Information algebra

<h2 id="information-and-its-operations">Information and its operations</h2>
<p>More precisely, in the two-sorted algebra 
  
    
      
        (
        Φ
        ,
        D
        )
      
    
    {\displaystyle (\Phi ,D)}
  
, the following operations are defined
</p>
<table><tbody><tr><td>Combination                    ⊗        :        Φ        ⊗        Φ        →        Φ        ,                 (        ϕ        ,        ψ        )        ↦        ϕ        ⊗        ψ              {\displaystyle \otimes :\Phi \otimes \Phi \rightarrow \Phi ,~(\phi ,\psi )\mapsto \phi \otimes \psi }  Focusing                    ⇒:        Φ        ⊗        D        →        Φ        ,                 (        ϕ        ,        x        )        ↦                  ϕ                      ⇒            x                                {\displaystyle \Rightarrow :\Phi \otimes D\rightarrow \Phi ,~(\phi ,x)\mapsto \phi ^{\Rightarrow x}}  </td><td>            </td></tr></tbody></table>
<p>Additionally, in 
  
    
      
        D
      
    
    {\displaystyle D}
  
 the usual lattice operations (meet and join) are defined.
</p>
<h2 id="axioms-and-definition">Axioms and definition</h2>
<p>The axioms of the two-sorted algebra 
  
    
      
        (
        Φ
        ,
        D
        )
      
    
    {\displaystyle (\Phi ,D)}
  
, in addition to the axioms of the lattice 
  
    
      
        D
      
    
    {\displaystyle D}
  
:
</p>
<table><tbody><tr><td>Semigroup                    Φ              {\displaystyle \Phi }   is a commutative semigroup under combination with a neutral element (representing vacuous information).Distributivity of Focusing over Combination                    (                  ϕ                      ⇒            x                          ⊗        ψ                  )                      ⇒            x                          =                  ϕ                      ⇒            x                          ⊗                  ψ                      ⇒            x                                {\displaystyle (\phi ^{\Rightarrow x}\otimes \psi )^{\Rightarrow x}=\phi ^{\Rightarrow x}\otimes \psi ^{\Rightarrow x}}  <p>To focus an information on                     x              {\displaystyle x}   combined with another information to domain                     x              {\displaystyle x}  , one may as well first focus the second information to                     x              {\displaystyle x}   and then combine.</p>Transitivity of Focusing                    (                  ϕ                      ⇒            x                                    )                      ⇒            y                          =                  ϕ                      ⇒            x            ∧            y                                {\displaystyle (\phi ^{\Rightarrow x})^{\Rightarrow y}=\phi ^{\Rightarrow x\wedge y}}  <p>To focus an information on                     x              {\displaystyle x}   and                     y              {\displaystyle y}  , one may focus it to                     x        ∧        y              {\displaystyle x\wedge y}  .</p>Idempotency                    ϕ        ⊗                  ϕ                      ⇒            x                          =        ϕ              {\displaystyle \phi \otimes \phi ^{\Rightarrow x}=\phi }  <p>An information combined with a part of itself gives nothing new.</p>Support                    ∀        ϕ        ∈        Φ        ,                 ∃        x        ∈        D              {\displaystyle \forall \phi \in \Phi ,~\exists x\in D}   such that                     ϕ        =                  ϕ                      ⇒            x                                {\displaystyle \phi =\phi ^{\Rightarrow x}}  <p>Each information refers to at least one domain (question).</p></td><td>            </td></tr></tbody></table>
<p>A two-sorted algebra 
  
    
      
        (
        Φ
        ,
        D
        )
      
    
    {\displaystyle (\Phi ,D)}
  
 satisfying these axioms is called an Information Algebra.
</p>
<h2 id="order-of-information">Order of information</h2>
<p>A partial order of information can be introduced by defining 
  
    
      
        ϕ
        ≤
        ψ
      
    
    {\displaystyle \phi \leq \psi }
  
 if 
  
    
      
        ϕ
        ⊗
        ψ
        =
        ψ
      
    
    {\displaystyle \phi \otimes \psi =\psi }
  
. This means that 
  
    
      
        ϕ
      
    
    {\displaystyle \phi }
  
 is less informative than 
  
    
      
        ψ
      
    
    {\displaystyle \psi }
  
 if it adds no new information to 
  
    
      
        ψ
      
    
    {\displaystyle \psi }
  
. The semigroup 
  
    
      
        Φ
      
    
    {\displaystyle \Phi }
  
 is a semilattice relative to this order, i.e. 
  
    
      
        ϕ
        ⊗
        ψ
        =
        ϕ
        ∨
        ψ
      
    
    {\displaystyle \phi \otimes \psi =\phi \vee \psi }
  
. Relative to any domain (question) 
  
    
      
        x
        ∈
        D
      
    
    {\displaystyle x\in D}
  
 a partial order can be introduced by defining 
  
    
      
        ϕ
        
          ≤
          
            x
          
        
        ψ
      
    
    {\displaystyle \phi \leq _{x}\psi }
  
  if 
  
    
      
        
          ϕ
          
            ⇒
            x
          
        
        ≤
        
          ψ
          
            ⇒
            x
          
        
      
    
    {\displaystyle \phi ^{\Rightarrow x}\leq \psi ^{\Rightarrow x}}
  
. It represents the order of information content of 
  
    
      
        ϕ
      
    
    {\displaystyle \phi }
  
 and 
  
    
      
        ψ
      
    
    {\displaystyle \psi }
  
 relative to the domain (question) 
  
    
      
        x
      
    
    {\displaystyle x}
  
.
</p>
<h2 id="labeled-information-algebra">Labeled information algebra</h2>
<p>The pairs 
  
    
      
        (
        ϕ
        ,
        x
        )
         
      
    
    {\displaystyle (\phi ,x)\ }
  
, where 
  
    
      
        ϕ
        ∈
        Φ
      
    
    {\displaystyle \phi \in \Phi }
  
 and 
  
    
      
        x
        ∈
        D
      
    
    {\displaystyle x\in D}
  
 such that 
  
    
      
        
          ϕ
          
            ⇒
            x
          
        
        =
        ϕ
      
    
    {\displaystyle \phi ^{\Rightarrow x}=\phi }
  
 form a labeled Information Algebra. More precisely, in the two-sorted algebra 
  
    
      
        (
        Φ
        ,
        D
        )
         
      
    
    {\displaystyle (\Phi ,D)\ }
  
, the following operations are defined
</p>
<table><tbody><tr><td>Labeling                    d        (        ϕ        ,        x        )        =        x                       {\displaystyle d(\phi ,x)=x\ }  Combination                    (        ϕ        ,        x        )        ⊗        (        ψ        ,        y        )        =        (        ϕ        ⊗        ψ        ,        x        ∨        y        )                                                  {\displaystyle (\phi ,x)\otimes (\psi ,y)=(\phi \otimes \psi ,x\vee y)~~~~}  Projection                    (        ϕ        ,        x                  )                      ↓            y                          =        (                  ϕ                      ⇒            y                          ,        y        )                   for                 y        ≤        x              {\displaystyle (\phi ,x)^{\downarrow y}=(\phi ^{\Rightarrow y},y){\text{ for }}y\leq x}  </td><td>            </td></tr></tbody></table>
<h2 id="models-of-information-algebras">Models of information algebras</h2>
<p>Here follows an incomplete list of instances of information algebras:
</p>
<ul><li><a href="/facts/Relational_algebra/7tjTFXdf">Relational algebra</a>: The reduct of a relational algebra with natural join as combination and the usual projection is a labeled information algebra, see Example.</li>
<li>Constraint systems: Constraints form an information algebra (Jaffar & Maher 1994).</li>
<li>Semiring valued algebras: C-Semirings induce information algebras (Bistarelli, Montanari & Rossi1997);(Bistarelli et al. 1999);(Kohlas & Wilson 2006).</li>
<li><a href="/facts/Logic/8JnpJLtt">Logic</a>: Many logic systems induce information algebras (Wilson & Mengin 1999). Reducts of <a href="/facts/Cylindric_algebra/yn3kwKTk">cylindric algebras</a> (Henkin, Monk & Tarski 1971) or <a href="/facts/Polyadic_algebra/LvMxoUPl">polyadic algebras</a> are information algebras related to <a href="/facts/Predicate_logic/lcISqWIg">predicate logic</a> (Halmos 2000).</li>
<li><a href="/facts/Module_(mathematics)/jP0LIhFL">Module algebras</a>: (Bergstra, Heering & Klint 1990);(de Lavalette 1992).</li>
<li><a href="/facts/Linear_system/uEl20Kc1">Linear systems</a>: Systems of linear equations or linear inequalities induce information algebras (Kohlas 2003).</li></ul>
<h3>Worked-out example: relational algebra</h3>

<p>Let 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathcal {A}}}
  
 be a set of symbols, called <i>attributes</i> (or <i>column names</i>). For each 
  
    
      
        α
        ∈
        
          
            A
          
        
      
    
    {\displaystyle \alpha \in {\mathcal {A}}}
  
 let 
  
    
      
        
          U
          
            α
          
        
      
    
    {\displaystyle U_{\alpha }}
  
 be a non-empty set, the set of all possible values of the attribute 
  
    
      
        α
      
    
    {\displaystyle \alpha }
  
. For example, if

A
          
        
        =
        {
        
          
            name
          
        
        ,
        
          
            age
          
        
        ,
        
          
            income
          
        
        }
      
    
    {\displaystyle {\mathcal {A}}=\{{\texttt {name}},{\texttt {age}},{\texttt {income}}\}}
  
, then 
  
    
      
        
          U
          
            
              name
            
          
        
      
    
    {\displaystyle U_{\texttt {name}}}
  
 could
be the set of strings, whereas 
  
    
      
        
          U
          
            
              age
            
          
        
      
    
    {\displaystyle U_{\texttt {age}}}
  
 and 
  
    
      
        
          U
          
            
              income
            
          
        
      
    
    {\displaystyle U_{\texttt {income}}}
  
 are both the set of non-negative integers.
</p><p>Let 
  
    
      
        x
        ⊆
        
          
            A
          
        
      
    
    {\displaystyle x\subseteq {\mathcal {A}}}
  
. An <i>
  
    
      
        x
      
    
    {\displaystyle x}
  
-tuple</i> is a function 
  
    
      
        f
      
    
    {\displaystyle f}
  
 so that

dom
          
        
        (
        f
        )
        =
        x
      
    
    {\displaystyle {\hbox{dom}}(f)=x}
  
 and 
  
    
      
        f
        (
        α
        )
        ∈
        
          U
          
            α
          
        
      
    
    {\displaystyle f(\alpha )\in U_{\alpha }}
  
 for each 
  
    
      
        α
        ∈
        x
      
    
    {\displaystyle \alpha \in x}
  
 The set
of all 
  
    
      
        x
      
    
    {\displaystyle x}
  
-tuples is denoted by 
  
    
      
        
          E
          
            x
          
        
      
    
    {\displaystyle E_{x}}
  
. For an 
  
    
      
        x
      
    
    {\displaystyle x}
  
-tuple 
  
    
      
        f
      
    
    {\displaystyle f}
  
 and a subset

y
        ⊆
        x
      
    
    {\displaystyle y\subseteq x}
  
 the restriction 
  
    
      
        f
        [
        y
        ]
      
    
    {\displaystyle f[y]}
  
 is defined to be the

y
      
    
    {\displaystyle y}
  
-tuple 
  
    
      
        g
      
    
    {\displaystyle g}
  
 so that 
  
    
      
        g
        (
        α
        )
        =
        f
        (
        α
        )
      
    
    {\displaystyle g(\alpha )=f(\alpha )}
  
 for all 
  
    
      
        α
        ∈
        y
      
    
    {\displaystyle \alpha \in y}
  
.
</p><p>A <i>relation 
  
    
      
        R
      
    
    {\displaystyle R}
  
 over 
  
    
      
        x
      
    
    {\displaystyle x}
  
</i> is a set of 
  
    
      
        x
      
    
    {\displaystyle x}
  
-tuples, i.e. a subset of 
  
    
      
        
          E
          
            x
          
        
      
    
    {\displaystyle E_{x}}
  
.
The set of attributes 
  
    
      
        x
      
    
    {\displaystyle x}
  
 is called the <i>domain</i> of 
  
    
      
        R
      
    
    {\displaystyle R}
  
 and denoted by

d
        (
        R
        )
      
    
    {\displaystyle d(R)}
  
. For 
  
    
      
        y
        ⊆
        d
        (
        R
        )
      
    
    {\displaystyle y\subseteq d(R)}
  
 the <i>projection</i> of 
  
    
      
        R
      
    
    {\displaystyle R}
  
 onto 
  
    
      
        y
      
    
    {\displaystyle y}
  
 is defined
as follows:
</p>

π
          
            y
          
        
        (
        R
        )
        :=
        {
        f
        [
        y
        ]
        ∣
        f
        ∈
        R
        }
        .
      
    
    {\displaystyle \pi _{y}(R):=\{f[y]\mid f\in R\}.}

<p>The <i>join</i> of a relation 
  
    
      
        R
      
    
    {\displaystyle R}
  
 over 
  
    
      
        x
      
    
    {\displaystyle x}
  
 and a relation 
  
    
      
        S
      
    
    {\displaystyle S}
  
 over 
  
    
      
        y
      
    
    {\displaystyle y}
  
 is
defined as follows:
</p>

R
        ⋈
        S
        :=
        {
        f
        ∣
        f
        
        (
        x
        ∪
        y
        )
        
          
            -tuple
          
        
        ,
        
        f
        [
        x
        ]
        ∈
        R
        ,
        
        f
        [
        y
        ]
        ∈
        S
        }
        .
      
    
    {\displaystyle R\bowtie S:=\{f\mid f\quad (x\cup y){\hbox{-tuple}},\quad f[x]\in R,\;f[y]\in S\}.}

<p>As an example, let 
  
    
      
        R
      
    
    {\displaystyle R}
  
 and 
  
    
      
        S
      
    
    {\displaystyle S}
  
 be the following relations:
</p>

R
        =
        
          
            
              
                
                  
                    name
                  
                
              
              
                
                  
                    age
                  
                
              
            
            
              
                
                  
                    A
                  
                
              
              
                
                  
                    34
                  
                
              
            
            
              
                
                  
                    B
                  
                
              
              
                
                  
                    47
                  
                
              
            
          
        
        
        S
        =
        
          
            
              
                
                  
                    name
                  
                
              
              
                
                  
                    income
                  
                
              
            
            
              
                
                  
                    A
                  
                
              
              
                
                  
                    20'000
                  
                
              
            
            
              
                
                  
                    B
                  
                
              
              
                
                  
                    32'000
                  
                
              
            
          
        
      
    
    {\displaystyle R={\begin{matrix}{\texttt {name}}&{\texttt {age}}\\{\texttt {A}}&{\texttt {34}}\\{\texttt {B}}&{\texttt {47}}\\\end{matrix}}\qquad S={\begin{matrix}{\texttt {name}}&{\texttt {income}}\\{\texttt {A}}&{\texttt {20'000}}\\{\texttt {B}}&{\texttt {32'000}}\\\end{matrix}}}

<p>Then the join of 
  
    
      
        R
      
    
    {\displaystyle R}
  
 and 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is:
</p>

R
        ⋈
        S
        =
        
          
            
              
                
                  
                    name
                  
                
              
              
                
                  
                    age
                  
                
              
              
                
                  
                    income
                  
                
              
            
            
              
                
                  
                    A
                  
                
              
              
                
                  
                    34
                  
                
              
              
                
                  
                    20'000
                  
                
              
            
            
              
                
                  
                    B
                  
                
              
              
                
                  
                    47
                  
                
              
              
                
                  
                    32'000
                  
                
              
            
          
        
      
    
    {\displaystyle R\bowtie S={\begin{matrix}{\texttt {name}}&{\texttt {age}}&{\texttt {income}}\\{\texttt {A}}&{\texttt {34}}&{\texttt {20'000}}\\{\texttt {B}}&{\texttt {47}}&{\texttt {32'000}}\\\end{matrix}}}

<p>A relational database with natural join 
  
    
      
        ⋈
      
    
    {\displaystyle \bowtie }
  
 as combination and the usual projection 
  
    
      
        π
      
    
    {\displaystyle \pi }
  
 is an information algebra.
The operations are well defined since
</p>
<ul><li>
  
    
      
        d
        (
        R
        ⋈
        S
        )
        =
        d
        (
        R
        )
        ∪
        d
        (
        S
        )
      
    
    {\displaystyle d(R\bowtie S)=d(R)\cup d(S)}
  
</li>
<li>If 
  
    
      
        x
        ⊆
        d
        (
        R
        )
      
    
    {\displaystyle x\subseteq d(R)}
  
, then 
  
    
      
        d
        (
        
          π
          
            x
          
        
        (
        R
        )
        )
        =
        x
      
    
    {\displaystyle d(\pi _{x}(R))=x}
  
.</li></ul>
<p>It is easy to see that relational databases satisfy the axioms of a labeled
information algebra:
</p>
semigroup

(
        
          R
          
            1
          
        
        ⋈
        
          R
          
            2
          
        
        )
        ⋈
        
          R
          
            3
          
        
        =
        
          R
          
            1
          
        
        ⋈
        (
        
          R
          
            2
          
        
        ⋈
        
          R
          
            3
          
        
        )
      
    
    {\displaystyle (R_{1}\bowtie R_{2})\bowtie R_{3}=R_{1}\bowtie (R_{2}\bowtie R_{3})}
  
 and 
  
    
      
        R
        ⋈
        S
        =
        S
        ⋈
        R
      
    
    {\displaystyle R\bowtie S=S\bowtie R}

transitivity
If 
  
    
      
        x
        ⊆
        y
        ⊆
        d
        (
        R
        )
      
    
    {\displaystyle x\subseteq y\subseteq d(R)}
  
, then 
  
    
      
        
          π
          
            x
          
        
        (
        
          π
          
            y
          
        
        (
        R
        )
        )
        =
        
          π
          
            x
          
        
        (
        R
        )
      
    
    {\displaystyle \pi _{x}(\pi _{y}(R))=\pi _{x}(R)}
  
.
combination
If 
  
    
      
        d
        (
        R
        )
        =
        x
      
    
    {\displaystyle d(R)=x}
  
 and 
  
    
      
        d
        (
        S
        )
        =
        y
      
    
    {\displaystyle d(S)=y}
  
, then 
  
    
      
        
          π
          
            x
          
        
        (
        R
        ⋈
        S
        )
        =
        R
        ⋈
        
          π
          
            x
            ∩
            y
          
        
        (
        S
        )
      
    
    {\displaystyle \pi _{x}(R\bowtie S)=R\bowtie \pi _{x\cap y}(S)}
  
.
idempotency
If 
  
    
      
        x
        ⊆
        d
        (
        R
        )
      
    
    {\displaystyle x\subseteq d(R)}
  
, then 
  
    
      
        R
        ⋈
        
          π
          
            x
          
        
        (
        R
        )
        =
        R
      
    
    {\displaystyle R\bowtie \pi _{x}(R)=R}
  
.
support
If 
  
    
      
        x
        =
        d
        (
        R
        )
      
    
    {\displaystyle x=d(R)}
  
, then 
  
    
      
        
          π
          
            x
          
        
        (
        R
        )
        =
        R
      
    
    {\displaystyle \pi _{x}(R)=R}
  
.
<h2 id="connections">Connections</h2>

Valuation algebras
Dropping the idempotency axiom leads to <a href="/facts/Valuation_algebra/W3W33SyE">valuation algebras</a>. These axioms have been introduced by (Shenoy & Shafer 1990) to generalize <i>local computation schemes</i> (Lauritzen & Spiegelhalter 1988) from Bayesian networks to more general formalisms, including belief function, possibility potentials, etc. (Kohlas & Shenoy 2000). For a book-length exposition on the topic see Pouly & Kohlas (2011).
Domains and information systems
<i>Compact Information Algebras</i> (Kohlas 2003) are related to <a href="/facts/Scott_domain/mFGtPq3E">Scott domains</a> and <a href="/facts/Scott_information_system/Q7YSE2mI">Scott information systems</a>  (Scott 1970);(Scott 1982);(Larsen & Winskel 1984).
Uncertain information
Random variables with values in information algebras represent <i><a href="/facts/Probabilistic_argumentation/KGnU8MCB">probabilistic argumentation</a> systems</i> (Haenni, Kohlas & Lehmann 2000).
Semantic information
Information algebras introduce semantics by relating information to questions through focusing and combination (Groenendijk & Stokhof 1984);(Floridi 2004).
Information flow
Information algebras are related to information flow, in particular classifications  (Barwise & Seligman 1997).
Tree decomposition
Information algebras are organized into a hierarchical tree structure, and decomposed into smaller problems.
Semigroup theory
...
Compositional models
Such models may be defined within the framework of information algebras: <a href="https://arxiv.org/abs/1612.02587">https://arxiv.org/abs/1612.02587</a>
Extended axiomatic foundations of information and valuation algebras
The concept of conditional independence is basic for information algebras and a new axiomatic foundation of information algebras, based on conditional independence, extending the old one (see above) is available: <a href="https://arxiv.org/abs/1701.02658">https://arxiv.org/abs/1701.02658</a>
<h2 id="historical-roots">Historical Roots</h2>
<p>The axioms for information algebras are derived from 
the axiom system proposed in (Shenoy and Shafer, 1990), see also (Shafer, 1991).
</p>

<ul><li><a href="/facts/Jon_Barwise/EoL8nn26">Barwise, J.</a>; Seligman, J. (1997), <i>Information Flow: The Logic of Distributed Systems</i>, Cambridge U.K.: Number 44 in Cambridge Tracts in Theoretical Computer Science, Cambridge University Press</li>
<li>Bergstra, J.A.; Heering, J.; Klint, P. (1990), "Module algebra", <i>Journal of the ACM</i>, 73 (2): 335–372, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1145%2F77600.77621">10.1145/77600.77621</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:7910431">7910431</a></li>
<li>Bistarelli, S.; <a href="/facts/H%C3%A9l%C3%A8ne_Fargier/zPeH3wzc">Fargier, H.</a>; Montanari, U.; Rossi, F.; Schiex, T.; Verfaillie, G. (1999), <a href="https://web.archive.org/web/20220310181822/https://aiolatest.com/application-of-abstract-algebra-in-real-life/">"Semiring-based CSPs and valued CSPs: Frameworks, properties, and comparison"</a>, <i>Constraints</i>, 4 (3): 199–240, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1023%2FA%3A1026441215081">10.1023/A:1026441215081</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:17232456">17232456</a>, archived from <a href="https://aiolatest.com/application-of-abstract-algebra-in-real-life/">the original</a> on March 10, 2022</li>
<li>Bistarelli, Stefano; Montanari, Ugo; Rossi, Francesca (1997), "Semiring-based constraint satisfaction and optimization", <i>Journal of the ACM</i>, 44 (2): 201–236, <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.5110">10.1.1.45.5110</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1145%2F256303.256306">10.1145/256303.256306</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:4003767">4003767</a></li>
<li>de Lavalette, Gerard R. Renardel (1992), <a href="http://citeseer.ist.psu.edu/484529.html">"Logical semantics of modularisation"</a>, in Egon Börger; Gerhard Jäger; Hans Kleine Büning; Michael M. Richter (eds.), <i>CSL: 5th Workshop on Computer Science Logic</i>, Volume 626 of Lecture Notes in Computer Science, Springer, pp. 306–315, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-55789-0</li>
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Information algebra open-in-new

Information algebra