A Compositional Typed Higher-Order Logic with
Definitions
Ingmar Dasseville, Matthias van der Hallen, Bart Bogaerts, Gerda Janssens,
Marc Denecker
Timeline
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
Building a language 101
What does an expression denote?
set of syntactical expressions
BNF
executable sequence of instructions
C
Logic
???
Infons
Infon
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
=
Language constructs
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?
Compositionality (Frege)
The meaning of a complex expression is determined by the meanings of its constituent expressions and the rules used to combine them
Compositionality (Frege)
Sem_L(C(e_1,...,e_n)) = Sem_C(Sem_L(e_1),...,Sem_L(e_n))
SemL(C(e1,...,en))=SemC(SemL(e1),...,SemL(en))
The semantics for a logic L and a language constructs C must satisfy:
And also typing:
Typ_L(C(e_1,...,e_n)) = Typ_C(Typ_L(e_1),...,Typ_L(e_n))
TypL(C(e1,...,en))=TypC(TypL(e1),...,TypL(en))
Language Constructs
Apply a function to an argument
Application
Introduce an anonymous function
Lambda Abstraction
Definitions
Well-founded definitions
Constants
Numbers, Strings, ...
Applications
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))}.
}
Bird's-eye view
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
iclp16
By Ingmar Dasseville
iclp16
- 929
