simplified the programming of database applications, we ask whether we can do the same for GUI programming? Can we then import a large body of knowledge from database research? We answer these questions by describing our experience in building nontrivial GUI applications initially using C++ programming and subsequently using Logic++, a higher order Horn clause logic language on complex objects with object-oriented features. We abstract a GUI application as a set of event handlers. Each event handler can be conceptualized as a transition from the old screen/program state to a new screen/program state. We use a data centric view of the screen/program state ( i.e., every entity on the screen corresponds to proxy datum in the program) and express each event handler as a query dependent update, albeit a complicated one. To express such complicated updates we use Logic++. The proxy data are expressed as derived views that are materialized on the screen. Therefore, the system must be active in maintaining these materialized views. Consequently, each event handler is conceptually an update followed by a fixpoint computation of the proxy data. Based on our experience in building the GUI system, we observe that many database techniques such as view maintenance, active DB, concurrency control, recovery, optimization as well as language concepts such as higher order logic are useful in the context of GUI programming.

A high-level Petri net for goal-directed semantics of Horn clause logic

A new high level Petri net (HLPN) model is introduced as a graphical syntax for Horn clause logic (HCL) programs. We call these nets: Horn clause logic goal directed nets (HCLGNs). It is shown that there is a bijection between the queried definite programs and the class of HCLGNs. In addition, a visualization of SLD resolution is realized through the enabling and firing rules and net markings. The correctness of these rules with respect to SLD resolution is also proven. We model SLD refutations and failing computations. It is shown how HCLGNs can be used to model built in atoms and provide a new AND/OR parallel execution model. Recently, several software packages (graphical editors) have become available for editing and executing HLPNs. The simulation capabilities of the HLPN software offer opportunities to perform automated, interactive code walk throughs and also have potential for providing a framework for visual debugging environments. However, HCLGNs differ from the major classes of HLPNs for which software tools have been developed in primarily two ways: the tokens in the markings can have variables; and the firing of a transition may not only update the marking of the adjacent places, but may instantiate variables in tokens in the markings of places that are non adjacent to the fired transition. Thus, the existing packages can only provide graphical syntax editing and are not appropriate for graphical simulation of HCLGNs. We provide an algebraic characterization of HCLGNs that can serve as a design guideline for implementing HCLGNs.

On computability by logic programs

The problem of computational completeness of Horn clause logic programs is revisited. The standard results on representability of all computable predicates by Horn clause logic programs are not related to the real universe on which logic programs operate. SLD-resolution, which is the main mechanism to execute logic programs, may give answer substitutions with variables. As the main result we prove that computability by Horn clause logic programs is equivalent to standard computability over the Herbrand universe with variables. The semantics we use isS-semantics introduced by Falaschi et al. [3]. As an application of the main result we prove the existence of a metainterpreter for a sublanguage of real Prolog, written in the language of Horn clauses with the S-semantics. We also show that the traditional semantics of Prolog do not reflect its computational behavior from the viewpoint of recursion theory.

Towards a fuzzy logic programming system: a clausal form fuzzy logic

Traditional logic and logic programming languages cannot handle uncertainty. Fuzzy logic can, but it requires a readily computable form. One possible way to achieve this is to define a propositional fuzzy logic, extend this to a first order form, convert it to Horn clause form, and, finally, to devise a theorem prover to manipulate the Horn clauses. The first two steps have already been published. The paper completes the exercise by deriving a fuzzy Horn clause logic from the first order logic already published, and proposing a theorem prover to manipulate these clauses.

Equivalence and difference between institutions: simulating Horn Clause Logic with based algebras

In this paper, we investigate several logical frameworks whose expressiveness lies between Conditional Equational Logic and Horn Clause Logic. The main result deals with thePART-construction, which interprets total based algebras as partial algebras. This construction can be viewed as a simulation of Horn Clause Theories by means of Conditional Equational Theories. Other constructions in other frameworks are extendable to simulations in a similar way. The notion of categorical retractive simulation captures some essential properties of these, which allows us to measure the equivalence and difference between institutions.

Completeness of category-based equational deduction

Equational deduction is generalised within a category-based abstract model theory framework, and proved complete under a hypothesis of quantifier projectivity, using a semantic treatment that regards quantifiers as models rather than variables, and valuations as model morphisms rather than functions. Applications include many- and order-sorted (conditional) equational logics, Horn clause logic, equational deduction modulo a theory, constraint logics, and more, as well as any possible combination among them. In the cases of equational deduction modulo a theory and of constraint logic the completeness result is new. One important consequence is an abstract version of Herbrand's Theorem, which provides an abstract model theoretic foundation for equational and constraint logic programming.

Suprema of Open and Closed Formulas and Their Application to Resolution

We discuss an algebra of open formulas, SOAF. The algebra is equipped with the operator mgi-o , which computes suprema of sets of open formulas, ordered according to a weak form of instance ordering. Rules to compute such points are provided. These rules supply a computation procedure to unify open formulas. Next, we consider an algebra of closed formulas, SCAF. It has an operator mgi-c , which computes suprema of set of formulas ordered according to the instance preordering. We give rules to compute such points. These rules supply a computation procedure to unify closed formulas. Relations between these operators and unification in open and closed formulas, and weak unification in closed formulas, are addressed. Also, we consider clausal formulas and apply these considerations to the resolution process. This provides a rule alternative to resolution and shows that resolvents can be expressed and effectively obtained as suprema of closed formulas. The new rule need not resort to renaming, unification. and instantiation as first class operations. Finally, applications to resolution in Horn clause logic and logic programming are discussed. This leads to a more compact theory of logic programming, in which, for instance, the domain of substitutions is eliminated.

CHICA, An Abductive Planning System Based on Event Calculus

This article presents the theory and implementation of an artificial intelligence planner, CHICA. CHICA is a non-linear, domain independent planner based on techniques of computational logic. The representation language of the planner is Horn clause logic which is used to model event calculus, a logical theory of changing properties over time. The reasoning component is an abductive extension of SLDNF resolution for generating assumptions to prove a given goal. In event calculus, this procedure generates a plan of events and temporal relations necessary to prove the planning goal. CHICA uses domain contraints and techniques from contraint logic programming to efficiently implement inequality, as well as a specialized module to evaluate temporal relations. CHICA's generic search algorithm lets the implementor of a planning domain define a particular search strategy and specify domain heuristics to prune the search space. CHICA has solved a number of planning problems successfully: multiple robot block world problems, the assembly of a flashlight, and a room decoration problem. Extensions to classical Al-planning can be solved within the same framework, such as plan execution and replanning.

A semantics for λProlog

We give a model-theoretic semantics for the logic of higher-order Horn clauses, the basis of a form of the λProlog higher-order logic programming language. We define certain intensional general models and show that higher-order Horn clause logic is sound and complete with respect to them.

Preference logics: Towards a unified approach to nonmonotonicity in deductive reasoning

The notion ofpreference is central to most forms of nonmonotonic reasoning. Shoham, in his dissertation, used this notion to give a single semantical point of view from which most nonmonotonic reasoning systems could be studied. In this paper, we study the notion of preference closely and devise a class of logics of preference that extract the logicalcore of the notion of preference. Earlier attempts have been largely unsuccessful, because of adoption as matters of logic of certain theory-specific preference principles such asasymmetry andtransitivity. Soundness, completeness and decidability proofs for the logics are given. We define the notion of a preferential theory and reframe nonmonotonicity as a symbolic optimization problem where defaults are coded aspreference criteria which place preference orders on the models of a first-order theory. We study the relationship between normal default theories and show the correspondence between models of extensions and optimal worlds. We give a preferential account of some forms of circumscription. Thelocal nature of preference logic is contrasted with the global notion of normality and preference that is used by conditional logics of normality and cumulative inference operations. In related papers, we give a completely declarative semantics for the stable models of normal logic programs, a deontic logic based on preferences that is free of the anomalies of standard deontic logic, and extend Horn clause logic programming to impose partial orders on the bodies of clauses as declarative specification of the relaxation criteria for the truth-hood of the heads.

High-level Petri net model of logic program with negation

The formalism of nonmonotonic reasoning has been integrated into logic programming to define semantics for logic program with negation. Because a Petri net provides a uniform model for both the logic of knowledge and the control of inference, the class of high-level Petri nets called predicate/transition nets (PrT-nets) has been employed to study production rule based expert systems and Horn clause logic programs. We show that a PrT-net can implement the nonmonotonicity associated with a logic program with negation as well as the monotonicity of Horn clause logic program. In particular, we define a semantics for a normal logic program and implement it with PrT-net. We demonstrate that in the presence of inconsistency in a normal logic program, the semantics still works well by deducing meaningful answers. The variations and potential applications of the PrT-net are also addressed. >

The computational complexity of the satisfiability of modal Horn clauses for modal propositional logics

This paper presents complexity results about the satisfiability of modal Horn clauses for several modal propositional logics. Almost all these results are negative in the sense that restricting the input formula to modal Horn clauses does not decrease the inherent complexity of the satisfiability problem. We first show that, when restricted to modal Horn clauses, the satisfiability problem for any modal logic clauses as well as the satisfiability of unrestricted formulas for any of K, T, B, ans S4 is PSPACE-complete. This result refutes the expectation (Fariñas del Cerro and Penttonen 1987) of getting a polynomial-time algorithm for the satisfiability of modal Horn clauses for these logics as long as P ≠ PSPACE. Next, we consider S4.3 and extensions of K5 including K5, KD5, K45, KD45 and S5, the satisfiability problem for each of which in general is known to be NP-complete, and show that for each extension of K5, a polynomial-time algorithm for the satisfiability of modal Horn clauses can be obtained; but for S4.3, together with some linear tense logics closely related to S4.3 like Cl, SL and PL, the satisfiability of modal Horn clauses still remains NP-complete.

Probabilistic estimates for the generalized maximum satisfiability problem

The generalized maximum satisfiability problem (GMAXSAT) deals with variables taking their values in a finite set. A set of logical clauses is given and the goal is to find an assignment of values to the variables, minimizing the number β of unsatisfied clauses. For randomly generated instances of a uniform type we study the distribution of β, as well as the distribution of the maximal size α of a satisfiable subproblem, by means of the first and second moment method (Spencer, 1987). Numerical estimates for the distribution of α and β are given for some instances. In relation with the asymptotic behavior, we show that α has almost surely three possible values only. Furthermore, in the spirit of Burkard and Fincke (1985), we show that for some sequences of random instances, the size of which tends to ∞, the relative error of any algorithm for GMAXSAT tends almost surely towards zero.