Siva Anantharaman

Automated proofs of the Moufang identities in alternatives rings

In this paper we present automatic proofs of the Moufang identities in alternative rings. Our approach is based on the term rewriting (Knuth-Bendix completion) method, enforced with various features. Our proofs seem to be the first computer proofs of these problems done by a general purpose theorem prover. We also present a direct proof of a certain property of alternative rings without employing any auxiliary functions. To our knowledge our computer proof seems to be the first direct proof of this property, by human or by a computer.

Closure properties and decision problems of dag automata

Tree automata are widely used in various contexts. They are closed under boolean operations and their emptiness problem is decidable in polynomial time. Dag automata are natural extensions of tree automata, operating on dags instead of on trees; they can also be used for solving problems. Our purpose in this paper is to show that algebraically they behave differently: the class of dag automata is not closed under complementation, dag automata are not determinizable, their membership problem is NP-complete, the universality problem is undecidable, and the emptiness problem is NP-complete even for deterministic labeled dag automata.

Heuristical Criteria in Refutational Theorem Proving

We hope to have convinced the reader by now, that the extra time spent on the calculations of the measures of the various rules is not wasted, but serves instead as the key factor in the reduction of the size of the search-space. It would seem however that there exists no index function γ, and no sorting strategy Sort, such that γ-Sort (or Sort-γ) be uniformly efficient in all cases. This is the main reason why the software SBR3 has opted for a lexicographic combination of more than one filtration-sorting, giving an appreciable overall efficiency.

Unification Modulo ACUI Plus Distributivity Axioms

AbstractE-unification problems are central in automated deduction. In this work, we consider unification modulo theories that extend the well-known ACI or ACUI by adding a binary symbol “*” that distributes over the AC(U)I-symbol “+.” If this distributivity is one-sided (say, to the left), we get the theory denoted AC(U)IDl; we show that AC(U)IDl-unification is DEXPTIME-complete. If “*” is assumed two-sided distributive over “+,” we get the theory denoted AC(U)ID; we show unification modulo AC(U)ID to be NEXPTIME-decidable and DEXPTIME-hard. Both AC(U)IDl and AC(U)ID seem to be of practical interest, for example, in the analysis of programs modeled in terms of process algebras. Our results, for the two theories considered, are obtained through two entirely different lines of reasoning. A consequence of our methods of proof is that, modulo the theory that adds to AC(U)ID the assumption that “*” is associative-commutative, or just associative, unification is undecidable.

Cap unification: application to protocol security modulo homomorphic encryption

We address the insecurity problem for cryptographic protocols, for an active intruder and a bounded number of sessions. The protocol steps are modeled as rigid Horn clauses, and the intruder abilities as an equational theory. The problem of active intrusion -- such as whether a secret term can be derived, possibly via interaction with the honest participants of the protocol -- is then formulated as a Cap Unification problem. Cap Unification is an extension of Equational Unification: look for a cap to be placed on a given set of terms, so as to unify it with a given term modulo the equational theory. We give a decision procedure for Cap Unification, when the intruder capabilities are modeled as homomorphic encryption theory. Our procedure can be employed in a simple manner to detect attacks exploiting some properties of block ciphers.

An Application of Automated Equational Reasoning to Many-valued Logic

In this paper we present a new set of axioms of an algebaric nature, for the many-valued logic of Lukasiewicz. These axioms are similar to those given by J.Hsiang for the Boolean Algebra. The equivalence of our set of axioms with those given by Lukasiewicz himself is proved mechanically, by resorting to the Automatic UKB-based Equational Theorem Prover SBR3. These new axioms may be helpful for further equational reasoning in such logics, or for interpreting the ‘equality symbol’ of linear logic. This paper is organized as follows. Section 1 presents briefly the many-valued logic. Section 2 indicates the proof-steps leading to our new set of axioms. Section 3 presents the equational prover SBR3.

Automata for positive core XPath queries on compressed documents

Given any dag t representing a fully or partially compressed XML document, we present a method for evaluating any positive unary query expressed in terms of Core XPath axes, on t, without unfolding t into a tree. To each Core XPath query of a certain basic type, we associate a word automaton; these automata run on the graph of dependency between the non-terminals of the straightline regular tree grammar associated to the given dag, or along complete sibling chains in this grammar. Any given Core XPath query can be decomposed into queries of the basic type, and the answer to the query, on the dag t, can then be expressed as a sub-dag of t suitably labeled under the runs of such automata.

Unification modulo synchronous distributivity

Unification modulo the theory defined by a single equation which specifies that a binary operator distributes synchronously over another binary operator is shown to be undecidable. It is the simplest known theory, to our knowledge, for which unification is undecidable: it has only one defining axiom and moreover, every congruence class is finite (so the matching problem is decidable).

ACID-Unification Is NEXPTIME-Decidable

We consider the unification problem for the equational theory AC(U)ID obtained by adjoining a binary ‘*’ which is distributive over an associative-commutative idempotent operator ‘+’, possibly admitting a unit element U. We formulate the problem as a particular class of set constraints, and propose a method for solving it by using the dag automata introduced by W. Charatonik, that we enrich with labels for our purposes. AC(U)ID-unification is thus shown to be in NEXPTIME.

Unification Modulo ACUI Plus Homomorphisms/Distributivity

In this paper, we consider the unification problem over theories that are extensions of ACI or ACUI, obtained by adding finitely many homomorphism symbols, or a symbol ‘*’ that distributes over the ACUI-symbol denoted ‘+’. We first show that when we adjoin a set of commuting homomorphisms to ACUI, unification is undecidable. We then consider the ACUID l -unification problem, i.e., unification modulo ACUI plus left-distributivity of a given ‘*’ w.r.t. ‘+’, and prove its NEXPTIME-decidability. When we assume the symbol ‘*’ to be 2-sided distributive w.r.t. ‘+’, we get the theory ACUID, for which the unification problem remains decidable. But when equations of associativity-commutativity, or just of associativity, on ‘*’ are added on to ACUID, the unification problem becomes undecidable.