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Corresponding author is Everardo Bárcenas.
1 Introduction
Path planning of agents (mobile devices including autonomous robots) consists in finding a sequence of motion tasks to reach a final goal in a known environment. Usually, constraints such as avoiding obstacles and collisions must also be satisfied. Constraints or specifications are usually given in a mathematical language, which traditionally has been a geometric one. It is then a natural research direction to study specification languages with more expressive power in order to give more complex instructions to agents.
Specifications involving sequentiality has recently been studied with great success in the setting of linear temporal logic [1, 23, 13, 14, 6, 25]. Specification are written as linear temporal logic formulas, and plan generation corresponds to counterexample generation in a model checking tool: given a formula (specification) and a model (environment), find a path in the model not satisfying (the counterexample) the negation of a formula.
In the context of mobile robotics, it is often required to implement continuous trajectories. This is achieved by first giving a discrete abstraction of the robot environment, which can clearly be obtained by well known efficient decomposition methods for polygonal environments [21]. A discrete plan is then obtained from the discrete abstraction of the environment and the specification by means of model checking techniques. A continuous implementation of the discrete plan can be obtained with a hybrid control approach [11].
As suggested by its name, linear temporal logic (LTL) formulas can express linear plans only. In linear plans, agents execute tasks in a linear sequential order. Consider for instance the following formula in linear temporal logic:
This expression requires a agent to visit p after one step, and after two steps to visit q. Hence, in a linear model, the above expression implies q is after p. However, in many cases it may not be true, say for instance that q is in a different path than p. Then, the resulting plan has a two branch tree form. With these kind of non-linear plans, more precisely branching plans, it is then easy to see that a team of agents may concurrently execute the plan, one agent each branch. In addition, since plans are tree shaped, collision-free plans are guaranteed and agents may be executed asynchronously.
In the community of context-aware computing, it has recently emerged the notion of context-aware path planning [24], which concerns the path planning problem with particular context variables included in the set of constraints, as for instance, location, time, weather, and community sense.
For instance, for this last variable, a well-known scenario arises in the generation of evacuation plans. It is often the case that evacuation plans need to take in consideration a team of people (with an smart phone each or some). In this scenario, it is desirable, whenever possible, each (or at least some) people follow a different evacuation plan in order to avoid bottlenecks and expedite the evacuation. Branching tree planning in the evacuation scenario offers several distinct and collision-free evacuation plans.
In the current work, we study the µ-calculus, which is an expressive modal logic, in the context of branching path planning for mobile agents (devices, robots, etc.). The µ-calculus is one of the most expressive, yet computable, logic languages. The expressive power of the µ-calculus corresponds the bisimulation invariant fragment of monadic second order logic [5], and hence, by the Rabin's Tree Theorem [20], also corresponds to tree automata and regular languages (grammars).
The µ-calculus is also known as the queen of the program logics, because it subsumes the linear temporal logic LTL, the propositional dynamic logic PDL, the computational tree logic CTL, and many expressive descriptive logics [7].
1.1 Motivations and Related Work
Planning or scheduling is one of the first fundamental problems in the Artificial Intelligence research community. This problem concerns the automation of strategies or action sequences of intelligent agents or autonomous robots. In this context, formal models described by action languages has a long tradition which dates back to the McCarthy's situation calculus [18].
Although efficient reasoning algorithm associated to modal languages are now mature enough at the industrial level [8, 15], there has been surprisingly relatively little research on planning in the setting of modal logic [12]. The mobile robotics community is another story, very recently, there is a major increasing research interest on modal (temporal) languages as specification language for path planning [1, 23, 13, 14, 6, 25].
Since linear temporal logic is not expressive enough for denoting multi-branching, there are still few studies on modal languages in the context of multi-agent systems, particularly mobile robots [16, 23]. In [16], it is presented a multi-agent approach to motion planning with respect to a common global goal. Specifications are written in linear temporal logic. Hence, robots cooperate to achieve a common goal. However, specifications are still restricted to produce linear plans. Another issue is that this approach is not complete, that is, some times a solution is not found even if it exists. In [23], multi-agent motion planning is studied in the setting of optimal plans. Specifications are also described in terms of linear temporal logic formulas. The team of agents is controlled by a timed automaton. Also, the team of agents cooperate to achieve a common goal. In contrast with the approach reported in [16], agents can move asynchronously.
In this work, we take a different approach from works in [16, 23], instead of complementing the linear temporal logic formalism with a multi-robot solution, we extend the expressive power of temporal logic so that in a unifying framework, multi-agent and expressive specifications can be handled. Models in µ-calculus produce multi-branching plans that can be executed by a team of agents, one each branch.
In contrast with other branching modal logics as CTL, the µ-calculus subsumes LTL, which has been largely studied and benchmarked before in the context of planning [1, 23, 13, 14, 6, 25]. Moreover, due to the correspondence of the µ-calculus and regular languages [5, 20], many other complex specifications not available in LTL, CTL and PDL, as for instance regular (tree) expressions (well known in programming languages), can be succinctly denoted.
Previous studies of path planning in the setting of LTL [1, 23, 13, 14, 6, 25] are based in counterexample generation in a model checking tool. The model checking problem consists in deciding whether a given formula (specification) is satisfied by a given model (agent environments). Counterexample generation consists in finding a an example where the specification fails to be satisfied. Hence, in the setting of planning, a plan is obtained by a counterexample of the specification negated. However, often in model checking algorithms, counterexample generation is a blind spot in the corresponding tools [10]. In the current work, we propose a sequent based derivation system for the model checking problem of the µ-calculus. The planning algorithm, based on logical flow graphs, extracts non-linear (multi-branching) plans directly from the proofs produced by the derivation system.
1.2 Contributions and Outline
In Section 2 the propositional modal µ-calculus is introduced. Also, we introduce a formal definition of the problem of path planning in terms of logic formulas and Kripke structures (transition systems). Intuitively, Kripke structures are the discrete abstraction of agent environments. We also show in this Section there is plan for any satisfiable formula. This notion of planning is used in the correctness proof the path planner proposed in Section 3.
In a contribution of independent interest, we also introduce in Section 3 a sequent based derivation system for the model checking problem of the alternation-free fragment of the modal µ-calculus. The planning algorithm extracts, using logical flow graphs, branching plans directly from the proofs produced by the derivation system. It is also shown our proposal is sound and complete, that is, the planner returns a satisfying path, if and only if, there is a solution. We conclude in Section 4 with a discussion of the current work and a brief summary of further research perspectives.
2 A Modal Logic for Path Planning
In this Section, we present the propositional modal µ-calculus, which is an expressive modal logic equipped with least and greatest fixed-points. We also show how to use logic formulas as specifications for path planning. It is also formally defined the problem of path planning in terms of logic formulas and Kripke structures (graph models).
Through this paper, we consider countable sets only, hence, when written set, we denote countable set.
2.1 Syntax and Semantics
An alphabet or signature is a pair (P , X), such that
Unless otherwise stated, we consider a fixed alphabet.
Definition 1 (Syntax). Given an alphabet, the set of µ-calculus formulas is given by the following grammar:
Formulas are interpreted as sets of nodes in Kripke structures (Definition 2). Intuitively, Kripke structures are graphs with labeled nodes, sometimes called transition systems. Propositions are true or false at each node, whereas negation and disjunction of formulas are interpreted as set complement and the union of sets, respectively. Modal formulas ⋄φ hold at nodes, such that there is at least one adjacent node where φ is true. µx.φ formulas are interpreted as least fixed-points. Intuitively, least fixed-points are used as operators for finite recursion.
Consider for instance formula ⋄p3. In a graph model, this expression denotes nodes, such that there is at least one accessible node where p3 is satisfied. Graphical representation of this example is depicted in Figure 1. ⋄p3 holds in n1 because n3 is accessible from n1 and p3 is satisfied by n3. Now consider the fixed-point formula µx.p1 ∨ ⋄x. This formula is true at nodes where p1 holds or at nodes that can access, through finite recursive navigation on edges, to at least one node where p1 is true. In Figure 1, µx.p1 ∨ ⋄x holds at nodes n1, n2, . . . , nk. If we now consider formulas as specifications for mobile agents, then µx.p1 ∨ ⋄x may be interpreted as a task for an agent, such that it must visit p1 no matters how far p1 is. Then the path traced by the formula can be seen as the path plan for the agent. However, it is natural to ask for another task, as for instance, also visit p2 no matters how far it is, that is, µx.p2 ∨ ⋄x. This two requirements may be easily expressed by conjunction:
But it may also be the case that p2 is not on the path to p1, or the other way around (see Figure 1), that is, the plan is not linear. These two tasks may be performed by a team of two agents. In Figure 1, this conjunction of tasks holds at n1 only, but µx.p2 ∨ ⋄x is satisfied by n1, n3, . . . , nl. This is the path to p2 from n1. Both paths, the one to p1 and the one to p2, form the non-linear plan required to do the tasks. Through this paper, we will show how to compute non-linear path plans. First, we give a formal description of formula semantics on Kripke structures.
Given an alphabet (P , X), a Kripke structure is a tuple (N, R, L), such that
— N is a non-empty set of nodes;
— R : N × N is a binary relations of nodes; and
— L : N ↦ 2P \ ∅ is a total label function.
A total Kripke structure is a Kripke structure where each node is connected by at least one transition relation, that is, for every node n ∈ N there is a relation R and a node n′ ∈ N, such that (n, n′) ∈ R. In this work, we consider total Kripke structures only.
Before giving an operational semantics of µ-calculus formulas, we need some notation. A valuation in a Kripke structure K is a function from the set of variables to the power set of nodes in K, that is, V : X ↦ 2N. We write , when V(x) = S.
Definition 2 (Semantics). Given a Kripke structure K = (N, R, L) and a valuation V , formulas are interpreted as follows.
If , we say K satisfies φ. The model checking problem consists in automatically deciding whether or not a given Kripke structure satisfies a given formula. We say that a formula is satisfiable, if there is a Kripke structure satisfying the formula. In case there is no Kripke structure satisfying the formula, then it is said that the formula is unsatisfiable. When a formula is satisfied by any Kripke structure, the formula is valid.
Other traditional logic operators are defined as follows:
Conjunction is interpreted as expected, as set intersection. ⊥ is a contradiction. ◻φ holds at nodes where each adjacent node satisfies φ. Consider for instance formula ◻p2, which denote nodes such that all its accessible nodes satisfies p2.
In Figure 2, ◻p2 does not hold at n2, because although n5 and n4 are accessible nodes from n2 and they satisfy p2, there is one accessible node where p2 is not satisfied and it is n3.
A greatest fixed point can also be defined as dual of the least fixed point.
This operator is intuitively used for non-finite recursion. Consider for instance formula νx.p1 ∧ ⋄x. This formula denotes paths (finite or infinite) where p1 is always true. In the context of path planning, it is natural to require a task to be performed permanently, such as surveillance. In Figure 2, νx.p1 ∧ ⋄x denotes the cycle composed by n1, n2, n3, . . . , nk.
Due to Knaster and Tarski we know µ and ν formulas are fixed points. Before describing this well known result, we need some notation.
Definition 3 (Finite expansions). We define the finite expansion (approximants) of fixed-points as follows, for a natural number k.
We now recall the Knaster-Tarski Theorem in terms of finite expansions.
Theorem 1 (Knaster-Tarski Fixed-Point Theorem [22, 17]). For any Kripke structure K = (N, R, L) and a valuation V,
Other known operators in LTL and CTL can also be easily defined.
The until formula φ U ψ holds in each node of paths (adjacent nodes) where φ is true in all but he last node, where φ is satisfied. Release formula φ R ψ holds in nodes of paths where ψ is true in all but the last node, where φ is true. Also, release formula holds in each node of non-finite paths where ψ is always true.
2.2 Plans
Path plans for mobile agents are defined as the resulting trees of nodes satisfying a formula and its subformulas in a given Kripke structure. Consider for example the following formula:
This expression denotes a plan with two sub-tasks, the first one consists in, starting from p0, visiting p1 and p2, in that order, whereas the second task also starts from p0, but requires to visit p3 and p4, also in that order. In Figure 3, the Kripke structure satisfies the formula, the tree is the plan extracted from the Kripke structure that satisfies the formula together with its subformulas.
As another example, consider now formula νx.p1 ∧ ⋄x and its model displayed in Figure 2. The plan of the formula is then the one branched infinite tree composed by the infinite occurrence of paths composed by n1, n2, n3, . . . , nk.
That a satisfiable formula is satisfied by a tree shaped Kripke structure is a well known result.
Theorem 2 (Tree model property [7]). If a formula φ is satisfiable, that is, there is a Kripke structure K, such that for any valuation V we have that , then φ is satisfied by a tree shaped Kripke structure K′, that is, .
Before defining a path plan in terms of Kripke structures and formulas, we need some notation.
Definition 4. A tree on a Kripke structure K = (N, R, L), written TK, is inductively defined as follows:
— the empty tree ∅ is a tree; and
— the tuple (n, T1, T2, . . . , Tk) is a tree, such that
- n ∈ N is a node called the root,
- for i = 1, . . . , k, Ti are trees, and
- in case Ti is not empty, then R(n, ni), where ni is the roots of Ti, respectively.
When clear from context, we often simply write T for a tree.
We now define the notion of plan. Intuitively, a plan is a tree whose nodes satisfy, following the paths of the tree, a formula together with its subformulas.
Definition 5 (Plan). Given a Kripke structure K = (N, R, L) and a valuation V , we say a tree TK on K is a plan for a formula φ if and only if TK ⊨ φ, such that relation ⊨ is inductively defined as follows.
Relation ⊭ is defined as follows.
Because of the tree model property of µ-calculus (Theorem 2), we know that if there is model satisfying a formula, then there is also a plan (tree) satisfying the formula.
Corollary 1. If a formula φ is satisfiable, that is, there is a Kripke structure K, such that for any valuation V we have that , then there is a plan TK for φ, that is, TK ⊨ φ.
3 Path Planning
In this Section, we introduce a sequent-like system for the model checking problem, that is, given a formula and a Kripke structure, decide whether or not the Kripke structure satisfies the formula. Plans can be extracted from the proofs produced by the sequent system. It is also proven that the system is sound and complete.
The sequent system considered in this Section works with fixed-point alternation-free formulas in negation normal form. Occurrences of greatest and least fixed-points cannot then alternate. The negation normal form of a formula is defined as follows.
For the rest of the paper, when we refer to the negation of a formula ¬φ, we mean its negation normal form nnf(φ).
The set of subformulas of a formula is inductively defined as follows: if φ is a proposition, a variable or constant ⊤, then subformula(φ) = {φ}; if φ is a conjunction ψ ∧ ψ or a disjunction ψ ∨ ϕ, then subformula(φ) = {φ} ∪ subformula(ψ) ∪ subformula(ϕ); and if φ is a modal formula ⋄ψ or ◻ψ, or a negation ¬ψ, or a fixed point formula µx.ψ or νx.ψ, then subformula(φ) = {φ} ∪ subformula(ψ). The set of formulas in a modality is inductively defined as follows: mod(p) = mod(x) = mod(top) = {}; mod(⋄φ) = mod(◻φ) = subformulas(φ); mod(φ ∨ ψ) = mod(φ ∧ ψ) = mod(φ) ∪ mod(ψ); mod(¬φ) = mod(µx.φ) = mod(ν(.φ)) = mod(φ).
We say a formula φ occurs under the scope of a modality in another formula ψ, whenever φ ∈ mod(ψ). The set of free variables in a formula is inductively defined as follows: free(x) = {x}; free(⊤) = free(p) = {}; free(¬φ) = free(⋄φ) = free(◻φ) = free(φ); free(φ ∨ ψ) = free(φ ∧ ψ) = free(φ) ∪ free(ψ); and free(µx.φ) = free(νx.φ) = free(φ) \ {x}, provided x occurs in φ. We say a variable x occurs free in a formula φ, whenever x ∈ free(φ). Without loss of generality, we consider formulas where variables can only occur under the scope of a modality, and not free [5].
Sequents are defined as pairs, written n ⊢ ∆, where n is a node and ∆ is a (possibly empty) set of formulas. Intuitively, node n syntactically satisfies each formula in ∆. We say a sequent is empty if its set of formulas is empty.
Rules are pairs (C, A), where C is a set of sequents, called consequent, and A is a family of sequent sets, called premises. Rules are usually written as follows:
Intuitively, from premises A, we derive C. If A contains only empty sequents, then we say A is empty and there are no premises. If a rule has no premises, then we say this rule is an axiom. A derivation system is composed by a set of rules.
Definition 6 (Planning derivation system). Given a Kripke structure K = (N, R, L), we define the following rules with respect to a given formula.
Please note that for readability we use commas for the union of formulas in sequents, and for the union of sequents we use semi-colons. That is, for a formula φ and a set of formulas ∆, we write φ, ∆ instead of {φ}∪∆, and for a sequent n ⊢ ∆ and a set of sequents Λ, we write n ⊢ ∆; Λ instead of {n ⊢ ∆} ∪ Λ.
Before defining derivation trees formally, consider the following formula evaluated in the Kripke structure displayed in Figure 3:
The corresponding derivation tree is the depicted in Figure 4. Inference goes bottom-up, so the formula in the bottom is concluded from the axiom (empty sequents). We thus read that n0 ⊢ p0 ∧ ⋄ (p1 ∧ ⋄p2) ∧ ⋄ (p3 ∧ ⋄p4) is concluded from n0 ⊢ p0, ⋄ (p1 ∧ ⋄p2) ∧ ⋄ (p3 ∧ ⋄p4) by applying the inference rule corresponding to conjunction of formulas. This last expression is concluded from n0 ⊢ ⋄ (p1 ∧ ⋄p2) ∧ ⋄ (p3 ∧ ⋄p4) by applying the inference rule corresponding to propositions.
In this case, we know that p0 ∈ L(n0). It is now applied again the rule for conjunction, so for the premise we obtain n0 ⊢ ⋄ (p1 ∧ ⋄p2) , ⋄ (p3 ∧ ⋄p4) Then, we apply the rule for diamond formulas, so the premise is n2 ⊢p1 ∧ ⋄p2; n0 ⊢ ⋄ (p3 ∧ ⋄p4). This is because we know that R(n0, n2).
Using conjunction rule, then the premise is now n2 ⊢ p1, ⋄p2; n0 ⊢ ⋄ (p3 ∧ ⋄p4), which is concluded from n2 ⊢ ⋄p2; n0 ⊢ ⋄ (p3 ∧ ⋄p4) due to the proposition rule, in particular, because p1 ∈ L(n2).
Using again the diamond rule, the premise is then n6 ⊢ p2; n2 ⊢; n0 ⊢ ⋄ (p3 ∧ ⋄p4), because R(n2, n6). For the next step we obtain the axiom n6 ⊢; n2 ⊢; n0 ⊢ ⋄ (p3 ∧ ⋄p4), because of proposition rule, more precisely, because p2 ∈ L(n6).
Analogously as just described above, we now apply successively the diamond, the conjunction and the diamond rule over n0 ⊢ ⋄ (p3 ∧ ⋄p4) to finally obtain as premise n6 ⊢; n2 ⊢; n7 ⊢; n4 ⊢; n0 ⊢, which turns to be an axiom.
As another example, consider the corresponding proof tree in Figure 5 for νx.p1 ∧ ⋄x in the Kripke structure in Figure 2.
In order to formally define derivation trees, we will consider trees of sequents, that is, trees with sequents instead of nodes. Formally, given a sequent Λ and a derivation system, a derivation tree of Λ is a tree (Λ, T1, T2, . . . , Tk), such that
— T1, . . . , Tk are all derivation trees of their respective sequent roots Λ1, . . . , Λk, and
— is an instance of a rule in the derivation system.
We are now ready to formally define derivation trees satisfying a sequent for a given Kripke structure. Proof trees are derivation trees for a given sequent, such that a given Kripke structure satisfies the formula, intuitively, a proof tree containing finite branches with at least one ending with axioms.
Definition 7 (Proof tree). Consider the planning derivation system, a proof tree of a sequent Λ is a derivation tree of Λ, such that there is at least one branch ending with only empty sequents.
We are now ready to state the main result of this work, which is that the derivation system is sound and complete. This means that whenever there is a Kripke structure satisfying a formula, the derivation system obtains a proof tree.
Theorem 3 (Model Checking). A formula φ is satisfiable by a Kripke structure K, if and only if, there is a proof tree of n ⊢ φ, such that n is a node in K.
To prove the sequent system is correct, we need to show both directions of the double implication.
Theorem 4 (Completeness). If a formula φ is satisfiable by a Kripke structure K, then there is a node n in K, such that there is a proof tree of n ⊢ φ.
Proof. The proof goes by structural induction on the input formula.
First consider the input formula is a proposition p. We then know p ∈ L(n). Hence, n ⊢ p.
For the case of negation, recall formulas are negated normal form, hence, the scope of the negation symbol can only involve a proposition, say p. Since we know n satisfies ¬p, then p ∉ L(n), then n ⊢ p.
In case the input formula is a disjunction with form ψ ∨ ϕ, we know at least one of ψ or ϕ is satisfied by n. Without loss of generality, assume ψ is satisfied by n. Then by induction we know n ⊢ ψ, and hence n ⊢ ψ ∨ ϕ.
The case when the input formula is a conjunction is proven analogously as the case for disjunction.
Assume now the formula has the form ⋄ψ. Since n satisfies ⋄ψ, then there is at least one node n′ satisfying ψ, such that R(n, n′). By induction we obtain n′ ⊢ ψ and hence n ⊢ ⋄ψ.
The for modal formulas is proven analogously as the case of diamond formulas ⋄ψ.
We consider now the case when a least fixed-point µx.ψ is satisfied by n. By the Knaster-Tarski Fixed-Point Theorem 1, we know µx.ψ is equivalent to its finite expansion µix.ψ where k is the number of nodes in K (Definition 3). We then obtain n satisfies the finite expansion. Hence, there is some i, such that n satisfies µix.ψ. A second induction on the structure of µix.ψ is applied in order to obtain n ⊢ µix.ψ. These cases are identical as the ones already proven above. We then obtain n ⊢ µix.ψ, hence n ⊢ µx.ψ.
Finally, for the case of greatest fixed points we proceed analogously as for least fixed points. By the Knaster-Tarski Fixed-Point Theorem 1, we know νx.ψ is equivalent to its finite expansion νix.ψ, where k is the number of nodes in K (Definition 3). In order to show there is a proof tree of the finite expansion we proceed by induction on i (the number of conjuncts). For each step of the induction on i, we also apply another induction on the structure of ψ. By assumption, we know n satisfies ν0x.ψ, the base cases (when ψ is a proposition or a negated proposition) are trivial. Consider now ψ is of the form ⟨m⟩ ϕ, then there is a m-connected node n′ to n satisfying ψ, by induction we obtain n′ ⊢ ψ. n ⊢ ⟨m⟩ ψ follows immediately. Since variables only occur under the scope of a modality and not free, and the fixed-points cannot alternate, other cases go smoothly by structural induction. We now consider the inductive step on i. In order to show n ⊢ νix.ψ, we again proceed by structural induction on ψ. Base cases are immediate. If ψ is ⟨m⟩ ϕ, it is not hard to see there is a m-connected node n′ to n, such that n′ ⊢ ϕ , which clearly implies n ⊢ ⟨m⟩ ϕ . n′ ⊢ ϕ is obtained by induction on ϕ. In the base case, n′ ⊢ follows from n′ ⊢ νi−1x.ϕ, which is known by induction. Other inductive cases goes straightforward by considering all possible occurrences of x. We then conclude n ⊢ νix.ψ.
We now show the converse, that is, if we have a proof tree, then the formula is satisfiable by the Kripke structure.
Theorem 5 (Soundness). Given a Kripke structure K and a formula φ, if there is a proof tree of n ⊢ φ for some node n in K, then n ∈ for any valuation V.
Proof. We proceed in this proof also by structural induction on the input formula.
If the formula is a proposition p, we then know p ∈ L(n), then n satisfies p.
When the formula is a negation, then it has the form ¬p (recall formulas are in negated normal form). We then know p ∉ L(n), and then n does not satisfy p, hence n satisfies ¬p.
Consider the case of n ⊢ ψ ∨ ϕ. The proof tree has then the following form:
By induction we know that n ∈ or n ∈ for any V , then n ∈ .
Conjunction is proven analogously as disjunction.
If the input formula is ◻ψ, then the proof has the form
for all n1, . . . , nk ∈ N, such that R(n, ni). Then by induction, ni=1,...,k ∈ for any V , and hence n ∈ .
When the input is a diamond formula we proceed analogously as with box formulas.
Now, consider the case of µx.ψ. The proof has the following form:
for natural number k, which is the cardinality of set of nodes in K. We know there is some i, such that n ⊢ µix.ψ. However, we cannot use direct induction to show that n ∈ for any valuation V, because µix.φ is not a subformula of µx.φ. We then proceed to use a second induction, now on the structure of µkx.φ. These cases are proven as the ones already proven above. Finally, we obtain n ∈ .
The case for greatest fixed-points is proven in an analogous manner as the case for least fixed points. We know by assumption
for natural number k, which is the cardinality of set of nodes in K. We proceed by a second induction, now on the structure of νix.φ, in order to show n ∈ for all i. These cases are proven as the ones already proven above (recall fixed points cannot alternate). We then conclude n ∈ .
Since nodes are syntactically represented in proof trees, and node transitions are denoted by derivation rules corresponding to modal formulas (diamond and box), it is not hard to see that plans can be extracted from proof trees. Consider for instance the proof tree in Figure 4. The nodes involved in the plan are n0, n2, n6, n4 and n7. Transitions are denoted by the application of rules involving modal subformulas, hence we obtain that R(n0, n2), and R(n2, n6), in that order, and R(n0, n4) and R(n4, n7).
It is then easy to see that the plan is a two-branch tree composed by n0, n2, and n6, and n0, n4, and n7. A graphical representation of the branching plan over the proof tree of Figure 4 is depicted in Figure 6, where the bold transitions (diamond rules) defines the topology of the branching plan. In case there are not finite branches to extract a plan, we then extract the plan from the branch producing a ν-cycle. For instance, in Figure 5, from the branch producing the cycle is easy to extract the plan, which involves n1, n2, . . . , nk, in that order.
In order to formalize the plan extraction process from proof trees, inspired from the notion of logical flow graphs [9], we now define a similar notion for nodes occurring in proof trees. From this logical graphs, we are then going to be able to directly obtain plans in the name of tree paths.
Definition 8 (Logical graphs). Given a proof tree of a certain formula, we define a logical graph as a tuple (N, E), such that
The logical graph of the proof tree displayed in Figure 3 is shown in Figure 6.
A tree path is simply the resulting tree of a logical graph without considering repetitions.
Definition 9 (Tree path). Given a logical graph (N, E) of a proof tree of a formula φ, such that n ⊢ φ is the root of the proof tree, we inductively define a tree path φ starting from n as follows:
— the root is n;
— if a node n′ is in the tree path, (n′, n″) ∈ E, and n′ ≠ n″, then n′ is the parent of n″.
The tree path corresponding to the logical graph of proof tree in Figure 3 is graphically represented by the bold transitions in Figure 6.
Now, it is easy to imply from Theorem 3 that tree paths (Definition 9) are also plans (Definition 5).
Theorem 6 (Path planning). If there is a proof tree of a formula φ, then a tree path of φ is also a plan.
Proof. The proof goes by induction on the structure of the input formula φ.
If the input formula is a proposition p, then we know there is a node n, such that n ⊢ p. Hence, p ∈ L(n), and then n ⊨ p.
Negation, since it only occurs in front of propositions, is proven analogously as the case of propositions.
Consider now the input formula is a disjunction ψ ∨ ϕ. We then know that there is a proof tree for either n ⊢ ψ or n ⊢ ϕ. Without loss of generality assume there is a proof tree for n ⊢ ψ. By induction we then obtain there is plan (n, T1, . . . , Tk) for ψ, that is, (n, T1, . . . , Tk) ⊨ ψ. By definition of plan (Definition 5), then (n, T1, . . . , Tk) ⊨ ψ ∨ ϕ.
The proof for the case of conjunctions is analogous as the proof for the case of disjunctions.
If the formula in question is ⋄ψ, and we know n ⊢ ⋄ψ, then there is a node n′ ⊢ ψ, such that R(n, n′). By induction, it is known there is a plan T′ = (n′, T1, . . . , Tk) ⊨ ψ. Then, by Definition 5, we obtain a plan (n, T′) ⊨ ⋄ψ.
Due to the well known equivalence (Theorem 1, for the case of least fixed points µx.ψ, we prove for the corresponding finite expansion µix.ψ, where k is the number of nodes of the Kripke structure satisfying the input formula. By assumption we know there is some i, such that n ⊢ µix.ψ. A plan (n, T1, . . . , Tm) ⊨ µix.ψ is obtained by applying a second structural induction on µix.ψ. This proof is the same as the cases already proven above. Now, by Definition 5, we obtain (n, T1, . . . , Tm) ⊨ µix.ψ.
Similarly as it is proven the least fixed point case, it is for the greatest fixed point.
4 Conclusions
Branching path planning for mobile agents are studied in the setting of the modal µ-calculus. In contrast with linear plans, where tasks are performed in a linear sequential order, in branching plans, several linear plans can be performed concurrently and asynchronously by a team of agents. Since plans are tree shaped, it is then guaranteed agents never collide. Since the µ-calculus is one of the most expressive, yet computable, formalisms, complex specifications for agents can be denoted by logic formulas. In particular, specifications involving finite and infinite recursion are nicely capture by the least and greatest fixed-points of the µ-calculus. In contrast with traditional plan generation in the context of temporal logics, where plans are obtained from counterexample generation in model checking tools, in the current work, we develop a model checking algorithm based on a sequent derivation system. This system produces proof trees whenever a formula is satisfied by the model. Plans are then directly extracted from the proof trees with the use of logical flow graphs. We provide correctness proofs for both, model checking and plan generation.
The inference system proposed in the current work is a general approach for non-linear plan generation. It can then be applied in motion planning for mobile robots by means of well-known discrete abstraction of the continuous environments typically occurring in robotics applications. We are also interested in the study of the application of our proposal in the setting of context-aware navigation systems for mobile devices.
Other further immediate research perspectives concerns the study and development of implementation techniques for the sequent system provided in this paper. In particular, we are interested in the representation of models based in binary decision diagrams [19]. We also expect to extend the approach proposed in the current paper, for the case of more expressive logics involving arithmetical constraints [4, 3, 2].


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English football is missing the belonging, identity and sense of self in Europe that Howard Kendall's Everton took for granted
- The entire country used to be proud of English teams in Europe
- But we no longer see clubs representing us in European competitions
- Craig Joubert made a mistake and was then hung out to dry
- Sir Ian McGeechan is part of the England panel - but we know his view
In April 1985, I went to work for The People in Manchester. I had one room at the Brown Bull pub in Salford, a Triumph Acclaim with no radio and a heater on the blink and, suddenly, a free pass to the best football matches in the country. They were always in the north, in those days.
Midweek fixtures are the best for a Sunday newspaper man. While the daily guys struggle to hit deadlines, you watch the game. And what games. The first one I saw up there was Manchester United beating Liverpool 2-1 in an FA Cup semi-final replay. Seven days later: Everton 3 Bayern Munich 1 in a European Cup-winners' Cup semi-final at Goodison Park.
That competition has passed now and so has that time. Little more than a month later I sat in my rented flat in Prestwich and watched the Heysel Stadium disaster unfold on a small screen. English football was never the same again.
The Heysel Stadium disaster in 1985 led to all English clubs being banned from Europe for five years
Thirty nine people were killed and 600 injured when escaping fans were pressed against a collapsing wall
By the time the first English club returned to Europe in 1990, there was already talk of a fracture along financial lines. The formation of the Premier League was less than two years away and it was clear everybody was in it for themselves. The days when the entire country felt a sense of pride, of connection, with English clubs in Europe were over.
Watching Everton beat Munich that night they briefly became my club, too; it was the same when grainy footage of Liverpool's exploits in the European Cup flickered away on Sportsnight. No more. Now, people delight in Chelsea's comeuppance in Europe, Arsenal's continued failure or another meltdown at Manchester City. When Manchester United scored twice to beat Bayern Munich in 1999, a fair proportion of the country would have been cheering on the Germans.
We no longer see English clubs as representing us, our football, our people: and nowhere is that disconnection more apparent than the Etihad Stadium.
When the Champions League anthem is played before City's game with Sevilla, it will be booed. Understandably, perhaps. It's a daft pretension, more corporate branding masquerading as ceremony.
Manchester City fans will boo the Champions League anthem before the match against Sevilla
City fans boo because this is a UEFA competition, a UEFA fanfare and UEFA tried to strangle their attempt to join the elite with ill-conceived and unjust Financial Fair Play rules. We get that, too. But it doesn't feel right. It feels like a contributory factor to City's continued disappointment. We hate your competition; we don't care about your competition; we're not bothered if we're out.
A gentler view is that domestic success is too new for City fans to worry about European domination. Manchester United have been at England's top table for years. With 13 Premier League titles they can claim supremacy over every domestic rival. They are on to stage two.
City have a distance to travel before they can worry about Barcelona and Juventus. They have never retained the Premier League title, never won it before the last day of the season, have beaten Chelsea home and away just once since 1957. There are many milestones before prioritising a win at Nou Camp.
Yet City's ambivalence is partly reflected throughout English football. The death of Howard Kendall brought back memories of 1985, of what it felt like to be at Goodison Park — as furiously loud as I've ever heard it. We were all on the same side back then.
Everton beat Bayern Munich at Goodison Park on their way to winning the European Cup Winners Cup in 1985
Andy Gray scores for Everton past Bayern Munich goalkeeper Jean-Marie Pfaff
When Andy Gray and Graeme Sharp were as good as beating up Munich's defence — their coaching staff were standing and banging the roof of the dug-out in frustration because they thought the referee should be offering hulking great defenders more protection — they pretty much had the country with them.
Everton under the brilliant Kendall played a high tempo, highly skilled game, typical of English club sides at the time. They had a style, and a team of English, Scots, Welsh and an Irishman in Kevin Sheedy. They were resolutely from here.
Where are Manchester City from? All over, the same as every English club. Their manager is South American. Their team is spread across continents. They play in the modern, universal style of the expensively assembled elite, no different to Manchester United, Arsenal and Chelsea.
Multiple entrants in the Champions League dilute the emotion of an English champion, alone. Yet even when a sole English club remains — Chelsea last season — there is no depth of feeling, no flying the flag, not really.
It isn't us against them anymore. We talk of our clubs representing the Premier League, not England, and that league is plainly a mongrel entity. Better for it, in terms of the standard, and a huge financial success, too, but the commercial victory has come at a cost.
Andy Gray and Graeme Sharp celebrate the former's goal (left) as Howard Kendall poses with the trophy
Bayern Munich have retained their German roots, so too Barcelona in Spain, Juventus in Italy, Ajax in Holland. They are international clubs but they know where they're from and what they are about.
If The People took on a young reporter from an agency in London now and he walked into the Etihad Stadium on a big European night, he wouldn't feel the same affiliation or elation as at Goodison Park 30 years ago.
He wouldn't feel the same excitement when Gray finally got Everton's noses in front after 78 minutes — Munich had scored first, the big away goal — and he wouldn't feel, instantly, part of it, that sense of ownership English football fans enjoyed in the last century.
'By God, we showed them we could play a bit,' said Gray in the aftermath. He meant Everton; but Everton were us, in a way English teams no longer are. Look at the joy on the message boards when an English team is eliminated.
If Everton replayed the Munich match now, they would be accused of being crude and unsophisticated because the first two goals were from mighty throws into the box — just as Manchester United are always tagged lucky and all Chelsea ever do is park a bus. So, at City, the locals will know who they hate. Them. UEFA. The powers that be. But who do we love? Not ourselves. And while we grope for explanations, maybe that belonging, that identity, the sense of self in Europe that Kendall's Everton took completely for granted, is what is truly missing.
Can you remember when referees made mistakes? Just mistakes. Not actions that were classed as despicable, or disgraceful. Not as part of any conspiracy. They didn't deserve to be severely punished, or threatened with lifetime bans. They erred, as people do, felt rotten about it and then moved on.
Nobody would argue Craig Joubert was in top form at Twickenham on Sunday. He erroneously gave a late penalty offence against Scotland that cost the game. He knows it now. Yet what followed was frenzied.
'You are a disgrace and should never referee again,' tweeted former England scrum-half Matt Dawson. Presumably he advocates the same consequence for any player making an honest error under pressure, too. Richie Gray, who called the over-ambitious line-out for Scotland; or Fraser Brown who delivered it inexpertly, setting off a fateful chain of events.
Referee Craig Joubert awards a penalty against Scotland's Jon Welsh (left) in the last minute of the game
Joubert has been hung out to dry by World Rugby but there are some respected commentators whose reputations are also in the mud — men such as Gavin Hastings and Kenny Logan, who had more time to think than the official, but instead chose to react with intemperate language.
This used to be football's thing. When Urs Meier disallowed Sol Campbell's goal for England against Portugal in the 2004 European Championship, he received 16,000 abusive messages including death threats. The Sun travelled to his home in Switzerland and, inexplicably, placed a large English flag in a nearby field. Asda offered free eye tests to all Swiss nationals. The irony being that Meier was probably right. John Terry did look to have committed a foul in the build-up to the goal.
And Joubert wasn't as woefully incompetent as he is being painted, either. In real time, few called the correct decision and the experts in the television studios took several reruns to see the action clearly. The worst that can be said is he acted hastily.
The sequence of play was fast and muddled, making any call uncertain.
If there was doubt, Joubert should have done nothing, or taken the least controversial option — which was certainly not the award of a match-winning penalty to Australia.
That was his big mistake. But that's all it was. A mistake. We all make them. And when they look back on this spiteful little episode, even Dawson, Hastings and Logan may concede it one day, too.
Joubert leaves the field at the end of the game as Greg Laidlaw of Scotland looks on in dismay
'This coach is the best person for me and for the team,' Leeds United owner Massimo Cellino told the Yorkshire Evening Post. 'I chose him and if you gave me the choice I would choose him again.'
That was on September 30. Nineteen days later, Cellino stopped choosing Uwe Rosler and sacked him instead. 'In the end, I could not even see a patch of blue in the sky, just cloud, cloud, cloud,' he said. Cellino is certainly a very colourful speaker; but as what comes out of his mouth is entirely without veracity, why does anyone even bother listening?
When, by virtue of an administrative error, Legia Warsaw fielded Bartosz Bereszynski for the last three minutes of a Champions League qualifier in 2014, it cost them a place in the tournament. Bereszynski had received a red card in a previous round and was banned for two games. Having left him out of two matches, Warsaw thought the suspension served. Yet a player has to be named in the team roster, and then omitted, for a ban to count.
Despite Legia leading 6-1 on aggregate, and with only minutes of the second leg remaining when Bereszynski arrived, UEFA awarded Celtic a 3-0 win on the night, making the aggregate score 4-3.
Arijan Ademi (left) is awaiting the result of his B sample to determine whether he has failed a drugs test
Yet if Arijan Ademi's B sample comes back positive, and he fails a UEFA drugs test taken after Dinamo Zagreb defeated Arsenal 2-1 in September, the result is likely to be unaffected. How so?
Ademi did not play some meaningless cameo in a game long over. He was present for 96 minutes and five seconds in the heart of Zagreb's midfield. He made 97 per cent of his 34 passes, the best return of any player in the Zagreb team. He won two free-kicks. He had a good game.
If he is found to have doped, how is this irrelevant to the outcome? Only at UEFA could drug cheats be indulged and an administrative error carry the weight of a capital offence.
If Jamie Vardy scores for Leicester against Crystal Palace on Saturday, he will join an illustrious band of seven players who have done so in seven consecutive Premier League games. They are: Ruud van Nistelrooy (twice), Daniel Sturridge, Ian Wright, Alan Shearer, Thierry Henry, Mark Stein and Emmanuel Adebayor. Not bad company for a lad from Fleetwood Town, that lot.
Jamie Vardy could join seven players who have scored in seven consecutive Premier League games
Some review this is going to be. The RFU have decided to canvass the opinions of those whose thoughts on England's World Cup demise are already known. Sir Ian McGeechan, for instance, who recommended Stuart Lancaster as head coach in 2012 and on the morning after England's exit at their home tournament, wrote in his Daily Telegraph column: 'England have a strong coaching team in Lancaster, Andy Farrell and Graham Rowntree. I cannot see anything better out there which is readily available.'
So we know Sir Ian's view. Nobody could have got England out of that World Cup group. Strange then, that a supposedly inferior Scotland side ran Australia far closer than England, and that South Africa promptly beat Wales. Face it, any talk of significant English improvement under Lancaster is bunk and the world can see it.
The hosts were the worst of the top-tier nations, beside France. The rest, even those removed at the quarter-final stage, have turned in at least one marquee performance here. New Zealand (against France), Australia (against Wales and England), South Africa (against Wales), Argentina (against Ireland), Wales (against England), Ireland (against France) and Scotland (against Australia) — all have brought something to the party. And not discounting the mighty performances from some second-tier teams, such as Fiji and Japan.
England were absent. Yet we already know that one of the men charged with exploring a way forward has a vested interest in inertia. If McGeechan cannot see the possibilities, or what more is out there, he isn't the man for this job. And he's not the only one.
Sir Ian McGeechan is part of the panel which will look into England's exit from the Rugby World Cup
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12 Methods to create Blog site Backlinks
From DPFWiki
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Attending ACCREDITCON, guests will get to listen to and have conversations with experts, interact with CoAEMSP staff and board, plug into best practices, network with peers and colleagues, and get answers to pertinent questions regarding initial and continuing CAAHEP accreditation for Paramedic educational programs.
Learn from top educators while earning up to 28 hours of CE (12 hours for the Accreditation Workshop, 2 hours for the Consortium Sponsorships, 16 hours for ACCREDITCON). *Accreditation Workshop and Consortium Sponsorships overlap
This CE activity is accredited for up to 28 education CEH* by the Committee on Accreditation of Educational Programs for the EMS Professions, an organization accredited by Commission on Accreditation for Prehospital Continuing Education.
Participants should contact their state EMS and/or the NREMT to verify acceptance of credits for recertification or relicensing.
The CAAHEP Accreditation Workshop for Paramedic Programs on May 29 and 30 is SOLD OUT!
Online registration will close at 11:59 pm Central Time on May 17, 2019.
Payment of registration fees may be made by credit card or check. Credit card payments are accepted through the online registration system.
To pay by check, make checks payable to:
Sladek Conference Services, Inc.
and include on the memo line 'ACCREDITCON'
Send checks to:
PO Box 187
Lampasas, TX 76550
REFUNDS may be requested through May 3, 2019, subject to a 20% administrative fee. No refunds will be given after May 3, 2019.
ACCREDITCON has contracted with the Omni Louisville Hotel to make your stay in Louisville more affordable. Discounted rates will be honored until May 3, 2019. After this date, rooms are subject to availability and are not guaranteed at the discounted rate. Please make your hotel reservations directly with the hotel.
Interested in HOSTING a User Group meeting or Focus Group during ACCREDITCON?
If you have questions, please see the contact information below.
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text:
News Headlines - 23 November 2018
▽Nissan board fires Ghosn as chairman following arrest:The Asahi Shimbun
Nissan Motor Co. fired Carlos Ghosn as chairman Thursday, curtailing the powerful executive's nearly two-decade reign at the Japanese automaker after his arrest for alleged financial improprieties.
In an hourslong meeting, the company's board of directors voted unanimously to dismiss Ghosn as chairman and as a representative director, Nissan said in a statement.
▽Washington Asks Allies to Drop Huawei - WSJ
The U.S. government has initiated an extraordinary outreach campaign to foreign allies, trying to persuade wireless and internet providers in these countries to avoid telecommunications equipment from China's Huawei Technologies Co., according to people familiar with the situation.
American officials have briefed their government counterparts and telecom executives in friendly countries where Huawei equipment is already in wide use, including Germany, Italy and Japan, about what they see as cybersecurity risks...
▽Facebook under fire for posts on auction of child bride - CNN
Facebook has come under fire after posts discussing the sale of a child bride were made and shared on its site.
An auction was discussed on the social media platform for a girl aged 16 or 17 in South Sudan that sought payment for her hand in marriage.
Facebook said the post was taken down as soon as the company learned of it, but that wasn't until after the girl involved was married.
▽Nintendo investigates conduct of Nintendo Russia CEO following Mario Kart live s | Pocket Gamer.biz
Nintendo is investigating the conduct of its Russian general manager Yasha Haddaji after footage of him verbally abusing staff become widely-circulated online.
The video in question involves Haddaji becoming agitated as he didn't believe the Mario Kart live stream was adequately promoted. Following on he then appears bewildered as to whether the stream is live or not.
While the video was pulled from Nintendo of Russia's social media, fans managed to archive it and share it to YouTube, where it then spread to Reddit and Resetera.
▽BBC 100 Women 2018: Who is on the list? - BBC News
BBC 100 Women has announced its list of 100 inspiring and influential women from around the world for 2018.
With their help we will explore a variety of themes, including using anger to spark action, and uncovering women from the shadows of history.
Ranging in age from 15 to 94, and from more than 60 countries, the BBC 100 Women list includes leaders, trailblazers and everyday heroes.
| 固定リンク
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- News Headlines - 10 March 2019(2019.03.10)
- News Headlines - 11 March 2019(2019.03.11)
- News Headlines - 09 March 2019(2019.03.09)
- News Headlines - 08 March 2019(2019.03.08)


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i'm trying to install a 3rd party software, and when try to install the program, the very last button says, run and lauch....I have done both choice but the result are the same. I have said yes, and then tried no and then tried to click on the icon to start the program but I get the same problem.
I get this error:
Program has shut down:
1. search internet for a solution, or (result, always is that it can't find any and it just shuts down the program)
2. close program.
I have run this same program on other win7 computers, but no one can find the problem?
Could you tell us what this secret application is and where you got it from?
Why did it take me over a year to phone in a problem to ATT?
the program is not really an issue, it's just like any other 3rd party software. why does it matter? i'm just having problems with installing a .exe file and nothing more.
"the program is not really an issue,"
Have it your way, google is your friend ;-)
Some HELP in posting on Computing.net plus free progs and instructions Cheers


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Trinity Chapel
Trinity Chapel
Trinity Chapel was designed by Ronald W. Buelow of The Associated Architects. The chapel was planned to be an enduring symbol of the Christian foundation of the college. Christian imagery abounds in the chapel.
The interior and exterior brick and stone walls are banded in a pattern based upon the mathematical Fibonacci Series which leads up to a series of proportions appearing and reappearing in virtually all of God's creations – man, animal, plant, mineral, and even the cosmos. Symbols of the Trinity are evidenced in the three-sided spire and in the steel railing and girders that adorn the bridge to the balcony. Read more.
Events at this location
Please note: There may be additional events at this location. Please see the event calendar for a full listing of events on and off campus.
|Trinity Chapel
|ProMusica Chamber Music Festival, opening recital
|Sunday, June 16, 2019
|Trinity Chapel
|ProMusica Chamber Music Festival, Young Artist Recital
|Monday, June 17, 2019
|Trinity Chapel
|ProMusica Chamber Music Festival, Circle of Friends recital
|Wednesday, June 19, 2019
|Trinity Chapel
|ProMusica Chamber Music Festival, C Minor Moods recital
|Friday, June 21, 2019
|Trinity Chapel
|Apologetics and Worldview Seminar 2019
|Thursday, June 27, 2019
|Trinity Chapel
|Concert in Trinity Chapel
|Tuesday, September 24, 2019
Employees in Trinity Chapel
Rooms in Trinity Chapel
|Location Name
|Location type
|Trinity Chapel 100 - Language Lab
|Trinity Chapel 101 - Classroom
|Trinity Chapel 110 - Staff Office
|Trinity Chapel 110 - Vestry
|Equipment room
|Trinity Chapel 111 - Chaplain Office
|Trinity Chapel 112 - Staff Office
|Trinity Chapel 113 - Staff Office
|Trinity Chapel 114 - Choir Robe Room
|Equipment room
|Trinity Chapel LOFT - LOFT


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Last time, I had the honor of speaking with Tiffany Gerstmar. Her work with the US Navy led to her become a cybersecurity policy professional. In this final interview of the current series, I got to speak with Roselle Safran. Not unlike Tiffany, work in US government agencies also helped her to get where she […]… Read More
The post Women in Information Security: Roselle Safran appeared first on The State of Security.
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Rabbits are easy to raise and a very efficient source of protein. While there are many recipes intended specifically for rabbit meat, it also can be used in any recipe that calls for chicken.
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For example, mold can be used to make the enzyme rennet that curds milk for cheese production. Molds in the Environment: Decay - Molds play a key role in the break down of organic materials and recycle nutrients back through our ecosystem.
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28/08/2008В В· The difference between "arrivederci" and "arrivederla" is one of formality. "arrivederla" is how you might say goodbye to someone with whom you are on formal speaking terms : a senior boss, your professor at university, an elderly person not personally known to you, a client in a purely professional relationship.
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Australian Capital Territory: Oaks Estate ACT, City ACT, Bonner ACT, Charnwood ACT, Brisbane ACT, ACT Australia 2658
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England: Warrington ENG, Christchurch ENG, Weston-super-Mare ENG, Mansfield ENG, Crosby ENG, ENG United Kingdom W1U 1A1
Northern Ireland: Bangor NIR, Belfast NIR, Craigavon (incl. Lurgan, Portadown) NIR, Craigavon (incl. Lurgan, Portadown) NIR, Newtownabbey NIR, NIR United Kingdom BT2 1H4
Scotland: Paisley SCO, Edinburgh SCO, Glasgow SCO, Glasgow SCO, Dundee SCO, SCO United Kingdom EH10 7B3
Wales: Barry WAL, Barry WAL, Cardiff WAL, Wrexham WAL, Neath WAL, WAL United Kingdom CF24 2D9


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Viking Ocean Cruises has begun its first-ever Caribbean sailings with its Viking Star cruise ship on an Eastern Caribbean itinerary.
Source: Viking Ocean Cruises Begins Caribbean Sailings Caribbean Journal
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Fatal Pedestrian Involved Vehicle Crash
At 0525 hrs on 15 September 2018, East Lampeter Township Police responded for a report of a pedestrian struck by a motor vehicle in the 2500 block of Siegrist Rd in East Lampeter Township.
It was determined that a 2002 Hyundai Sonata operated by Craig Gomez of North Ronks Rd, Ronks, was traveling eastbound and struck a pedestrian along the south side of the roadway. The pedestrian, Thelma Rineer of Siegrist Rd, Ronks, was declared deceased at the scene by the Lancaster County Coroner.
The roadway between Lynwood and Eastbrook Rd was shut down to traffic for approximately five (5) hours as part of the initial crash reconstruction. The investigation is ongoing. Any witnesses are asked to contact the East Lampeter Township Police Department at ph# (717) 291-4676.
The East Lampeter Township Police Department was assisted at the scene by the Manor-Hempfield Crash Team as well as the following agencies: Gordonville Ambulance, Bird-In-Hand Fire Company, LEMSA, and Lancaster County Coroner's Office.


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Family Recovery Specialist (FRS) will help families move into and through the recovery process. A FRS understands the stigma associated with opioid and substance use disorder and its impact on the family. The FRS provides support and enhancement that promotes positive outcomes. The FRS are individuals with lived experience who provide non-clinical support as a fully integrated treatment team member and educator for families. The FRS shares their lived experience with other families to provide recovery support services. The FRS are role models of recovery values and principles by helping families to identify and remove barriers and triggers to the recovery process. The FRS works full time as part of a dedicated multi-disciplinary team providing emotional and informational support.
ATTENTION: In order to qualify for this position, you must be self-identified as having lived recovery along with demonstrated sobriety under non-residential, independent living conditions in the immediate past two years.
Licensure, Certifications, and Clearances:
Salary Range: $12.40 to $18.88 / hour
At UPMC, our shared goal is to create a cohesive, positive, experience for our employees, patients, health plan members, and community. If you too are driven by these values, you may be a great fit at UPMC!QUALITY & SAFETY
UPMC provides a total rewards package that can help you achieve the goals you have for your career and your personal life. Whether you want to learn a new skill through a training course, reach personal health and wellness targets, become more involved in your community, or follow a career path that provides you with the right experience to be successful, UPMC can help you get to where you want to be.
Now more than ever, YOU can help us shape our communities and UPMC into a better place for everyone to work, study, play, and thrive.
Learn more about working here and check out our policies and recent updates.


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GRAPHTEC FC8000 ARMS inconsistent read
barakusa replied to barakusa's topic in GraphTec Cutting Plotter Support RequestsThis happens on any media now and on any position on the media, whether it is 1" from the edge or right in the middle. It has just seemed to get worse over time. At one point we printed and cut on clear static cling with no errors, now I can't even cut any matte material. Is there a way to clean the sensor? I didn't see it in the manual. Thanks again
Weird J with some cuts... CE6000-60
barakusa replied to theishkid's topic in GraphTec Cutting Plotter Support RequestsJust to confirm, I have an FC8000 and these settings were similar and solved a similar problem.
GRAPHTEC FC8000 ARMS inconsistent read
barakusa posted a topic in GraphTec Cutting Plotter Support RequestsHello, New to the forum and I appreciate any help in advance. I'd like to start by saying we've been using this cutter for about 6 months and have only recently been encountering this problem more often. We've tried on a range of media, such as: http://www.advantagesgs.com/Optima-Opti-Solve-Economy-Wall-Tex-6-mil-54-x-100 http://www.advantagesgs.com/Oracal-ORAJET-3165-Calendered-PVC-Inkjet-Media-Gloss-3-75-mil-54-x-164 http://www.advantagesgs.com/General-Formulations-Concept-207-Static-Cling-Vinyl-Clear-54-x-150 As well as generic white solvent printable vinyls. We use the Graphtec type 2 registratin marks at 20mm wide, 4 corners. Typically we are not cutting larger than 50" x 50" so there is not a skew concern. Whereas, 3 months ago I could print registration marks on any media and contour cut with ease, recently the ARMS system is not reading the marks. I have auto-adjusted the ARMS system as well as manually adjusted the sensitivity from 30 through 90 with no success. We have read through the entire (poorly worded) chapter on the ARMS system and nothing has worked. We have even tried cleaning it. Does anyone have a suggestion? Thank you,


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Queensland's best brokers revealed | Insurance Business
The National Insurance Brokers Association (NIBA) has named Queensland's best brokers this week as it moves closer to its annual industry convention.
Steven Hill, from Capital Innovation Australia, and Glen Ryan, from Austcover, were confirmed as the two finalists for the Queensland Broker of the Year Award.
Sponsored by QBE, the accolade is given to an individual broker who is deemed an inspirational role model for the wider broking community.
The eventual winner of the regional competition will then be considered for the national Stephen Ball Memorial Award for Insurance Broker of the Year.
NIBA CEO Dallas Booth congratulated both finalists and said there had been a high standard of entries for this year's award.
"We are delighted to recognise Steven and Glen who have both demonstrated excellence in broking practice," he added.
Hill, who has over 30 years' experience in the industry, said he was surprised to be named as a finalist but appreciated the recognition.
"It certainly will help improve my profile professionally and will see me continue to promote what we do at every opportunity," he said.
Ryan was also honoured to make the cut – "What a privilege to be named a state finalist – even my children are thrilled and proud!" he said.
The Queensland region winner will be announced at the NIBA Queensland Gala Lunch on July 18 at the Royal International Convention Centre in Brisbane. The state winner will go on to compete for Insurance Broker of the Year, which will be announced at the 2018 NIBA Convention in Hobart, Tasmania.
Related stories:
NIBA announces latest award finalists
NIBA reveals best young brokers in NSW/ACT


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Half door and two person lockers use as much space as a single full door locker, but they are split into two units, which doubles the number of lockers overall. These are extremely popular because they are a nice middle ground between full door and box lockers. They provide a moderate amount of storage space while also being more efficient on overall space than full door lockers. Used half door lockers are one unit over the other, each unit being the same with as a full door locker. Two person lockers are two vertical units side by side, which are the same width as one full door locker as well. They also have two small shelves overhead for storage of small items.
Tiers: may be specified as single-tier (full height), two-tier, three-tier, etc., meaning that the lockers are stacked on top of each other in layers two high, three high, etc. Tiers are commonly up to eight high; on occasion, even more tiers may be found, in the case of very small lockers for such purposes as storing laptop computers. The most common numbers of tiers found in lockers are, in order, one, two, and four; three-tier lockers are rather less common, and other numbers such as five, six, or eight even less common still - seven almost non-existent. Since locker cabinets are most commonly 6 feet (182.9 cm.) high (although there are exceptions), the height of individual lockers varies according to how many tiers are accommodated within the cabinet. The height of individual lockers is usually approximately 6 feet (182.9 cm.) divided by the number of tiers, so that two-tier lockers are about 3 feet (91.4 cm.) high, three-tier lockers 2 feet (61 cm.) high, four-tier lockers 1.5 feet (45.7 cm.) high, and so on. Standard features often vary according to the number of tiers: single-tier lockers usually include a shelf about a foot (roughly 30 cm.) from the top, and a hanging rail (sometimes with one or two hooks) immediately underneath that, at the top of the large compartment beneath the shelf; two- or three-tier lockers usually lack the shelf, but include the hanging rail; lockers with four or more tiers usually have none of these fittings, but consist of just the bare compartment.
So college students now actually need therapy dogs to accompany them to their therapy sessions?CU's 2017 total enrollment was over 34,000 students, and CU's mental health wellness programs that year saw a whopping 52,093 students.Grow the hell up and snap out of it you poor fragile little delicate flowers.You're never going to make it in the real world if you're adversely offended by everything around you.
Source 4 Industries is a trusted supplier for material handling equipment and a variety of associated industrial products. We have proven over 40 years that we can meet or exceed your commercial and industrial product needs. Please allow us the opportunity to show you how we distinguish ourselves from other suppliers through our commitment to service and supply. We keep product specialists on hand to answer your most perplexing questions, so give us a try and let us be the solution to your material handling and industrial supply requirements.
The one-stop-shop Glosshouz is a sleek, modern, bright space with the entire spectrum of beauty and spa care available in one place. Whether you need a relaxing full-day escape or a quick pick-me-up, everything is on the menu here: nails, makeup, hair extensions, blow outs, spray tanning and waxing to facials, body treatments, massages, soaking spa and more. Start at the bar, where you can order up a healthy smoothie, glass of bubbly, cup of tea or whatever suits your mood. Then move on to an especially indulgent (but at a great value) spa or beauty package. Try the Benefit Spa Package, which runs two hours and 15 minutes and includes a massage, reflexology for hands and feet, a custom facial and a complimentary beverage.
A variety of floor plans to suit a variety of needs and preferences - all available on a lease basis with one bedroom apartments ranging from $3,250 to $6,495 per month and two bedrooms from $5,995 to $8,515 per month. No upfront entrance fee required. For information about Assisted Living and Memory Care residence options, please fill out the form below.
AIS Shelving Division handles high quality locker systems designed for use in athletic facilities, schools, retail environments and other commercial, military, industrial and institutional applications. For the best selection (featuring steel, plastic, and laminate) and pricing of storage lockers, cabinets, chests, accessories, furniture, shelves, and locks call AIS today to speak with our knowledgeable staff.
The U.S. Postal Service delivers mail Monday through Saturday any time between 6:00am-6:00pm. Outgoing mail boxes are located in the mailrooms of Tower I and Tower II. Mail is collected daily according to The U.S. Postal Service's pick up schedule. If you are located in Tower I or Tower II, please see the Tenant Services Coordinator to receive an assigned mailbox and key.
Est. in 1936, Salsbury Industries is home to both our lockers.com & mailboxes.com websites. The industry leader in the manufacturing & distributing of quality metal lockers, wood lockers, plastic lockers & storage solutions. Salsbury Industries manufacturing facilities are vertically integrated. Headquartered just minutes from downtown Los Angeles.
In the event of a life threatening emergency, please call 911 immediately. If there is a disastrous emergency within the building or property, please, call our office at (303) 893-1886. If you are calling for after-hours assistance, please listen to the voicemail prompts to page an on call engineer. You will be asked to leave a message, be sure to leave all pertinent information such as your name, tenant name, building location, phone number and the nature of the emergency. After an engineer has been paged, you will receive a call back immediately. Out of courtesy to our engineers, please restrict your calls to emergencies only.
Downtown Denver's crown jewel, the historic Brown Palace Hotel, has one of the city's most luxurious spas, the Spa at the Brown Palace. Tucked away from the hustle of the Brown's opulent lobby, the Spa at the Brown welcomes visitors with a soothing waterfall that sets the stage for the spa's signature soaks that draw from a natural artesian well 750 feet below the hotel.
Try the Colorado River Stone massage, a must for weary travelers as they acclimate to The Mile High City's altitude, and follow it with traditional afternoon tea in the lobby. Although this historic hotel was built in 1892, the spa opened in 2006 with special thought to male business travelers, providing a well-appointed men's locker room and several treatments designed just for him.
Additionally, we're creating downtown's first commuter bike station, located near the corner of Wewatta and 16th Street. The building itself is designed with bikes in mind, right down to the bike frame-inspired architecture. The bike station will offer lockers and bike racks for day use as well as memberships for daily commuters. Amenities include heated locker rooms with toilets and showers and a bike shop selling necessities like tubes, pumps and lights.
The Oxford Hotel, located in the heart of Lower Downtown (LoDo), houses a sophisticated spa that pampers visitors with skin and body care tailored to Colorado's elements. Long a destination for locals craving expert skincare, The Oxford Club, Spa & Salon offers a full range of services such as the unique Herbal Steam, featuring a lymphatic dry brushing of the body, followed by time spent in the spa's detoxifying steam tent, a cool stone face massage and an aromatherapy massage to encourage circulation. It's a cleansing treatment any spa junkie will appreciate. Or try the Signature Almond Joy Experience using Sanitas Coconut Sugar Scrub from Boulder, Colorado, along with a chocolate body masque and warm almond oil for an amazing 75-minute pampering.
Lyon makes metal lockers of prime, high grade class 1 mild annealed, cold-rolled steel. Door frame corners are lapped and welded of 16 gauge steel to form a strong, rigid assembly. Furthermore Lyon fits all steel locker doors on these American school lockers for sale with nylon lockbar guides. This reduces clanging as well as provides a smoother, quieter operation.
Our double lockers are a highly popular choice among our customers and for this reason, they rarely stick around our inventory for long. The main advantage of this style is you can save twice the floor space per compartment when compared to traditional, single tier compartments. Double tier columns are frequently utilized as school lockers because they make it easy for users to share a single locker without too much interference.
It's the easiest, fastest way to get from Denver's outlying areas and back. The new RTD rail system will consist of over 120 miles of track and over 70 park-n-ride lots where you can drop your car and jump onboard. It's fast—up to 55 mph—and stress free, letting you bypass your morning traffic and bad weather to get you where you need to go safely and efficiently.
You are eligible for a full refund if no ShippingPass-eligible orders have been placed. You cannot receive a refund if you have placed a ShippingPass-eligible order. In this case, the Customer Care team will remove your account from auto-renewal to ensure you are not charged for an additional year and you can continue to use the subscription until the end of your subscription term.
Grocery and meal delivery kits are one of the hottest new services on the market; if you work in an apartment or other residential facilities, then you might have noticed an increase in these types of deliveries. Since many of these deliveries include perishable items, they can be an even bigger headache than regular packages especially if you're still using manual systems. Many of these applications have found a solution in computerized grocery pickup refrigerated and freezer lockers in apartments, universities, and student housing in Denver, Colorado Springs, Aurora, Fort Collins, Boulder, Pueblo, and throughout the state of Colorado. See below to learn more. Click here to watch a video showing how the computerized pickup lockers work.
All countries United Kingdom Italy United States Spain Netherlands France Canada Australia Hungary Czech Republic Germany Brazil Portugal Slovakia Ireland Croatia Bulgaria Denmark Poland Greece Serbia Thailand Philippines Nepal Mexico Russia Finland Belgium Malaysia Lithuania Malta South Korea Norway Estonia Hong Kong Japan Switzerland Sweden Austria Latvia Slovenia Singapore Romania
Locking options: various types of key locking or padlocking facility are available now. Key locking options include flush locks, cam locks, or locks incorporated into a rotating handle; padlocking facilities may be a simple hasp and staple, or else a padlocking hole may be included in a handle, often called a latchlock. More modern designs include keyless operation, either by coin deposit (which may or may not be returned when use of the locker terminates), or by using electronic keypads to enter passwords for later reopening the locker. Some older lockers used a drop-latch which was incorporated into the door handle, and slid up and down and could be padlocked at the bottom in the "down" position, but these are less used now. Three-point locking is not possible with this type of latch, because it needs to be operated by means of a latch that rotates rather than slides up and down; so this drop-latch is probably a less secure locking option, which may be why it is little used nowadays. Prefect Combination locks are very popular in school lockers used in the UK due to their ease of use and the time and cost saved in the removal of locker keys.


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Voiceover Just moments earlier, Barrister Davis had called, all the way from Nigeria, to check how the deal was progressing.
Don't send money to tide someone over after a mugging or robbery, and don't do anyone a favor by making an online purchase or forwarding a package to another country.For weeks, even months, you may chat back and forth with one another, forming a connection. But ultimately, it's going to happen—your new-found "friend" is going to ask you for money.So you send money..rest assured the requests won't stop there.In addition to losing your money to someone who had no intention of ever visiting you, you may also have unknowingly taken part in a money laundering scheme by cashing phony checks and sending the money overseas and by shipping stolen merchandise (the forwarded package).In another recently reported dating extortion scam, victims usually met someone on an online dating site and then were asked to move the conversation to a particular social networking site, where the talk often turned intimate.


text: bearrfoot 0 Report post Posted November 10, 2018 I am desperate to back up my Win 10 desktop...! MS tools will NOT work (can create system backup image BUT restore always hangs...!) I have downloaded and attempted cloning with AMEIO Backupper, but the clone it creates will never boot up. My machine now has not been backed up in 3 years...! So I just downloaded a Win 10 version (2.08.0) of XXCLONE, which was the last version since creator Pixielabs folded up. In the past I have successfully used XXCLONE on previous computers from Win2000 thru WinXP. So I was hopeful this would work. But noooo... now Win 10 will not even allow me to install it, giving error message that it is obsolete and giving me advice to visit the Pixielab website... which of course no longer exists....!! Does anyone here know of a work-around to get past this MS foolishness....? Pixielabs did have 4 releases of XXCLONE prior to folding up in 2017, so there's a good chance it would work on mine dating from 2015.... Thanks....! Share this post Link to post Share on other sites