Unification theory

Unification is an important concept. It is used in Prolog, resolution, term rewriting and natural language understanding. As the use of formal methods increases, unification will be part of formally developed systems. Hence a theory of unification is desirable. This paper demonstrates the use of VDM to develop a theory of unification. Substitution application is defined recursively, a theory of non-circular substitutions is developed, and an implicit specification of unification is written. A proof of a particular unification algorithm is outlined with respect to this specification. The algorithm considered is more space-efficient than that which Manna and Waldinger prove correct. The theory developed is also compared with that of Manna and Waldinger.

Equational unification is a generalization of syntactic unification in which semantic properties of function symbols are taken into account. For example, assume that the function symbol '+' is known to be commutative. Given the unication problem x + y ≐ a + b (where x and y are variables, and a and b are constants), an algorithm for syntactic unification would return the substitution {x ↦ a; y ↦ b} as the only (and most general) unifier: to make x + y and a + b syntactically equal, one must replace the variable x by a and y by b. However, commutativity of '+' implies that {x ↦ b; y ↦ b} also is a unifier in the sense that the terms obtained by its application, namely b + a and a + b, are equal modulo commutativity of '+'. More generally, equational unification is concerned with the problem of how to make terms equal modulo a given equational theory, which specifies semantic properties of the function symbols that occur in the terms to be unified.

We study the unification problem for a class of equational theories that comprises important examples like abelian monoids (AC), idempotent abelian monoids (ACI), and abelian groups. Monoidal theories have the common characteristic that unification algorithms are based on solving linear equation systems over a semiring.The close correspondence between unification and linear algebra can be used to characterize the unification type of monoidal theories in purely algebraic terms, and an application of Hilbert's Basis Theorem gives a sufficient criterion for a monoidal theory to be unitary.

Unification in the Union of Disjoint Equational Theories: Combining Decision Procedures

We study decision procedures for two knowledge problems critical to the verification of security protocols, namely the intruder deduction and the static equivalence problems. These problems can be related to particular forms of context matching and context unification. Both problems are defined with respect to an equational theory and are known to be decidable when the equational theory is given by a subterm convergent term rewrite system (TRS). In this work, we extend this to consider a subterm convergent TRS defined modulo an equational theory, like Commutativity. We present two pairs of solutions for these important problems. The first solves the deduction and static equivalence problems in rewrite systems modulo shallow theories such as Commutativity. The second provides a general procedure that solves the deduction and static equivalence problems in subterm convergent systems modulo syntactic permutative theories, provided a finite measure is ensured. Several examples of such theories are also given.

Toyama’s theorem states that the union of two confluent term rewrite systems with disjoint signatures is again confluent. This is a fundamental result in term rewriting, and several proofs appear in the literature. The underlying proof technique has been adapted to prove further results like persistence of confluence (if a many-sorted term rewrite system is confluent, then the underlying unsorted system is confluent) or the preservation of confluence by currying.

On the number of steps in proofs

Unification in equational theories, that is solving equations in varieties, is of special relevance to automated deduction. Recent results in term rewriting systems, as in [Peterson and Stickel 81] and [Hsiang 82], depend on unification in presence of associative-commutative functions. Stickel [75,81] gave an associative-commutative unification algorithm, but its termination in the general case was still questioned. Here we give an abstract framework to present unification problems, and we prove the total correctness of Stickel’s algorithm.

Proximity-based unification theory

Nominal logic is a variant of first-order logic with special facilities for reasoning about names and binding based on the underlying concepts of swapping and freshness. It serves as the basis of logic programming and term rewriting techniques that provide similar advantages to, but remain simpler than, higher-order logic programming or term rewriting systems. Previous work on nominal rewriting and logic programming has relied on nominal unification, that is, unification up to equality in nominal logic. However, because of nominal logic's equivariance property, these applications require a stronger form of unification, which we call equivariant unification. Unfortunately, equivariant unification and matching are NP-hard decision problems. This paper presents an algorithm for equivariant unification that produces a complete set of finitely many solutions, as well as NP decision procedure and a version that enumerates solutions one at a time. In addition, we present a polynomial time algorithm for swapping-free equivariant matching, that is, for matching problems in which the swapping operation does not appear.

In the field of automated deduction, natural language understanding, theorem proving and rewriting theory, unification theory is a very fundamental process. Nowadays researchers have designed several efficient unification algorithms and many automated tools have also been developed. However, there is no efficient practice of an E-unifier generator. In this paper, we propose an efficient method to construct an E-unifier generator. Firstly we implement an algorithm for syntactic unification. Via the bijection between terms and shapes of trees, we can get the correlative set of terms by the transformation of shapes of trees. After these operations, we expand the syntactic unification solution to E-unification. This proposal is able to solve E-unification problems in which case the Diophantine Equations is unable to help. Hence, the method is practical enough for other research works. <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup>

Unification is a fundamental operation in various areas of computer science, in particular in automated theorem proving and logic programming. In this paper we establish a relation between unification theory and classical model theory. We show how model-theoretic methods can be used to investigate a generalized form of unification, namely the problem whether, given an equational theory E and a system of equations S, there is an extension of the free algebra in E in which S is solvable.

Dynamic typing: syntax and proof theory

Term rewriting systems provide a simple mechanism for computing in equations. An equation is converted into a directed rewrite rule by comparing both sides w.r.t. an ordering. However, there exist equations which are incomparable. The handling of such equations includes, for example, partitioning the given equational theory into a set R of rules and a set E of equations. The appropriate reduction relation allows reductions modulo the equations in E. The effective computation with this relation presumes E-termination. This report deals with a new ordering applicable to [R,E]-systems where E contains associative-commutative equations. The method is based on the Knuth-Bendix ordering and is AC-commuting, a property introduced by Jouannaud and Munoz.KeywordsFunction SymbolPolynomial InterpretationRecursive Path OrderingAssociative PathAssociative Pair ConditionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The subject of this work is the development of a nonlinear model of the interaction of an unsteady fluid flow with a structure and finding analytical solutions for the system of equations that correspond to the specified model. The convection effect of the fluid velocity field was already considered in the previous works of the authors of this paper. These studies are devoted to the water hammer without considering the "flow-structure" interaction. This work expands the possibilities of modeling and considers four equations instead of two equations of the theory of the water hammer (equations of conservation of mass and momentum), two of which relate to the motion of particles of a solid body (pipes or structures). The novelty of this work lies in the consideration of the model that describes the interaction of the flow with the structure, the convection in the velocity field, and the effect, together with the friction of the fluid against the solid wall, on the dynamics of the shock pulse propagation process both in the fluid and in the solid body. It should be noted that the solution of the nonlinear system of differential equations as a whole is carried out by an analytical method, which makes it possible to obtain an exact (rather than numerical) solution of the problem. Since the effects of various factors should be evaluated by comparison with the main components, dimensionless equations containing six parameters (dimensionless combinations) were obtained in this study. Two of these parameters were named after scientists – Darcy and Weisbach (steady friction) and Bruno (unsteady friction). Particular cases of the general (full) model were considered, and the effects of various factors on the dynamics of the interaction of the flow with the structure during the propagation of the shock pulse were determined. Research methods are purely theoretical. The concepts of a self-similar equation and a system of equations, balances of forces acting on particles of a fluid and a solid body, and a standard method of reducing a system of equations to a single equivalent equation are used. Conclusions. An extended model of the interaction between the unsteady fluid flow and the structure is proposed. The transition to a self-similar variable makes it possible to solve a nonlinear system of differential equations and obtain an analytical (exact) solution. The functions of longitudinal stress in a solid body, pressure disturbance, and velocity of motion of particles in a solid body (pipe) are linearly expressed by the velocity of shock pulse propagation in the fluid. It should also be noted that the results for the particular case of the linear model completely agree with the already known ones. The advantage of using a self-similar solution is that it is easy to obtain. The results of previous studies on the water hammer problem were qualitatively consistent. As the fluid viscosity increases, the shock pulse domain becomes more concentrative.