Ingmar Dasseville, Matthias van der Hallen, Bart Bogaerts, Gerda Janssens,
Marc Denecker
adding templates to languages
Last Year
framework for adding language constructs and building logics
Result
This Year
Building a very general logic
including compositionality principles
What does an expression denote?
set of syntactical expressions
BNF
executable sequence of instructions
C
Logic
???
Infons
Function of structures to values
Infon of
p v q
Maps {p} to True
Maps {} to False
Infon of
c+5
Maps {c->3} to 8
A quantum of information
=
Extend the set of valid expressions
Syntax
Not all expressions within the grammar are sensible (e.g. 1+"hello")
Typing
Semantics
What infon corresponds to the expression?
The meaning of a complex expression is determined by the meanings of its constituent expressions and the rules used to combine them
The semantics for a logic L and a language constructs C must satisfy:
And also typing:
Apply a function to an argument
Application
Introduce an anonymous function
Lambda Abstraction
Definitions
Well-founded definitions
Constants
Numbers, Strings, ...
Templates
Game Playing
{
$\forall$cur $\forall$Move $\forall$IsWon: win(cur, Move, IsWon) $\leftarrow$ IsWon(cur) $\lor$
$\exists$ nxt : Move(cur,nxt) $\land$ lose(nxt,Move,IsWon).
$\forall$cur $\forall$Move $\forall$IsWon: lose(cur,Move, IsWon) $\leftarrow$ $\neg$IsWon(cur) $\land$
$\forall$ nxt : Move(cur,nxt) $\Rightarrow$ win(nxt,Move,IsWon).
}
{
$\forall$Q $\forall$P: tc(P,Q) $\leftarrow$
{$\forall$x $\forall$y: Q(x,y) $\leftarrow$ P(x,y) $\lor$($\exists$ z: Q(x,z)$\land$Q(z,y))}.
}
Constraints on semantics by Frege
Compositionality
For free because of unifying infon objects
Integration of constructs
Applications
Current applications more natural +
loads of new ones (but we need solvers!)
Meaning of an expression
Infon