Tableau Rules Research Articles

Coalgebraic Satisfiability Checking for Arithmetic $\mu$-Calculi

The coalgebraic $\mu$-calculus provides a generic semantic framework for fixpoint logics over systems whose branching type goes beyond the standard relational setup, e.g. probabilistic, weighted, or game-based. Previous work on the coalgebraic $\mu$-calculus includes an exponential-time upper bound on satisfiability checking, which however relies on the availability of tableau rules for the next-step modalities that are sufficiently well-behaved in a formally defined sense; in particular, rule matches need to be representable by polynomial-sized codes, and the sequent duals of the rules need to absorb cut. While such rule sets have been identified for some important cases, they are not known to exist in all cases of interest, in particular ones involving either integer weights as in the graded $\mu$-calculus, or real-valued weights in combination with non-linear arithmetic. In the present work, we prove the same upper complexity bound under more general assumptions, specifically regarding the complexity of the (much simpler) satisfiability problem for the underlying one-step logic, roughly described as the nesting-free next-step fragment of the logic. The bound is realized by a generic global caching algorithm that supports on-the-fly satisfiability checking. Notably, our approach directly accommodates unguarded formulae, and thus avoids use of the guardedness transformation. Example applications include new exponential-time upper bounds for satisfiability checking in an extension of the graded $\mu$-calculus with polynomial inequalities (including positive Presburger arithmetic), as well as an extension of the (two-valued) probabilistic $\mu$-calculus with polynomial inequalities.

Coalgebraic Reasoning with Global Assumptions in Arithmetic Modal Logics

We establish a generic upper bound ExpTime for reasoning with global assumptions (also known as TBoxes) in coalgebraic modal logics. Unlike earlier results of this kind, our bound does not require a tractable set of tableau rules for the instance logics, so that the result applies to wider classes of logics. Examples are Presburger modal logic, which extends graded modal logic with linear inequalities over numbers of successors, and probabilistic modal logic with polynomial inequalities over probabilities. We establish the theoretical upper bound using a type elimination algorithm. We also provide a global caching algorithm that potentially avoids building the entire exponential-sized space of candidate states, and thus offers a basis for practical reasoning. This algorithm still involves frequent fixpoint computations; we show how these can be handled efficiently in a concrete algorithm modelled on Liu and Smolka’s linear-time fixpoint algorithm. Finally, we show that the upper complexity bound is preserved under adding nominals to the logic, i.e., in coalgebraic hybrid logic.

Optimizing a tableau reasoner and its implementation in Prolog

One of the foremost reasoning services for knowledge bases is finding all the justifications for a query. This is useful for debugging purpose and for coping with uncertainty. Among Description Logics (DLs) reasoners, the tableau algorithm is one of the most used. However, in order to collect the justifications, the reasoners must manage the non-determinism of the tableau method. For these reasons, a Prolog implementation can facilitate the management of such non-determinism.The TRILL framework contains three probabilistic reasoners written in Prolog: TRILL, TRILLP and TORNADO. Since they are all part of the same framework, the choice about which to use can be done easily via the framework settings. Each one of them uses different approaches for probabilistic inference and handles different DLs flavors. Our previous work showed that they can sometimes achieve better results than state-of-the-art (non-)probabilistic reasoners.In this paper we present two optimizations that improve the performances of the TRILL reasoners. The first one consists into identifying the fragment of the KB that allows to perform inference without losing the completeness. The second one modifies which tableau rule to apply and their order of application, in order to reduce the number of operations. Experimental results show the effectiveness of the introduced optimizations.

Optimizing a Tableau Reasoner and Its Implementation in Prolog

One of the foremost reasoning services for knowledge bases is finding all the justifications for a query. This is useful for debugging purpose and for coping with uncertainty. Among Description Logics (DLs) reasoners, the tableau algorithm is one of the most used. However, in order to collect the justifications, the reasoners must manage the non-determinism of the tableau method. For these reasons, a Prolog implementation can facilitate the management of such non-determinism. The TRILL framework contains three probabilistic reasoners written in Prolog: TRILL, TRILL and TORNADO. Since they are all part of the same framework, the choice about which to use can be done easily via the framework settings. Each one of them uses different approaches for probabilistic inference and handles different DLs flavours. Our previous work showed that they can sometimes achieve better results than state-of-the-art (non-)probabilistic reasoners. In this paper we present two optimizations that improve the performances of the TRILL reasoners. The first one consists into identifying the fragment of the KB that allows to perform inference without losing the completeness. The second one modifies which tableau rule to apply and their order of application, in order to reduce the number of operations. Experimental results show the effectiveness of the introduced optimizations.

Quantified Temporal Alethic Boulesic Doxastic Logic

The paper develops a set of quantified temporal alethic boulesic doxastic systems. Every system in this set consists of five parts: a ‘quantified’ part, a temporal part, a modal (alethic) part, a boulesic part and a doxastic part. There are no systems in the literature that combine all of these branches of logic. Hence, all systems in this paper are new. Every system is defined both semantically and proof-theoretically. The semantic apparatus consists of a kind of T times W models, and the proof-theoretical apparatus of semantic tableaux. The ‘quantified part’ of the systems includes relational predicates and the identity symbol. The quantifiers are, in effect, a kind of possibilist quantifiers that vary over every object in the domain. The tableaux rules are classical. The alethic part contains two types of modal operators for absolute and historical necessity and possibility. According to ‘boulesic logic’ (the logic of the will), ‘willing’ (‘consenting’, ‘rejecting’, ‘indifference’ and ‘non-indifference’) is a kind of modal operator. Doxastic logic is the logic of beliefs; it treats ‘believing’ (and ‘conceiving’) as a kind of modal operator. I will explore some possible relationships between these different parts, and investigate some principles that include more than one type of logical expression. I will show that every tableau system in the paper is sound and complete with respect to its semantics. Finally, I consider an example of a valid argument and an example of an invalid sentence. I show how one can use semantic tableaux to establish validity and invalidity and read off countermodels. These examples illustrate the philosophical usefulness of the systems that are introduced in this paper.

A machine learning approach for optimizing heuristic decision‐making in Web Ontology Language reasoners

AbstractDescription logics (DLs) are formalisms for representing knowledge bases of application domains. The Web Ontology Language (OWL) is a syntactic variant of a very expressive DL. OWL reasoners can infer implied information from OWL ontologies. The performance of OWL reasoners can be severely affected by situations that require decision‐making over many alternatives. Such a nondeterministic behavior is often controlled by heuristics that are based on insufficient information. This article proposes a novel OWL reasoning approach that applies machine learning (ML) to implement pragmatic and optimal decision‐making strategies in such situations. Disjunctions occurring in ontologies are one source of nondeterministic actions in reasoners. We propose two ML‐based approaches to reduce the nondeterminism caused by dealing with disjunctions. The first approach is restricted to propositional DL while the second one can deal with standard DL. Both approaches speed up our ML‐based reasoner by up to two orders of magnitude in comparison to the non‐ML reasoner. Another source of nondeterministic actions is the order in which tableau rules should be applied. On average, our ML‐based approach achieves a speedup of two orders of magnitude when compared to the most expensive rule ordering of the non‐ML reasoner.

Vanishing of Littlewood–Richardson polynomials is in P

J. DeLoera-T. McAllister and K. D. Mulmuley-H. Narayanan-M. Sohoni independently proved that determining the vanishing of Littlewood-Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert calculus numbers, we prove the generalization to the Littlewood-Richardson polynomials that control equivariant cohomology of Grassmannians. We construct a polytope using the edge-labeled tableau rule of H. Thomas-A. Yong. Our proof then combines a saturation theorem of D. Anderson-E. Richmond-A. Yong, a reading order independence property, and E. Tardos' algorithm for combinatorial linear programming.

Equivariant -theory of Grassmannians II: the Knutson–Vakil conjecture

In 2005, Knutson–Vakil conjectured apuzzlerule for equivariant$K$-theory of Grassmannians. We resolve this conjecture. After giving a correction, we establish a modified rule by combinatorially connecting it to the authors’ recently proved tableau rule for the same Schubert calculus problem.

Defining Cognitive Logics by Non-Classical Tableau Rules

Abstract In the paper we propose a new approach to formalization of cognitive logics. By cognitive logics we understand supraclassical, but non-trivial consequence operations, defined in a propositional language. We extend some paradigm of tableau methods, in which classical consequence Cn is defined, to stronger logics - monotonic, as well as non-monotonic ones - by specific use of non-classical tableau rules. So far, in that context tableaus have been treated as a way of formalizing other approaches to supraclassical logics, but we use them autonomically to generate various consequence operations. It requires a description of the hierarchy of non-classical tableau rules that result in different supraclassical consequence operations, so we give it.

A New Rule for LTL Tableaux

Propositional linear time temporal logic (LTL) is the standard temporal logic for computing applications and many reasoning techniques and tools have been developed for it. Tableaux for deciding satisfiability have existed since the 1980s. However, the tableaux for this logic do not look like traditional tree-shaped tableau systems and their processing is often quite complicated. In this paper, we introduce a novel style of tableau rule which supports a new simple traditional-style tree-shaped tableau for LTL. We prove that it is sound and complete. As well as being simple to understand, to introduce to students and to use, it is also simple to implement and is competitive against state of the art systems. It is particularly suitable for parallel implementations.

A fast transition of linear temporal logic formulae to transition-based Büchi automata

This paper presents an algorithm based on revised tableau rules, which converts linear temporal logic formulae to transition-based Buchi automata more efficiently. The algorithm is geared towards being used in model checking in on-the-fly fashion. The algorithm circumvents the intermediate steps and the simplification process that follows, and therefore performs more efficiently. By attaching the satisfaction information of the infinite sequence on states and transitions, we can judge whether the runs of the automata are acceptable by using only one acceptance condition set, but not multiple ones. On-the-fly simplifications as well as binary decision diagram presentation are adopted in the algorithm to gain significant reduction both on the size of product automata and on the computational complexity. It can expand the state nodes while detecting the validity of these, removing the invalid nodes and combining equivalent states and transitions. By comparative testing with other conversion tools, the algorithm runs faster, with fewer states and transitions of the automaton.

Degeneralization Algorithm for Generation of Büchi Automata Based on Contented Situation

We present on-the-fly degeneralization algorithm used to transform generalized Büchi automata (GBA) into Büchi Automata (BA) different from the standard degeneralization algorithm. Contented situation, which is used to record what acceptance conditions are satisfiable during expanding LTL formulae, is attached to the states and transitions in the BA. In order to get the deterministic BA, the Shannon expansion is used recursively when we expand LTL formulae by applying the tableau rules. On-the-fly degeneralization algorithm is carried out in each step of the expansion of LTL formulae. Ordered binary decision diagrams are used to represent the BA and simplify LTL formulae. The temporary automata are stored as syntax directed acyclic graph in order to save storage space. These ideas are implemented in a conversion algorithm used to build a property automaton corresponding to the given LTL formulae. We compare our method to previous work and show that it is more efficient for four sets of random formulae generated by LBTT.

Analytic Tableaux for all of SIXTEEN 3

In this paper we give an analytic tableau calculus PL16 for a functionally complete extension of Shramko and Wansing’s logic. The calculus is based on signed formulas and a single set of tableau rules is involved in axiomatising each of the four entailment relations ⊧t, ⊧f, ⊧i, and ⊧ under consideration—the differences only residing in initial assignments of signs to formulas. Proving that two sets of formulas are in one of the first three entailment relations will in general require developing four tableaux, while proving that they are in the ⊧ relation may require six.

SLAP: Specification logic of actions with probability

A logic for specifying probabilistic transition systems is presented. Our perspective is that of agents performing actions. A procedure for deciding whether sentences in this logic are valid is provided. One of the main contributions of the paper is the formulation of the decision procedure: a tableau system which appeals to solving systems of linear equations. The tableau rules eliminate propositional connectives, then, for all open branches of the tableau tree, systems of linear equations are generated and checked for feasibility. Proofs of soundness, completeness and termination of the decision procedure are provided.

Decision procedures for some strong hybrid logics

Hybrid logics are extensions of standard modal logics, which significantly increase the expressive power of the latter. Since most of hybrid logics are known to be decidable, decision procedures for them is a widely investigated field of research. So far, several tableau calculi for hybrid logics have been presented in the literature. In this paper we introduce a sound, complete and terminating tableau calculus T H(@,E,D, ♦ −) for hybrid logics with the satisfaction operators, the universal modality, the difference modality and the inverse modality as well as the corresponding sequent calculus S H(@,E,D, ♦ −) . They not only uniformly cover relatively wide range of various hybrid logics but they are also conceptually simple and enable effective search for a minimal model for a satisfiable formula. The main novelty is the exploitation of the unrestricted blocking mechanism introduced as an explicit, sound tableau rule.

Tableau systems for reasoning about risk

In order to evaluate the effectiveness of the security measures undertaken to protect a distributed system (e.g., protecting privacy of data in a network or in an information system) one should, among other things, perform a risk assessment. In this paper, we introduce a logical framework that allows one to reason about risk by means of operators that formalize causes, effects, preconditions, prevention and mitigation of events that may occur in the system. We give tableau rules and discuss a number of interesting variants that could be considered, prove soundness and completeness for some of the resulting tableau systems, and give an algorithm for satisfiability.

TRACTARIAN FIRST-ORDER LOGIC: IDENTITY AND THE N-OPERATOR

AbstractIn theTractatus, Wittgenstein advocates two major notational innovations in logic. First, identity is to be expressed by identity of the sign only, not by a sign for identity. Secondly, only one logical operator, called “N” by Wittgenstein, should be employed in the construction of compound formulas. We show that, despite claims to the contrary in the literature, both of these proposals can be realized, severally and jointly, in expressively complete systems of first-order logic. Building on early work of Hintikka’s, we identify three ways in which the first notational convention can be implemented, show that two of these are compatible with the text of theTractatus, and argue on systematic and historical grounds, adducing posthumous work of Ramsey’s, for one of these as Wittgenstein’s envisaged method. With respect to the second Tractarian proposal, we discuss how Wittgenstein distinguished between general and non-general propositions and argue that, claims to the contrary notwithstanding, an expressively adequate N-operator notation is implicit in theTractatuswhen taken in its intellectual environment. We finally introduce a variety of sound and complete tableau calculi for first-order logics formulated in a Wittgensteinian notation. The first of these is based on the contemporary notion of logical truth as truth in all structures. The others take into account the Tractarian notion of logical truth as truth in all structures over one fixed universe of objects. Here the appropriate tableau rules depend on whether this universe is infinite or finite in size, and in the latter case on its exact finite cardinality.As it is obviously easy to express how propositions can be constructed by means of this operation and how propositions are not to be constructed by means of it, this must be capable of exact expression.5.503

Automated Synthesis of Tableau Calculi

This paper presents a method for synthesising sound and complete tableau calculi. Given a specification of the formal semantics of a logic, the method generates a set of tableau inference rules that can then be used to reason within the logic. The method guarantees that the generated rules form a calculus which is sound and constructively complete. If the logic can be shown to admit finite filtration with respect to a well-defined first-order semantics then adding a general blocking mechanism provides a terminating tableau calculus. The process of generating tableau rules can be completely automated and produces, together with the blocking mechanism, an automated procedure for generating tableau decision procedures. For illustration we show the workability of the approach for a description logic with transitive roles and propositional intuitionistic logic.

The Tableau Workbench

The Tableau Workbench (TWB) is a generic framework for building automated theorem provers for arbitrary propositional logics. The TWB has a small core that defines its general architecture, some extra machinery to specify tableau-based provers and an abstraction language for expressing tableau rules. This language allows users to ''cut and paste'' tableau rules from textbooks and to specify a search strategy for applying those rules in a systematic manner. A new logic module defined by a user is translated and compiled with the proof engine to produce a specialized theorem prover for that logic. The TWB provides various hooks for implementing blocking techniques using histories and variables, as well as hooks for utilising/defining optimisation techniques. We describe the latest version of the TWB which has changed substantially since our system description in TABLEAUX 2003.

An Efficient Tableau Prover using Global Caching for the Description Logic ALC

We report on our implementation of a tableau prover for the description logic ALC, which is based on the EXPTIME tableau algorithm using global caching for ALC that was developed jointly by us and Gore [9]. The prover, called TGC for Tableaux with Global Caching, checks satisfiability of a set of concepts w.r.t. a set of global assumptions by constructing an and-or graph, using tableau rules for expanding nodes. We have implemented for TGC a special set of optimizations which co-operates very well with global caching and various search strategies. The test results on the test set T98-sat of DL'98 Systems Comparison indicate that TGC is an efficient prover for ALC. This suggests that global caching together with the set of optimizations used for TGC is worth implementing and experimenting also for other modal/description logics.