MULTILINEAR EQUATIONS IN AMALGAMS OF FINITE INVERSE SEMIGROUPS

2011 ◽  
Vol 21 (01n02) ◽  
pp. 35-59 ◽  
Author(s):  
A. CHERUBINI ◽  
C. NUCCIO ◽  
E. RODARO
Keyword(s):  
Normal Form ◽  
Inverse Semigroups ◽  
Satisfiability Problem ◽  
Finite Set ◽  
Context Free ◽  
Schützenberger Automaton ◽  
Amalgamated Product

Let S = S1 *U S2 = Inv〈X; R〉 be the free amalgamated product of the finite inverse semigroups S1, S2 and let Ξ be a finite set of unknowns. We consider the satisfiability problem for multilinear equations over S, i.e. equations wL ≡ wR with wL, wR ∈ (X ∪ X-1 ∪ Ξ ∪ Ξ-1)+ such that each x ∈ Ξ labels at most one edge in the Schützenberger automaton of either wL or wR relative to the presentation 〈X ∪ Ξ|R〉. We prove that the satisfiability problem for such equations is decidable using a normal form of the words wL, wR and the fact that the language recognized by the Schützenberger automaton of any word in (X ∪ X-1)+) relative to the presentation 〈X|R〉 is context-free.

Algorithm for Decision Procedure in Temporal Logic Treating Uncertainty, Plausibility, Knowledge and Interacting Agents

2012 ◽  
pp. 32-46
Author(s):  
V. Rybakov
Keyword(s):  
Normal Form ◽  
Temporal Logic ◽  
Decision Procedure ◽  
Linear Temporal Logic ◽  
Satisfiability Problem ◽  
Global Knowledge ◽  
Interacting Agents ◽  
Logical Operations ◽  
Multi Agent ◽  
Bounded Size

Our paper studies a logic UIALTL, which is a combination of the linear temporal logic LTL, a multi-agent logic with operation for passing knowledge via agents’ interaction, and a suggested logic based on operation of logical uncertainty. The logical operations of UIALTL also include (together with operations from LTL) operations of strong and weak until, agents’ knowledge operations, operation of knowledge via interaction, operation of logical uncertainty, the operations for environmental and global knowledge. UIALTL is defined as a set of all formulas valid at all Kripke-Hintikka like models NC. Any frame NC represents possible unbounded (in time) computation with multi-processors (parallel computational units) and agents’ channels for connections between computational units. The main aim of our paper is to determine possible ways for computation logical laws of UIALTL. Principal problems we are dealing with are decidability and the satisfiability problems for UIALTL. We find an algorithm which recognizes theorems of UIALTL (so we show that UIALTL is decidable) and solves satisfiability problem for UIALTL. As an instrument we use reduction of formulas to rules in the reduced normal form and a technique to contract models NC to special non-UIALTL-models, and, then, verification of validity these rules in models of bounded size. The paper uses standard results from non-classical logics based on Kripke-Hintikka models.

scholarly journals EMBEDDINGS IN COSET MONOIDS

2008 ◽  
Vol 85 (1) ◽  
pp. 75-80
Author(s):  
JAMES EAST
Keyword(s):  
Semigroup Forum ◽  
Inverse Semigroup ◽  
Simple Proof ◽  
Inverse Semigroups ◽  
Sufficient Condition ◽  
Simple Description ◽  
Group Of Units ◽  
Finite Monoids ◽  
Finite Set ◽  
Symmetric Inverse Semigroup

AbstractA submonoid S of a monoid M is said to be cofull if it contains the group of units of M. We extract from the work of Easdown, East and FitzGerald (2002) a sufficient condition for a monoid to embed as a cofull submonoid of the coset monoid of its group of units, and show further that this condition is necessary. This yields a simple description of the class of finite monoids which embed in the coset monoids of their group of units. We apply our results to give a simple proof of the result of McAlister [D. B. McAlister, ‘Embedding inverse semigroups in coset semigroups’, Semigroup Forum20 (1980), 255–267] which states that the symmetric inverse semigroup on a finite set X does not embed in the coset monoid of the symmetric group on X. We also explore examples, which are necessarily infinite, of embeddings whose images are not cofull.

On bijunctive predicates over a finite set

2019 ◽  
Vol 29 (1) ◽  
pp. 49-58 ◽  
Author(s):  
Svetlana N. Selezneva
Keyword(s):  
Normal Forms ◽  
Satisfiability Problem ◽  
Polynomial Algorithms ◽  
Majority Function ◽  
Finite Set

Abstract The paper is concerned with representations of predicates over a finite set in the form of generalized conjunctive normal forms (GCNF). Properties of predicates GCNF are found which are preserved by some majority function. Such predicates are called generalized bijunctive predicates. These properties are used to construct new faster polynomial algorithms for the generalized satisfiability problem in the case when some majority function preserves all the original predicates.

Conservative reduction classes of Krom formulas

1982 ◽  
Vol 47 (1) ◽  
pp. 110-130 ◽  
Author(s):  
Stål O. Aanderaa ◽  
Egon Börger ◽  
Harry R. Lewis
Keyword(s):  
Normal Form ◽  
Decision Problem ◽  
Conjunctive Normal Form ◽  
Satisfiability Problem ◽  
Quantification Theory ◽  
Finite Models ◽  
Reduction Class

AbstractA Krom formula of pure quantification theory is a formula in conjunctive normal form such that each conjunct is a disjunction of at most two atomic formulas or negations of atomic formulas. Every class of Krom formulas that is determined by the form of their quantifier prefixes and which is known to have an unsolvable decision problem for satisfiability is here shown to be a conservative reduction class. Therefore both the general satisfiability problem, and the problem of satisfiability in finite models, can be effectively reduced from arbitrary formulas to Krom formulas of these several prefix types.

HNN extensions of inverse semigroups with zero

2007 ◽  
Vol 142 (1) ◽  
pp. 25-39 ◽  
Author(s):  
E. R. DOMBI ◽  
N. D. GILBERT
Keyword(s):  
Normal Form ◽  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Universal Group ◽  
Main Application ◽  
Hnn Extensions ◽  
Hnn Extension ◽  
Natural Way

AbstractWe study a construction of an HNN extension for inverse semigroups with zero. We prove a normal form for the elements of the universal group of an inverse semigroup that is categorical at zero, and use it to establish structural results for the universal group of an HNN extension. Our main application of the HNN construction is to show that graph inverse semigroups –including the polycyclic monoids –admit HNN decompositions in a natural way, and that this leads to concise presentations for them.

A Survey of Normal form Covers for Regular Grammars

1981 ◽  
Vol 4 (4) ◽  
pp. 761-776
Author(s):  
Anton Nijholt
Keyword(s):  
Normal Form ◽  
Special Form ◽  
Normal Forms ◽  
Context Free Grammar ◽  
Left Regular ◽  
Regular Grammar ◽  
Left And Right ◽  
Context Free ◽  
Greibach Normal Form

An overview is given of cover results for normal forms of regular grammars. Due to the special form of regular grammars and due to the results which are obtained it is sufficient to consider covering grammars in Greibach normal form. Among other things it is proved that any left-regular grammar can be left covered with a context-free grammar in Greibach normal form. All the cover results concerning the left- and right-regular grammars are listed, with respect to several types of covers, in a cover table.

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