Hitoshi Ohsaki

Beyond Regularity: Equational Tree Automata for Associative and Commutative Theories

A new tree automata framework, called equational tree automata, is presented. In the newly introduced setting, congruence closures of recognizable tree languages are recognizable. Furthermore, we prove that in certain useful cases, recognizable tree languages are closed under union and intersection. To compare with early related work, e.g. [7], we discuss the relationship between linear bounded automata and equational tree automata. As a consequence, we obtain some (un)decidability results. We further present a hierarchy of 4 classes of tree languages.

Decidability and Closure Properties of Equational Tree Languages

Equational tree automata provide a powerful tree language framework that facilitates to recognize congruence closures of tree languages. In the paper we show the emptiness problem for AC-tree automata and the intersection-emptiness problem for regular AC-tree automata, each of which was open in our previous work, are decidable, by a straightforward reduction to the reachability problem for ground AC-term rewriting. The newly obtained results generalize decidability of so-called reachable property problem of Mayr and Rusinowitch. We then discuss complexity issue of AC-tree automata. Moreover, in order to solve some other questions about regular A- and AC-tree automata, we recall the basic connection between word languages and tree languages.

Transforming Termination by Self-Labelling

We introduce a new technique for proving termination of term rewriting systems. The technique, a specialization of Zantemas semantic labelling technique, is especially useful for establishing the correctness of transformation methods that attempt to prove termination by transforming term rewriting systems into systems whose termination is easier to prove. We apply the technique to modularity, distribution elimination, and currying, resulting in new results, shorter correctness proofs, and a positive solution to an open problem.

A sufficient completeness checker for linear order-sorted specifications modulo axioms

We present a tool for checking the sufficient completeness of left-linear, order-sorted equational specifications modulo associativity, commutativity, and identity. Our tool treats this problem as an equational tree automata decision problem using the tree automata library CETA, which we also introduce in this paper. CETA implements a semi-algorithm for checking the emptiness of a class of tree automata that are closed under Boolean operations and an equational theory containing associativity, commutativity and identity axioms. Though sufficient completeness for this class of specifications is undecidable in general, our tool is a decision procedure for subcases known to be decidable, and has specialized techniques that are effective in practice for the general case.

ACTAS: A System Design for Associative and Commutative Tree Automata Theory

ACTAS is an integrated system for manipulating associative and commutative tree automata (AC-tree automata for short), that has various functions such as for Boolean operations of AC-tree automata, computing rewrite descendants, and solv- ing emptiness and membership problems. In order to deal with high-complexity problems in reasonable time, over- and under-approximation algorithms are also equipped. Such functionality enables us automated verification of safety property in infinite state models, that is helpful in the domain of, e.g. network security, in particular, for security problems of cryptographic protocols allowing an equational property. In runtime of model construction, a tool support for analysis of state space expansion is provided. The intermediate status of the computation is dis- played in numerical data table, and also the line graphs are generated. Besides, a graphical user interface of the system provides us a user-friendly environment for handy use.

Formal Model-Based Test for AUTOSAR Multicore RTOS

AUTOSAR multicore RTOS is a safety-critical concurrent system, for which high quality is required. A conformance test is important to ensure the quality of the software, but the conventional test is low in coverage and high in cost. In this paper, we present a formal model-based test for multicore RTOS that supports AUTOSAR specifications. First, we developed a formal model. With the model, we developed a test case generator, from which an entire test suite can be extracted. Moreover, we proposed a test program generator, with which optimal executable test programs can be generated fully automatically. Both of the generators are assisted with model checking on the formal model. Bug analysis also becomes easy. Our method demonstrated its advantage over conventional testing by finding 33 test cases for three system service calls, whereas a conventional test carried out by a development team found only 10 test cases. Our method can improve the coverage of the test, clearly saving in cost and development time. It is expected to significantly improve the testing of the AUTOSAR multicore RTOS.

Monotone AC-Tree automata

We consider several questions about monotone AC-tree automata, a class of equational tree automata whose transition rules correspond to rules in Kuroda normal form of context-sensitive grammars. Whereas it has been proved that this class has a decision procedure to determine if, given a monotone AC-tree automaton, it accepts no terms, other important decidability or complexity results have not been well-investigated yet. In the paper, we prove that the membership problem for monotone AC-tree automata is PSPACE-complete. We then study the expressiveness of monotone AC-tree automata: precisely, we prove that the family of AC-regular tree languages is strictly subsumed in that of AC-monotone tree languages. The proof technique used in obtaining the above result yields the answers to two different questions, specifically that the family of monotone AC-tree languages is not closed under complementation, and that the inclusion problem for monotone AC-tree automata is undecidable.

Type Introduction for Equational Rewriting

Type introduction is a useful technique for simplifying the task of proving properties of rewrite systems by restricting the set of terms that have to be considered to the well-typed terms according to any many-sorted type discipline which is compatible with the rewrite system under consideration. A property of rewrite systems for which type introduction is correct is called persistent. Zantema showed that termination is a persistent property of non-collapsing rewrite systems and non-duplicating rewrite systems. We extend his result to the more complicated case of equational rewriting. As a simple application we prove the undecidability of AC-termination for terminating rewrite systems. We also present sufficient conditions for the persistence of acyclicity and non-loopingness, two properties which guarantee the absence of certain kinds of infinite rewrite sequences.

Type introduction for equational rewriting

Abstract. Type introduction is a useful technique for simplifying the task of proving properties of rewrite systems by restricting the set of terms that have to be considered to the well-typed terms according to any many-sorted type discipline which is compatible with the rewrite system under consideration. A property of rewrite systems for which type introduction is correct is called persistent. Zantema showed that termination is a persistent property of non-collapsing rewrite systems and non-duplicating rewrite systems. We extend his result to the more complicated case of equational rewriting. As a simple application we prove the undecidability of AC-termination for terminating rewrite systems. We also present sufficient conditions for the persistence of acyclicity and non-loopingness, two properties which guarantee the absence of certain kinds of infinite rewrite sequences. In the final part of the paper we show how our results on persistence give rise to new modularity results.

On Accelerating SMT-based Bounded Model Checking of HSTM Designs

Hierarchical State Transition Matrix (HSTM) is a table-based modeling language for developing designs of software systems. We have proposed a Satisfiability Modulo Theory (SMT) based Bounded Model Checking (BMC) approach in [1] to provide formal verification supports for conducting rigorous and automatic analysis to improve reliability of HSTM designs. In this paper, we continue that work by developing and evaluating approaches to accelerating BMC of HSTM designs. The approaches center around an unrolled Bounded Reach ability Tree (BRT) of a HSTM design that is built with stateless explicit state exploration. Specifically, reach ability of invalid cells (representing undesired states) of a HSTM design, which occurs within the bound concerned, could be discovered during construction of the BRT, and furthermore, if no such occurrence, the constructed BRT could be utilized to rule out unnecessary subformulas of a BMC instance for verification of LTL properties. We have implemented these approaches in a tool called Garakabu2 with the state-of-the-art SMT solver CVC3 as its back-ended solver. Our preliminary experiments show that verification could be accelerated substantially.