6,281 research outputs found
Non-commutative Unification in Brane World
We point out that in (open) string compactifications with non-zero NS-NS
B-field we can have large Kaluza-Klein thresholds even in the small volume
limit. In this limit the corresponding gauge theory description is in terms of
a compactification on a non-commutative space (e.g., a torus or an orbifold
thereof). Based on this observation we discuss a brane world scenario of
non-commutative unification via Kaluza-Klein thresholds. In this scenario, the
unification scale can be lowered down to the TeV-range, yet the corresponding
compactification radii are smaller than the string length. We discuss a
potential application of this scenario in the context of obtaining mixing
between different chiral generations which is not exponentially suppressed - as
we point out, such mixing is expected to be exponentially suppressed in certain
setups with large volume compactifications. We also point out that T-duality is
broken by certain non-perturbative twisted open string sectors which are
supposed to give rise to chiral generations, so that in the case of a small
volume compactification with a rational B-field we cannot T-dualize to a large
volume description. In this sense, the corresponding field theoretic picture of
unification via Kaluza-Klein thresholds in this setup is best described in the
non-commutative language.Comment: 15 pages, revtex; misprints corrected, clarifying remarks added (to
appear in Phys. Lett. B
Unification modulo a 2-sorted Equational theory for Cipher-Decipher Block Chaining
We investigate unification problems related to the Cipher Block Chaining
(CBC) mode of encryption. We first model chaining in terms of a simple,
convergent, rewrite system over a signature with two disjoint sorts: list and
element. By interpreting a particular symbol of this signature suitably, the
rewrite system can model several practical situations of interest. An inference
procedure is presented for deciding the unification problem modulo this rewrite
system. The procedure is modular in the following sense: any given problem is
handled by a system of `list-inferences', and the set of equations thus derived
between the element-terms of the problem is then handed over to any
(`black-box') procedure which is complete for solving these element-equations.
An example of application of this unification procedure is given, as attack
detection on a Needham-Schroeder like protocol, employing the CBC encryption
mode based on the associative-commutative (AC) operator XOR. The 2-sorted
convergent rewrite system is then extended into one that fully captures a block
chaining encryption-decryption mode at an abstract level, using no AC-symbols;
and unification modulo this extended system is also shown to be decidable.Comment: 26 page
Higher-Order Equational Pattern Anti-Unification
We consider anti-unification for simply typed lambda terms in associative, commutative, and associative-commutative theories and develop a sound and complete algorithm which takes two lambda terms and computes their generalizations in the form of higher-order patterns. The problem is finitary: the minimal complete set of generalizations contains finitely many elements. We define the notion of optimal solution and investigate special fragments of the problem for which the optimal solution can be computed in linear or polynomial time
The Role of Term Symmetry in E-Unification and E-Completion
A major portion of the work and time involved in completing an incomplete set of reductions using an E-completion procedure such as the one described by Knuth and Bendix [070] or its extension to associative-commutative equational theories as described by Peterson and Stickel [PS81] is spent calculating critical pairs and subsequently testing them for coherence. A pruning technique which removes from consideration those critical pairs that represent redundant or superfluous information, either before, during, or after their calculation, can therefore make a marked difference in the run time and efficiency of an E-completion procedure to which it is applied.
The exploitation of term symmetry is one such pruning technique. The calculation of redundant critical pairs can be avoided by detecting the term symmetries that can occur between the subterms of the left-hand side of the major reduction being used, and later between the unifiers of these subterms with the left-hand side of the minor reduction. After calculation, and even after reduction to normal form, the observation of term symmetries can lead to significant savings.
The results in this paper were achieved through the development and use of a flexible E-unification algorithm which is currently written to process pairs of terms which may contain any combination of Null-E, C (Commutative), AC (Associative-Commutative) and ACI (Associative-Commutative with Identity) operators. One characteristic of this E-unification algorithm that we have not observed in any other to date is the ability to process a pair of terms which have different ACI top-level operators. In addition, the algorithm is a modular design which is a variation of the Yelick model [Ye85], and is easily extended to process terms containing operators of additional equational theories by simply plugging in a unification module for the new theory
Unification Theory - An Introduction
Aus der Einleitung:
„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.
Unification in Abelian Semigroups
Unification in equational theories, i.e. solving of equations in varieties, is a basic operation in Computational Logic, in Artificial Intelligence (AI) and in many applications of Computer Science. In particular the unification of terms in the presence of an associative and commutative f unction, i.e. solving of equations in Abelian Semigroups, turned out to be of practical relevance for Term Rewriting Systems, Automated Theorem Provers and many AI-programming languages. The observation that unification under associativity and commutativity reduces to the solution of certain linear diophantine equations is the basis for a complete and minimal unification algorithm. The set of most general unifiers is closely related to the notion of a basis for the linear solution space of these equations.
These results are extended to unification in free term algebras combined with Abelian Semigroups
A Parallel Implementation of Stickel\u27s AC Unification Algorithm in a Message-Passing Environment
Unification algorithms are an essential component of automated reasoning and term rewriting systems. Unification finds a set of substitutions or unifiers that, when applied to variables in two or more terms, make those terms identical or equivalent. Most systems use Robinson\u27s unification algorithm or some variant of it. However, terms containing functions exhibiting properties such as associativity and commutativity may be made equivalent without appearing identical. Systems employing Robinson\u27s unification algorithm must use some mechanism separate from the unification algorithm to reason with such functions. Often this is done by incorporating the properties into a rule base and generating equivalent terms which can be unified by Robinson\u27s algorithm. However, rewriting the terms in this manner can generate large numbers of useless terms in the problem space of the system.
If the properties of the functions are incorporated into the unification algorithm itself, there is no need to rewrite the terms such that they appear identical. Stickel developed an algorithm to unify two terms containing associative and commutative functions. The unifiers (there may be more than one) are found by creating a homogeneous linear Diophantine equation with integer coefficients from the terms being unified. The unifiers can be constructed from solutions to this equation.
The unifiers generated from one solution of the Diophantine equation are independent of any other solution to the equation. Therefore, once the Diophantine equation has been solved, the unifiers can be calculated from the solutions in parallel. We have implemented Stickel\u27s AC unification algorithm to run in parallel on a local area network of Sun 4/110 workstations in an effort to improve the speed of AC unification
Nominal C-Unification
Nominal unification is an extension of first-order unification that takes
into account the \alpha-equivalence relation generated by binding operators,
following the nominal approach. We propose a sound and complete procedure for
nominal unification with commutative operators, or nominal C-unification for
short, which has been formalised in Coq. The procedure transforms nominal
C-unification problems into simpler (finite families) of fixpoint problems,
whose solutions can be generated by algebraic techniques on combinatorics of
permutations.Comment: Pre-proceedings paper presented at the 27th International Symposium
on Logic-Based Program Synthesis and Transformation (LOPSTR 2017), Namur,
Belgium, 10-12 October 2017 (arXiv:1708.07854
Preserving Filtering Unification by Adding Compatible Operations to Some Heyting Algebras
We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x ⋁ ¬¬x = 1, preserves filtering (or directed) unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To prove the results we apply the theorems of [9] on direct products of l-algebras and filtering unification. We consider examples of frontal Heyting algebras, in particular Heyting algebras with the successor, and G operations as well as expansions of some commutative integral residuated lattices with successor operations
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