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Commit 252f37c6 authored by Uli Fahrenberg's avatar Uli Fahrenberg
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<td align="left"><i >Frank Valencia, <u >Sergio Ramírez</u>, Santiago Quintero and Carlos Pinzón</i></td>
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<a href="javascript:toggleDiv('T4')">Computing Aggregated Knowledge as the Greatest Lower Bound of Knowledge</a>
<div style="display:none", id="T4"><br /><br />In economics and multi-agent systems, \emph{aggregated (or distributed) knowledge} of a group is the knowledge of someone who knows exactly what every member of the group knows. This notion can be used to analyse the implications of the knowledge of a community if they were to combine their knowledge. In this paper we use fundamental tools from the theory of distributive lattice to characterize and compute aggregate knowledge.We prove that for distributive lattices of size $n$, (1) the meet of join-endomorphisms can be computed in time $O(n)$. Previous upper bound was $O(n^2)$. We show that (2) aggregated knowledge of given group can be viewed as the meet of the knowledge of each member of the group. We also show that for state sets of size $n$, (3) aggregated knowledge can be computed in time $O(n^2)$ in the general case and (4) for $S5$ knowledge (or \emph{Aumman structures}), it can be computed in time $O(n\alpha(n))$ where $\alpha(n)$ is the inverse of the Ackermann function.</div>
<div style="display:none", id="T4"><br /><br />In economics and multi-agent systems, <i>aggregated (or distributed) knowledge</i> of a group is the knowledge of someone who knows exactly what every member of the group knows. This notion can be used to analyse the implications of the knowledge of a community if they were to combine their knowledge. In this paper we use fundamental tools from the theory of distributive lattice to characterize and compute aggregate knowledge.We prove that for distributive lattices of size $n$, (1) the meet of join-endomorphisms can be computed in time $O(n)$. Previous upper bound was $O(n^2)$. We show that (2) aggregated knowledge of given group can be viewed as the meet of the knowledge of each member of the group. We also show that for state sets of size $n$, (3) aggregated knowledge can be computed in time $O(n^2)$ in the general case and (4) for $S5$ knowledge (or <i>Aumman structures</i>), it can be computed in time $O(n\alpha(n))$ where $\alpha(n)$ is the inverse of the Ackermann function.</div>
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<td align="left"><i >Natanael Alpay, Peter Jipsen and <u >Melissa Sugimoto</u></i></td>
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<a href="javascript:toggleDiv('T8')">Unary-determined distributive l-magmas and bunched implication algebras</a>
<div style="display:none", id="T8"><br /><br />A distributive lattice-ordered magma ($d\ell$-magma) $(A,\wedge,\vee,\cdot)$ is a distributive lattice with a binary operation $\cdot$ that preserves joins in both arguments, and when $\cdot$ is associative then $(A,\vee,\cdot)$ is an idempotent semiring. A $d\ell$-magma with a top $\top$ is \emph{unary-determined} if $x\cdot y=(x\top\wedge y)\vee(x\wedge \top y)$. These algebras are term-equivalent to a subvariety of distributive lattices with $\top$ and two join-preserving unary operations $p,q$. We obtain simple conditions on $p,q$ such that $x\cdot y=(px\wedge y)\vee(x\wedge qy)$ is associative, commutative, idempotent and/or has an identity element. This generalizes previous results on the structure of doubly idempotent semirings and, in the case when the distributive lattice is a Heyting algebra, it provides structural insight into unary-determined algebraic models of bunched implication logic. We also provide Kripke semantics for the algebras under consideration, which leads to more efficient algorithms for constructing finite models.</div>
<div style="display:none", id="T8"><br /><br />A distributive lattice-ordered magma ($d\ell$-magma) $(A,\wedge,\vee,\cdot)$ is a distributive lattice with a binary operation $\cdot$ that preserves joins in both arguments, and when $\cdot$ is associative then $(A,\vee,\cdot)$ is an idempotent semiring. A $d\ell$-magma with a top $\top$ is <i>unary-determined</i> if $x\cdot y=(x\top\wedge y)\vee(x\wedge \top y)$. These algebras are term-equivalent to a subvariety of distributive lattices with $\top$ and two join-preserving unary operations $p,q$. We obtain simple conditions on $p,q$ such that $x\cdot y=(px\wedge y)\vee(x\wedge qy)$ is associative, commutative, idempotent and/or has an identity element. This generalizes previous results on the structure of doubly idempotent semirings and, in the case when the distributive lattice is a Heyting algebra, it provides structural insight into unary-determined algebraic models of bunched implication logic. We also provide Kripke semantics for the algebras under consideration, which leads to more efficient algorithms for constructing finite models.</div>
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<td align="left"><i ><u >Ralph Sarkis</u></i></td>
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<a href="javascript:toggleDiv('T23')">Quotienting a Monad via Projective Algebras (short talk)</a>
<div style="display:none", id="T23"><br /><br />Quotient types are quite common in mathematics but they are rather difficult to implement in a programming language. For instance, one can easily define the type of pairs of integers $\mathsf{Int}\times \mathsf{Int}$, but in order to define the type of rational numbers, one needs to quotient $\mathsf{Int}\times \mathsf{Int}$ by the relation $(p,q) \sim (r,s) \Leftrightarrow ps = rq$, which is easier said than done. In the framework of monadic programming, datatypes are free algebras for a monad. Not all algebras for a monad are free, but they are all quotients of free algebras. An algebra for a monad is said to be \emph{projective} if it is both a quotient and a subalgebra of a free algebra. We show that a natural family of projective algebras can be seen as algebras for a quotient monad. In other words, when a quotienting operation is nice enough that 1) the resulting algebra is a subalgebra of the free algebra and 2) it satisfies some naturality condition, then we obtain a monad that models the quotient type.</div>
<div style="display:none", id="T23"><br /><br />Quotient types are quite common in mathematics but they are rather difficult to implement in a programming language. For instance, one can easily define the type of pairs of integers $\mathsf{Int}\times \mathsf{Int}$, but in order to define the type of rational numbers, one needs to quotient $\mathsf{Int}\times \mathsf{Int}$ by the relation $(p,q) \sim (r,s) \Leftrightarrow ps = rq$, which is easier said than done. In the framework of monadic programming, datatypes are free algebras for a monad. Not all algebras for a monad are free, but they are all quotients of free algebras. An algebra for a monad is said to be <i>projective</i> if it is both a quotient and a subalgebra of a free algebra. We show that a natural family of projective algebras can be seen as algebras for a quotient monad. In other words, when a quotienting operation is nice enough that 1) the resulting algebra is a subalgebra of the free algebra and 2) it satisfies some naturality condition, then we obtain a monad that models the quotient type.</div>
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