Tableau Construction Research Articles

A tableau construction for finite linear-time temporal logic

This paper gives a tableau-based technique for converting formulas in finite propositional linear-time temporal logic (Finite LTL) into finite-state automata whose languages are the models of the given formula. Finite LTL differs from traditional LTL in that formulas are interpreted with respect to finite, rather than infinite, sequences of states; this fact means that traditional finite-state automata, rather than ω-automata such as those developed by Büchi and others, suffice for recognizing models of such formulas. The approach presented here works by associating with each state in the constructed automaton a Finite LTL formula satisfied by the sequences that can be accepted by the automaton when started in the given state. Such tableau-construction techniques are well-known in the setting of traditional LTL, where they are used to construct ω-automata; we adapt here for the setting of Finite LTL and finite-state automata. The resulting automata may be used as a basis for model checking, model synthesis, query checking and satisfiability testing.

Temporal prophecy for proving temporal properties of infinite-state systems

Various verification techniques for temporal properties transform temporal verification to safety verification. For infinite-state systems, these transformations are inherently imprecise. That is, for some instances, the temporal property holds, but the resulting safety property does not. This paper introduces a mechanism for tackling this imprecision. This mechanism, which we call temporal prophecy, is inspired by prophecy variables. Temporal prophecy refines an infinite-state system using first-order linear temporal logic formulas, via a suitable tableau construction. For a specific liveness-to-safety transformation based on first-order logic, we show that using temporal prophecy strictly increases the precision. Furthermore, temporal prophecy leads to robustness of the proof method, which is manifested by a cut elimination theorem. We integrate our approach into the Ivy deductive verification system, and show that it can handle challenging temporal verification examples.

THE LOGIC OF SEQUENCE FRAMES

AbstractThis paper investigates and develops generalizations of two-dimensional modal logics to any finite dimension. These logics are natural extensions of multidimensional systems known from the literature on logics for a priori knowledge. We prove a completeness theorem for propositional n-dimensional modal logics and show them to be decidable by means of a systematic tableau construction.

Branching-time logic [formula omitted] and its tree-style one-pass tableau: Extending fairness expressibility of [formula omitted

Temporal logic has become essential for various areas in computer science, most notably for the specification and verification of hardware and software systems. For the specification purposes rich temporal languages are required that, in particular, can express fairness constraints. For linear-time logics which deal with fairness in the linear-time setting, one-pass and two-pass tableau methods have been developed. In the repository of the CTL-type branching-time setting, the well-known logics ECTL and ECTL+ were developed to explicitly deal with fairness. However, due to the syntactical restrictions, these logics can only express restricted versions of fairness. The logic CTL⋆, often considered as ‘the full branching-time logic’ overcomes these restrictions on expressing fairness. However, CTL⋆ is extremely challenging for the application of verification techniques, and the tableau technique, in particular. For example, there is no one-pass tableau construction for CTL⋆, while one-pass tableau has an additional benefit enabling the formulation of dual sequent calculi that are often treated as more ‘natural’ being more friendly for human understanding. These two considerations lead to the following problem - are there logics that have richer expressiveness than ECTL+, allowing the formulation of a new range of fairness constraints with ‘until’ operator, yet ‘simpler’ than CTL⋆, and for which a one-pass tableau can be developed? Here we give a positive answer to this question, introducing a sub-logic of CTL⋆ called ECTL#, its tree-style one-pass tableau, and an algorithm for obtaining a systematic tableau, for any given admissible branching-time formulae. We prove the termination, soundness and completeness of the method. As tree-shaped one-pass tableaux are well suited for the automation and are amenable for the implementation and for the formulation of sequent calculi. Our results also open a prospect of relevant developments of the automation and implementation of the tableau method for ECTL#, and of a dual sequent calculi.

Efficient Decentralized LTL Monitoring Framework Using Tableau Technique

This paper presents a novel framework for decentralized monitoring of Linear Temporal Logic (LTL) formulas, under the situation where processes are synchronous and the formula is represented as a tableau. The tableau technique allows one to construct a semantic tree for the input LTL formula, which can be used to optimize the decentralized monitoring of LTL in various ways. Given a system P and an LTL formula φ, we construct a tableau T φ . The tableau T φ is used for two purposes: (a) to synthesize an efficient round-robin communication policy for processes, and (b) to find the minimal ways to decompose the formula and communicate observations of processes in an efficient way. In our framework, processes can propagate truth values of both atomic and compound formulas (non-atomic formulas) depending on the syntactic structure of the input LTL formula and the observation power of processes. We demonstrate that this approach of decentralized monitoring based on tableau construction is more straightforward, more flexible, and more likely to yield efficient solutions than alternative approaches.

Towards a Reasoning Model for Context-aware Systems: Modal Logic and the Tree Model Property

Modal logics forms a family of formalisms widely used as reasoning frameworks in diverse areas of computer science. Description logics and their application to the web semantic is a notable example. Also, description logics have been recently used as a reasoning model for context-aware systems. Most reasoning algorithms for modal (de- scription) logics are based on tableau constructions. In this work, we propose a reasoning (satisability) algorithm for the multi-modal Km with converse. The algorithm is based on the nite tree model property and a Fischer-Ladner construction. We show the algorithm is sound and complete, and we provide the corresponding complexity analysis. We also present some exploratory results of a preliminary implementation of the algorithm.

Reducing behavioural to structural properties of programs with procedures

There is an intimate link between program structure and behaviour. Exploiting this link to phrase program correctness problems in terms of the structural properties of a program graph rather than in terms of its unfoldings is a useful strategy for making analyses more tractable. The present paper presents a characterisation of behavioural program properties through sets of structural properties by means of a translation. The characterisation is given in the context of a program model based on control flow graphs of sequential programs with procedures, abstracting away completely from program data, and properties expressed in a fragment of the modal μ-calculus with boxes and greatest fixed-points only. The property translation is based on a tableau construction that conceptually amounts to symbolic execution of the behavioural formula, collecting structural constraints along the way. By keeping track of the subformulae that have been examined, recursion in the structural constraints can be identified and captured by fixed-point formulae. The tableau construction terminates, and the characterisation is exact, i.e., the translation is sound and complete. A prototype implementation has been developed. In addition, we show how the translation can be extended beyond the basic flow graph model and safety logic to richer behavioural models (such as open programs) and richer program models (including Boolean programs), and discuss possible extensions for more complex logics. We present several applications of the characterisation, in particular sound and complete compositional verification for behavioural properties based on maximal models.

A tableau-based decision procedure for CTL*

AbstractWe present a sound, complete and implementable tableau method for deciding satisfiability of formulas in the propositional version of computation tree logic CTL*. This is the first such tableau. CTL* is an exceptionally important temporal logic with applications from hardware design to agent reasoning, but there is no easily automated reasoning approach to CTL*. The tableau here is a traditional tree-shaped or top-down style tableau, and affords the possibility of reasonably quick decisions on the satisfiability of medium-sized formulas and construction of small models for them. A straightforward subroutine is given for determining when looping allows successful branch termination, but much needed further development is left as future work. In particular, a more general repetition prevention mechanism is needed to speed up the task of tableau construction.

An efficient approach to nominal equalities in hybrid logic tableaux

Basic hybrid logic extends modal logic with the possibility of naming worlds by means of a distinguished class of atoms (called nominals) and the so-called satisfaction operator, that allows one to state that a given formula holds at the world named a, for some nominal a. Hence, in particular, hybrid formulae include “equality” assertions, stating that two nominals are distinct names for the same world. The treatment of such nominal equalities in proof systems for hybrid logics may induce many redundancies. This paper introduces an internalized tableau system for basic hybrid logic, significantly reducing such redundancies. The calculus enjoys a strong termination property: tableau construction terminates without relying on any specific rule application strategy, and no loop-checking is needed. The treatment of nominal equalities specific of the proposed calculus is briefly compared to other approaches. Its practical advantages are demonstrated by empirical results obtained by use of implemented systems. Finally, it is briefly shown how to extend the calculus to include the global and converse modalities.

A Tableau Construction Approach to Control Synthesis of FSMs Using Simulation Relations

We propose a new tableau construction which builds an FSM, instead of a Kripke structure, from a formula in a class of temporal logic named ASTL. This FSM is a maximal model of the formula under the preorder derived from simulation relations. Additionally, we propose a method using the tableaus to build controllers in a certain topology of interconnected FSMs. We can use ASTL to describe the desired behaviors of the control system. This method is applicable to generating digital circuits. Moreover, this method accepts a wider range of specifications than conventional methods.

A Decision Procedure and Complete Axiomatization of Finite Interval Temporal Logic with Projection

This paper presents a complete axiomatization for propositional interval temporal logic (PITL) with projection. The axiomatization is based on a tableau decision procedure for the logic, which in turn is founded upon a normal form for PITL formulae. The construction of the axiomatization provides a general mechanism for generating axiomatizations thus: given a normal form for a new connective, axioms can be generated for the connective from the tableau construction using that normal form. The paper concludes with a discussion of aspects of compositionality for PITL with projection.

Naming Worlds in Modal and Temporal Logic

In this paper we suggest adding to predicate modal and temporal logic a locality predicate W which gives names to worlds (or time points). We also study an equal time predicate D(x, y) which states that two time points are at the same distance from the root. We provide the systems studied with complete axiomatizations and illustrate the expressive power gained for modal logic by simulating other logics. The completeness proofs rely on the fairly intuitive notion of a configuration in order to use a proof technique similar to a Henkin completion mixed with a tableau construction. The main elements of the completeness proofs are given for each case, while purely technical results are grouped in the appendix.

Ordered Semantic Hyper Tableaux

A family of tableau methods, called i>ordered semantic hyper (i>OSH) i>tableau methods for first-order theories with function symbols, is presented. These methods permit semantic information to guide the search for a proof. They also may make use of orderings on literals, clauses, and interpretations to guide the search. In a typical tableau, the branches represent conjunctions of literals, and the tableau represents the disjunction of the branches. An OSH tableau is as usual except that each branch i>B has an interpretation i>I0[i>B] associated with it, where i>I0 is an interpretation supplied at the beginning and i>I0[i>B] is the interpretation most like i>I0 that satisfies i>B. Only clauses that i>I0[i>B] falsifies may be used to expand the branch i>B, thus restricting the kinds of tableau that can be constructed. This restriction guarantees the i>goal sensitivity of these methods if i>I0 is properly chosen. Certain choices of i>I0 may produce a purely bottom-up tableau construction, while others may result in goal-oriented evaluation for a given query. The choices of which branch is selected for expansion and which clause is used to expand this branch are examined and their effects on the OSH tableau methods considered. A branch reordering method is also studied, as well as a branch pruning technique called i>complement modification, that adds additional literals to branches in a soundness-preserving manner. All members of the family of OSH tableaux are shown to be sound, complete, and proof convergent for refutations. Proof convergence means that any allowable sequence of operations will eventually find a proof, if one exists. OSH tableaux are powerful enough to be treated as a generalization of several classes of tableau discussed in the literature, including forward chaining and backward chaining procedures. Therefore, they can be used for efficient query processing.

Combining Local and Global Model Checking

The verification process of reactive systems in local model checking [1,7] and in explicit state model checking is[13,15] on-the-fly. Therefore only those states of a system have to be traversed that are necessary to prove a property. In addition, if the property does not hold, than often only a small subset of the state space has to be traversed to produce a counterexample. Global model checking [6,23] and, in particular, symbolic model checking [4,22] can utilize compact representations of the state space, e.g. BDDs [3], to handle much larger designs than what is possible with local and explicit model checking. We present a new model checking algorithm for LTL that combines both approaches. In essence, it is a generalization of the tableau construction of [1] that enables the use of BDDs but still is on-the-fly.

On the Construction of Monitors for Temporal Logic Properties

Temporal logic is a valuable tool for specifying correctness properties of reactive programs. With the advent of temporal logic model checkers, it has become an important aid for the verification of concurrent and reactive systems. In model checking the temporal logic properties are verified against models expressed in the tool's modelling language. In addition, model-checking techniques are useful to test actual implementations or to verify models of the system that are too detailed to be analysed by a model checker, by means of, for instance, simulation.A tableau construction is an algorithm that translates a temporal logic formula into a finite-state automaton that accepts precisely all the models of the formula. It is a key ingredient to checking satisfiability of a formula as well as to the automata-theoretic approach to model checking. An improvement to the efficiency of tableau constructions has been the development of on-the-fly versions.In this paper, we present a particular tableau construction for the incremental analysis of execution traces during test, simulation or model-checking. The automaton forms the basis of a monitor that detects both good and bad prefix of a particular kind, namely those that are informative for the property under investigation. We elaborate on the construction of the monitor and demonstrate its correctness.

Some generalizations of the criss-cross method for the linear complementarity problem of oriented matroids

Quadratic programming, symmetry, positive (semi) definiteness and the linear complementary problem were generalized by Morris and Todd to oriented matroids. Todd gave a constructive solution for the quadratic programming problem of oriented matroids. Using Las Vergnas' lexicographic extension and Bland's basic tableau construction Todd generalized Lemke's quadratic programming algorithm for this problem.

Multisectioned partitions of integers

This paper presents identities on generating functions for multisectioned partitions of integers by developing in the language of partitions some powerful and essentially combinatorial techniques from the literature of principal differential ideals. D. Mead has stated in Vol. 42 of this journal that one can obtain interesting combinatorial relations by constructing different vector space bases for a subspace of a differential ring and using the fact that the cardinality of all bases is the same. The results of the present paper are of this nature. In particular, we enumerate certain sets of ordered pairs of that have a central role in Mead's paper. Tableaux were used by A. Young and others to study the structure of the symmetric groups Sn. In [3], D. Knuth used an insertion into tableau construction of C. Schensted to give a direct 1-to-l correspondence between generalized permutations and ordered pairs of generalized Young tableaux having the same shape. In [5], Mead independently proved the existence of such a bisection while developing a new vector space basis for the ring of differential polynomials in n independent differential indeterminates. Mead's paper deals with principal differential ideals generated by Wronskians and used determinantal identities going back to Gayley. The ordered pairs of used by Mead appear in a more general setting in the paper [1] by Doubilet, Rota, and Stein.

Intuitive semantics for first-degree entailments and ?coupled trees?

Classically, an argument A therefore B is ‘valid’ (or A is said to ‘entail’ B) if and only if (iff) each situation (model) is such that either A is false or B is true. This fits well with so-called ‘tableau’ methods for showing that A entails B by working out the mutual inconsistency of A and ~B. But both the classical notion of validity and the corresponding tableau methods allow that A may entail B because of some feature of A alone, irrespective of B, and vice versa. Thus if A is a contradiction, then each situation is such that A is false, and so a fortiori is such that A is false or B is true. And if A is a contradiction, then a tableau construction will show that A is inconsistent, and so a fortiori that A and ~B are inconsistent. Of course, the same points can be made dually when B is a logical truth.