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On the Expressivity of Feature Logics with Negation, Functional Uncertainty, and Sort Equations
 JOURNAL OF LOGIC, LANGUAGE AND INFORMATION
, 1993
"... Feature logics are the logical basis for socalled unification grammars studied in computational linguistics. We investigate the expressivity of feature terms with negation and the functional uncertainty construct needed for the description of longdistance dependencies and obtain the following resu ..."
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Feature logics are the logical basis for socalled unification grammars studied in computational linguistics. We investigate the expressivity of feature terms with negation and the functional uncertainty construct needed for the description of longdistance dependencies and obtain the following results: satisfiability of feature terms is undecidable, sort equations can be internalized, consistency of sort equations is decidable if there is at least one atom, and consistency of sort equations is undecidable if there is no atom.
Normalised Rewriting and Normalised Completion
, 1994
"... We introduce normalised rewriting, a new rewrite relation. It generalises former notions of rewriting modulo E, dropping some conditions on E. For example, E can now be the theory of identity, idempotency, the theory of Abelian groups, the theory of commutative rings. We give a new completion algor ..."
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Cited by 19 (2 self)
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We introduce normalised rewriting, a new rewrite relation. It generalises former notions of rewriting modulo E, dropping some conditions on E. For example, E can now be the theory of identity, idempotency, the theory of Abelian groups, the theory of commutative rings. We give a new completion algorithm for normalised rewriting. It contains as an instance the usual AC completion algorithm, but also the wellknown Buchberger's algorithm for computing standard bases of polynomial ideals. We investigate the particular case of completion of ground equations, In this case we prove by a uniform method that completion modulo E terminates, for some interesting E. As a consequence, we obtain the decidability of the word problem for some classes of equational theories. We give implementation results which shows the efficiency of normalised completion with respect to completion modulo AC. 1 Introduction Equational axioms are very common in most sciences, including computer science. Equations can ...
Applications of Diagrams to Decision Problems
, 1993
"... Classical decision problems such as the word and conjugacy problem are introduced and methods are given for solving them in certain cases. All the methods we present involve VanKampen diagrams as one of the most powerful tools when dealing with the classical decision problems. 1. Introduction In ..."
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Cited by 4 (3 self)
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Classical decision problems such as the word and conjugacy problem are introduced and methods are given for solving them in certain cases. All the methods we present involve VanKampen diagrams as one of the most powerful tools when dealing with the classical decision problems. 1. Introduction In 1912 Max Dehn formulated in his article ,, Uber unendliche diskontinuierliche Gruppen" ("On infinite discontinuous groups") three fundamental problems for infinite groups given by finite presentations: the identity problem, the transformation problem, and the isomorphism problem. The following is a translation of Dehn's definition of the first two problems called in modern terms the word problem and the conjugacy problem: The identity problem (word problem): Let an arbitrary element of the group be given as a product of the generators. Find a method to decide in a finite number of steps whether or not this element equals the identity element. The transformation problem (conjugacy proble...
COMPLEXITY CLASSES AS MATHEMATICAL AXIOMS
, 810
"... Abstract. Treating a conjecture, P #P ̸ = NP, on the separation of complexity classes as an axiom, an implication is found in three manifold topology with little obvious connection to complexity theory. This is reminiscent of Harvey Friedman’s work on finitistic interpretations of large cardinal axi ..."
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Abstract. Treating a conjecture, P #P ̸ = NP, on the separation of complexity classes as an axiom, an implication is found in three manifold topology with little obvious connection to complexity theory. This is reminiscent of Harvey Friedman’s work on finitistic interpretations of large cardinal axioms. 1.
Transforming Curves on Surfaces Redux
, 2013
"... Almost exactly 100 years ago, Max Dehn described an algorithm to determine whether two given cycles on a compact surface are homotopic, meaning one cycle can be continuously deformed into the other without leaving the surface. We describe a simple variant of Dehn’s algorithm that runs in linear time ..."
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Almost exactly 100 years ago, Max Dehn described an algorithm to determine whether two given cycles on a compact surface are homotopic, meaning one cycle can be continuously deformed into the other without leaving the surface. We describe a simple variant of Dehn’s algorithm that runs in linear time, with no hidden dependence on the genus of the surface. Specifically, given two closed vertexedge walks of length ℓ and ℓ ′ in a combinatorial surface of complexity n, our algorithm determines whether the two walks are freely homotopic in O(n + ℓ + ℓ ′ ) time. Our algorithm simplifies and corrects a similar algorithm of Dey and Guha [JCSS 1999] and simplifies the more recent algorithm of Lazarus and Rivaud [FOCS 2012], who identified a subtle flaw in Dey and Guha’s results. Our algorithm combines components of these earlier algorithms, classical results in small cancellation theory by Gersten and Short [Inventiones 1990], and simple runlength encoding. Portions of this author’s work was partially supported by NSF grant CCF 0915519. Portions of this work were done while the authors were visiting IST Austria. See
On computing higherorder Alexander modules of knots, Preprint
, 2013
"... Abstract. Cochran defined the nthorder integral Alexander module of a knot in the three sphere as the first homology group of the knot’s (n+1)thiterated abelian cover. The case n = 0 gives the classical Alexander module (and polynomial). After a localization, one can get a finitely presented modul ..."
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Abstract. Cochran defined the nthorder integral Alexander module of a knot in the three sphere as the first homology group of the knot’s (n+1)thiterated abelian cover. The case n = 0 gives the classical Alexander module (and polynomial). After a localization, one can get a finitely presented module over a principal ideal domain, from which one can extract a higherorder Alexander polynomial. We present an algorithm to compute the firstorder Alexander module for any knot. As applications, we show that these higherorder Alexander polynomials provide a better bound on the knot genus than does the classical Alexander polynomial, and that they detect mutation. Included in this algorithm is a solution to the word problem in finitely presented Z[Z]modules. 1.
The Word Problem in Quandles
, 2013
"... A word over an algebra A is a finite sequence of elements of A, parentheses, and operations of A defined recursively: Given each nary operation ◦ of A, if a1, a2,..., an are words, then ◦(a1, a2,..., an) is also a word. The word problem over an algebra A is the following decision problem: ..."
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A word over an algebra A is a finite sequence of elements of A, parentheses, and operations of A defined recursively: Given each nary operation ◦ of A, if a1, a2,..., an are words, then ◦(a1, a2,..., an) is also a word. The word problem over an algebra A is the following decision problem:
WHAT IS AN ALMOST NORMAL SURFACE?
"... Abstract. A major breakthrough in the theory of topological algorithms occurred in 1992 when Hyam Rubinstein introduced the idea of an almost normal surface. We explain how almost normal surfaces emerged naturally from the study of geodesics and minimal surfaces. Patterns of stable and unstable ge ..."
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Abstract. A major breakthrough in the theory of topological algorithms occurred in 1992 when Hyam Rubinstein introduced the idea of an almost normal surface. We explain how almost normal surfaces emerged naturally from the study of geodesics and minimal surfaces. Patterns of stable and unstable geodesics can be used to characterize the 2sphere among surfaces, and similar patterns of normal and almost normal surfaces led Rubinstein to an algorithm for recognizing the 3sphere. 1. Normal Surfaces and Algorithms There is a long history of interaction between lowdimensional topology and the theory of algorithms. In 1910 Dehn posed the problem of finding an algorithm to recognize the unknot [3]. Dehn’s approach was to check whether the fundamental group of the complement of the knot, for which a finite presentation can easily be computed, is infinite cyclic. This led Dehn to pose some of the first decision problems in group theory, including asking for an algorithm to decide if a finitely presented group is infinite cyclic. It was shown about fifty years later that general