Tableau Calculus Research Articles

Extended ASP Tableaux and rule redundancy in normal logic programs

AbstractWe introduce an extended tableau calculus for answer set programming (ASP). The proof system is based on the ASP tableaux defined in the work by Gebser and Schaub (Tableau calculi for answer set programming. In Proceedings of the 22nd International Conference on Logic Programming (ICLP 2006), S. Etalle and M. Truszczynski, Eds. Lecture Notes in Computer Science, vol. 4079. Springer, 11–25) with an added extension rule. We investigate the power of Extended ASP Tableaux both theoretically and empirically. We study the relationship of Extended ASP Tableaux with the Extended Resolution proof system defined by Tseitin for sets of clauses, and separate Extended ASP Tableaux from ASP Tableaux by giving a polynomial-length proof for a family of normal logic programs {Φn} for which ASP Tableaux has exponential-length minimal proofs with respect to n. Additionally, Extended ASP Tableaux imply interesting insight into the effect of program simplification on the lengths of proofs in ASP. Closely related to Extended ASP Tableaux, we empirically investigate the effect of redundant rules on the efficiency of ASP solving.

Partial information basis for agent-based collaborative dialogue

We propose a partial information state-based framework for collaborative dialogue and argument between agents. We employ a three-valued based nonmonotonic logic, NML3, for representing and reasoning about Partial Information States (PIS). NML3 formalizes some aspects of revisable reasoning and it is sound and complete. Within the framework of NML3, we present a formalization of some basic dialogue moves and the rules of protocols of some types of dialogue. The rules of a protocol are nonmonotonic in the sense that the set of propositions to which an agent is committed and the validity of moves vary from one move to another. The use of PIS allows an agent to expand consistently its viewpoint with some of the propositions to which another agent, involved in a dialogue, is overtly committed. A proof method for the logic NML3 has been successfully implemented as an automatic theorem prover. We show, via some examples, that the tableau method employed to implement the theorem prover allows an agent, absolute access to every stage of a proof process. This access is useful for constructive argumentation and for finding cooperative and/or informative answers.

Connection Tableaux with Lazy Paramodulation

It is well known that the connection refinement of clause tableaux with paramodulation is incomplete (even with weak connections). In this paper, we present a new connection tableau calculus for logic with equality. This calculus is based on a lazy form of paramodulation where parts of the unification step become auxiliary subgoals in a tableau and may be subjected to subsequent paramodulations. Our calculus uses ordering constraints and a certain form of the basicness restriction.

New Inference Rules for Max-SAT

Exact Max-SAT solvers, compared with SAT solvers, apply little inference at each node of the proof tree. Commonly used SAT inference rules like unit propagation produce a simplified formula that preserves satisfiability but, unfortunately, solving the Max-SAT problem for the simplified formula is not equivalent to solving it for the original formula. In this paper, we define a number of original inference rules that, besides being applied efficiently, transform Max-SAT instances into equivalent Max-SAT instances which are easier to solve. The soundness of the rules, that can be seen as refinements of unit resolution adapted to Max-SAT, are proved in a novel and simple way via an integer programming transformation. With the aim of finding out how powerful the inference rules are in practice, we have developed a new Max-SAT solver, called MaxSatz, which incorporates those rules, and performed an experimental investigation. The results provide empirical evidence that MaxSatz is very competitive, at least, on random Max-2SAT, random Max-3SAT, Max-Cut, and Graph 3-coloring instances, as well as on the benchmarks from the Max-SAT Evaluation 2006.

An approach to automatic deductive synthesis of functional programs

The work deals with automatic deductive synthesis of functional programs. Formal specification of a program is taken as a mathematical existence theorem and while proving it, we derive a program and simultaneously prove that this program corresponds to given specification. Several problems have to be resolved for automatic synthesis: the choice of synthesis rules that allows us to derive the basic constructions of a functional program, order of rule application and choice of a particular induction rule. The method proposed here is based on the deductive tableau method. The basic method gives rules for functional program construction. To determine the proof strategy we use some external heuristics, including rippling. And for the induction hypothesis formation the combination of rippling and the deductive tableau method became very useful. The proposed techniques are implemented in the system ALISA (Automatic Lisp Synthesizer) and used for automatic synthesis of several functions in the Lisp language.

Operational Semantics for Lazy Evaluation

An operational semantics for lazy evaluation of a calculus without higher order functions was defined. Although it optimizes many aspects of implementation, e.g. there is a sharing in the recursive computation, there is no α conversion, the heap is automatically reclaimed, and an attempt to evaluate an argument is done at most once. It is still suitable for reasoning about program behavior and proofs of program correctness; this is primarily due to the definition via inferences and axioms which allows for proofs by induction on the height of the proof tree. We also proved the correctness of this operational semantics by showing that it is equivalent with respect to the values calculated to the operational semantics of LAZY-PCF+SHAR due to S. Purushothaman Iyer and Jill Seaman.

Inconsistency-tolerant description logic. Part II: A tableau algorithm for [formula omitted

In the first part of this paper, we motivated and defined three systems of constructive and inconsistency-tolerant description logic. The variety of arising systems is conditioned by the variety of approaches to defining modalities in the constructive setting. We also presented sound and complete tableau calculi for the logics under consideration. Whereas these calculi were not meant to give rise to tableau algorithms, in the present second part of the paper, after providing some motivation and recalling the main definitions, we adapt methods developed by R. Dyckhoff and by I. Horrocks and U. Sattler in order to define a tableau algorithm for our basic four-valued constructive description logic CALC C . Notice that among the three logics defined in the first part of the paper, CALC C is the only logic which lends itself to applications, because for the other logics it is unknown whether they are elementarily decidable. The presented algorithm for CALC C is the first example of an elementary decision procedure for a constructive description logic.

Testing XML constraint satisfiability

In a previous paper, we have showed that Hybrid Modal Logic can be successfully used to model semistructured data and provides a simple and well suited formalism for capturing “well typed” references and of course a powerful language for expressing constraint. This paper builds on the previous one and provides a tableau proof technique for constraint satisfiability testing in the presence of schemas.

Modelling of a hybrid system through differential predicate transition Petri nets model and proof tree

PurposeThe aim of this work is to propose a logical‐based approach for deriving the critical scenarii from a differential predicate transition Petri nets model and proof tree, to evaluate diagnosis and safety of hybrid systems.Design/methodology/approachThe new approach integrates the Petri net model and linear logic. The design process is twofold: first, it considers the specific translation from Petri net to linear logic, and, second, it studies the relationship between the order of the rule application in the sequent proof and the processes associated with the same reachability problem.FindingsWith linear logic and Petri net it is proved that, from a canonical proof tree, a partial order can be derived (based on causality point of view) of transition firing and focusing the search of the parts of the model that are interesting for safety analysis. Each scenario is given by a partial order between the events necessary for the occurrence of the feared event.Originality/valueThis paper deals with safety in design of hybrid systems. For this purpose, it is important to characterize feared behaviours (which are critical) in the early design stage.

A New Modal Approach to the Logic of Intervals

In Artificial Intelligence there is a need for reasoning about continuous processes, where assertions refer to time intervals rather than time points. Taking our lead from van Benthem’s treatment of interval temporal structures and Halpern and Shoham’s work on intervals, we present a interval temporal logic with two binary relations, precedence and inclusion. We study the logic in its full generality without making any assumptions about the underlying nature of time, be it discrete or dense, linear or branching. We identify two general classes of interval temporal structures, minimal interval structures and van Benthem minimal interval structures. We show that in our interval temporal language, the two classes in fact have the same logic. We go on to prove that the logic of minimal interval structures is complete and decidable, possessing the finite model property, and that the satisfiability problem is PSPACEcomplete. In order to establish the complexity result we extend the tableau method introduced by Horrocks et al., which treats transitive and inverse relations only, to also incorporate interaction between relations. We go on to identify some important limitations in the expressive power of our logic, before concluding the paper by raising a number of interesting questions that follow from our work.

Analisis Kinerja Reksa Dana Nusantara PT Bhakti Asset Management dan Penyusunan Portfolio Optimum Teoritis Periode Januari – April 2005

The research objective is to find theoritical optimum portfolio of the Reksa Dana Nusantara (RDN) of BAM, using data from January – April 2005 (60 workdays). The coefficient of variance (CV) will be used to determine the stock rank, then it will be optimize with 3 methods ; linear programing, trial-error, and tableau method. The research found that RDN has Sharpe Performance Index (SPI) of 1.15 but the market can reach 1.60. The linear programing can not find optimum solution of portfolios consist of 9-5 stocks, but it can find the best portfolio consist in 3 stocks (TLKM, TKIM an CMNP) with SPI of 2.19. Overall, the research summarizes that RDN is still below the market performance.

Linear clause generation by Tableaux and DAGs

Clause generation is a preliminary step in theorem proving since most of the state-of-the-art theorem proving methods act on clause sets. Several clause generating algorithms are known. Most of them rewrite a formula according to well-known logical equivalences, thus they are quite complicated and produce not very understandable information on their functioning for humans. There are other methods that can be considered as ones based on tableaux, but only in propositional logic. In this paper, we propose a new method for clause generation in first-order logic. Since it inherits rules from analytic tableaux, analytic dual tableaux, and free-variable tableaux, this method is called clause generating tableaux (CGT). All of the known clause generating algorithms are exponential, so is CGT. However, by switching to directed acyclic graphs (DAGs) from trees, we propose a linear CGT method. Another advantageous feature is the detection of valid clauses only by the closing of CGT branches. Last but not least, CGT generates a graph as output, which is visual and easy-to-understand. Thus, CGT can also be used in teaching logic and theorem proving.

外延的真理様相の論理としての部分論理

In [Kachi 2002a] I made comparison between several definitions of validity in Simple Partial Logic (SPL) and adopted two of them as most appropriate. In this paper, after elaborating more on these two definitions than in my previous paper and considering the characteristics of Partial Semantics, in which these definitions are given, I construct a tableau proof theory and prove its soundness and completeness. Then, based on the characterization of Partial Semantics, I will show that we can regard SPL as a logic of extensional alethic modality.

A new Young tableau method of obtaining equivalent-electron Young basis

In this paper, the projection function of regular Young tableau of equivalent-electrons is proved to be a Slauer function, and the operation rules of Young's permutation operators operating on the Slater function are given. The Young operators can be distinguished as elimination operator and effective permutation operator by the normalization of Young basis, then the constructive rules of permutation operators of the orthogonal normalization of Young basis is given. By this method, the number of Young's permutation operators is reduced and the new Young tableau method of obtaining equivalent-electron Young basis is represented.

Comparing Instance Generation Methods for Automated Reasoning

The clause-linking technique of Lee and Plaisted proves the unsatisfiability of a set of first-order clauses by generating a sufficiently large set of instances of these clauses that can be shown to be propositionally unsatisfiable. In recent years, this approach has been refined in several directions, leading to both tableau-based methods, such as the disconnection tableau calculus, and saturation-based methods, such as primal partial instantiation and resolution-based instance generation. We investigate the relationship between these calculi and answer the question to what extent refutation or consistency proofs in one calculus can be simulated in another one.

The Disconnection Tableau Calculus

In this paper we give a comprehensive presentation of the disconnection tableau calculus, a proof method for formulas in classical first-order clause logic. The distinguishing property of this calculus is that it uses unification in such a manner that important proof-theoretic advantages of the classical (i.e., Smullyan-style) tableau calculus are preserved, specifically the termination and model generation characteristics for certain formula classes. Additionally, the calculus is well suited for fully automated proof search. The calculus is described in detail with soundness and completeness proofs, and a number of important calculus refinements developed over the past years are presented. Referring to the model-finding abilities of the disconnection calculus, we explain the extraction and representation of models. We also describe the integration of paramodulation-based equality handling. Finally, we give an overview of related methods.

Superposition-based Equality Handling for Analytic Tableaux

We present a variant of the basic ordered superposition rules to handle equality in an analytic free-variable tableau calculus. We prove completeness of this calculus by an adaptation of the model generation technique commonly used for completeness proofs of superposition in the context of resolution calculi. The calculi and the completeness proof are compared to earlier results of Degtyarev and Voronkov. Some variations and refinements are discussed.

The complexity of analytic tableaux

AbstractThe method of analytic tableaux is employed in many introductory texts and has also been used quite extensively as a basis for automated theorem proving. In this paper, we discuss the complexity of the system as a method for refuting contradictory sets of clauses, and resolve several open questions. We discuss the three forms of analytic tableaux: clausal tableaux, generalized clausal tableaux, and binary tableaux. We resolve the relative complexity of these three forms of tableaux proofs and also resolve the relative complexity of analytic tableaux versus resolution. We show that there is a quasi-polynomial simulation of tree resolution by analytic tableaux; this simulation is close to optimal, since we give a matching lower bound that is tight to within a polynomial.

Anytime Approximations of Classical Logic from Above

In this article we present s1, a family of logics that is useful to disprove propositional formulas by means of an anytime approximation process. The systems follows the paradigm of a parameterized family of logics established by Schaerf’s and Cadoli’s system S1. We show that s1 inherits several of the nice properties of S1, while presenting several attractive new properties. The family s1 deals with the full propositional language, has a complete tableau proof system which provides for incremental approximations; furthermore, it constitutes a full approximation of classical logic from above, with an approximation process with better relevance and locality properties than S1. When applied to clausal inference, s1 provides a strong simplification method. An application of s1 to model-based diagnosis is presented, demonstrating how the solution to this problem can benefit from the use of s1 approximations.

A Tableau for Bundled CTL

We present a sound, complete and relatively straightforward tableau method for deciding valid formulas in the propositional version of the bundled (or suffix and fusion closed) computation tree logic BCTL*. This proves that BCTL* is decidable. It is also moderately useful to have a tableau available for a reasonably expressive branching-time temporal logic. However, the main interest in this should be that it leads us closer to being able to devise a tableau-based technique for theorem-proving in the important full computational tree logic CTL*.