Title: Theory / Unimportant / CS / Fundamental / Automata / Finite / Pin{ / D Advanced tools= / 19 Sequential functions+32
Tags: examplesSwitchExample1 hasSeenConjDef

Name: A pure sequential transducer
Definition: \(Q\) a finite set of states
Page: 251
Chapter: XVI
Section: 1.1
Definition2: \(A\) the input alphabet
Conjdef: A tuple
Definition3: \(R\) the output semigroup
Definition4: \(q_0\in Q\) initial state
Denoted by: \(\mathcal T\)
Definition5: [$]\defFun{Q\times A}{Q}{(q,a)}{q\cdot a}[/$] the transition partial function
Definition6: [$]\defFun{Q\times A}{R}{(q,a)}{q\otimes a}[/$] the output partial function

Name: A transition in \(\mathcal T\)
Page: 251
Chapter: XVI
Section: 1.1
Variables: \(\mathcal T=(Q,A,R,q_0,\cdot,\times)\)

Name: The extension of \(\cdot\)
Definition: The partial function [$]\defFun{Q\times A^*}{Q}{a_1\dots a_n}{(\dots(q a_1)\dots)a_n}[/$]
Page: 251
Chapter: XVI
Section: 1.1
Variables: \(\mathcal T=(Q,A,R,q_0,\cdot,\times)\)

Name: The extension of \(\star\)
Definition: The partial function [$]\defFun{Q\times A^*}{R}{a_1\dots a_n}{\prod_{i=1}^n(q\star (a_1\dots a_{i-1}))\star a_i}[/$]
Page: 251
Chapter: XVI
Section: 1.1
Variables: \(\mathcal T=(Q,A,R,q_0,\cdot,\times)\)

Name: The function realised by \(\mathcal T\)
Definition: Partial function [$]\defFun{A^*}{R^1}{u}{q_0\star u}[/$]
Page: 252
Chapter: XVI
Section: 1.1
Variables: \(\mathcal T=(Q,A,R,q_0,\cdot,\times)\) a pure sequential transducer,
Index: 1
Kind: Prop
Denoted by: \(\phi\)