We could say that two proofs are the same if theynormalise to the same proof. This `works' for intuitionistic logic (it is basically computation in theλ-calculus).
Perhaps OK! But:
I we equate proofs ofwildly different sizes, because cut-elimination is at least exponential;
I the decision cost corresponds to the computation cost,i.e.it can behuge;
I this ideafails for classical logic, because cut-elimination is non-con uent.
←−In Gentzen!
One big problem is that semantics usually does not takesizeinto consideration (only results matter).
Another big problem (in AG's opinion) is that we should design
languages starting fromhowwe compute with them, not fromwhatwe compute.Gentzen was mostly interested in the `what' and his view has not been suf ciently challenged.
The problem of proof identity: Example 1
We could say that two proofs are the same if theynormalise to the same proof. This `works' for intuitionistic logic (it is basically computation in theλ-calculus). Perhaps OK! But:
I we equate proofs ofwildly different sizes, because cut-elimination is at least exponential;
I the decision cost corresponds to the computation cost,i.e.it can behuge;
I this ideafails for classical logic, because cut-elimination is non-con uent.
←−In Gentzen!
One big problem is that semantics usually does not takesizeinto consideration (only results matter).
Another big problem (in AG's opinion) is that we should design
languages starting fromhowwe compute with them, not fromwhatwe compute.Gentzen was mostly interested in the `what' and his view has not been suf ciently challenged.
The problem of proof identity: Example 1
We could say that two proofs are the same if theynormalise to the same proof. This `works' for intuitionistic logic (it is basically computation in theλ-calculus). Perhaps OK! But:
I we equate proofs ofwildly different sizes, because cut-elimination is at least exponential;
I the decision cost corresponds to the computation cost,i.e.it can behuge;
I this ideafails for classical logic, because cut-elimination is non-con uent.
←−In Gentzen!
One big problem is that semantics usually does not takesizeinto consideration (only results matter).
Another big problem (in AG's opinion) is that we should design
languages starting fromhowwe compute with them, not fromwhatwe compute.Gentzen was mostly interested in the `what' and his view has not been suf ciently challenged.
The problem of proof identity: Example 1
We could say that two proofs are the same if theynormalise to the same proof. This `works' for intuitionistic logic (it is basically computation in theλ-calculus). Perhaps OK! But:
I we equate proofs ofwildly different sizes, because cut-elimination is at least exponential;
I the decision cost corresponds to the computation cost,i.e.it can behuge;
I this ideafails for classical logic, because cut-elimination is non-con uent.
←−In Gentzen!
One big problem is that semantics usually does not takesizeinto consideration (only results matter).
Another big problem (in AG's opinion) is that we should design
languages starting fromhowwe compute with them, not fromwhatwe compute.Gentzen was mostly interested in the `what' and his view has not been suf ciently challenged.
The problem of proof identity: Example 1
We could say that two proofs are the same if theynormalise to the same proof. This `works' for intuitionistic logic (it is basically computation in theλ-calculus). Perhaps OK! But:
I we equate proofs ofwildly different sizes, because cut-elimination is at least exponential;
I the decision cost corresponds to the computation cost,i.e.it can behuge;
I this ideafails for classical logic, because cut-elimination is non-con uent. ←−In Gentzen!
One big problem is that semantics usually does not takesizeinto consideration (only results matter).
Another big problem (in AG's opinion) is that we should design
languages starting fromhowwe compute with them, not fromwhatwe compute.Gentzen was mostly interested in the `what' and his view has not been suf ciently challenged.
The problem of proof identity: Example 1
We could say that two proofs are the same if theynormalise to the same proof. This `works' for intuitionistic logic (it is basically computation in theλ-calculus). Perhaps OK! But:
I we equate proofs ofwildly different sizes, because cut-elimination is at least exponential;
I the decision cost corresponds to the computation cost,i.e.it can behuge;
I this ideafails for classical logic, because cut-elimination is non-con uent. ←−In Gentzen!
One big problem is that semantics usually does not takesizeinto consideration (only results matter).
Another big problem (in AG's opinion) is that we should design
languages starting fromhowwe compute with them, not fromwhatwe compute.Gentzen was mostly interested in the `what' and his view has not been suf ciently challenged.
The problem of proof identity: Example 1
We could say that two proofs are the same if theynormalise to the same proof. This `works' for intuitionistic logic (it is basically computation in theλ-calculus). Perhaps OK! But:
I we equate proofs ofwildly different sizes, because cut-elimination is at least exponential;
I the decision cost corresponds to the computation cost,i.e.it can behuge;
I this ideafails for classical logic, because cut-elimination is non-con uent. ←−In Gentzen!
One big problem is that semantics usually does not takesizeinto consideration (only results matter).
Another big problem (in AG's opinion) is that we should design
languages starting fromhowwe compute with them, not fromwhatwe compute.
Gentzen was mostly interested in the `what' and his view has not been suf ciently challenged.
The problem of proof identity: Example 1
We could say that two proofs are the same if theynormalise to the same proof. This `works' for intuitionistic logic (it is basically computation in theλ-calculus). Perhaps OK! But:
I we equate proofs ofwildly different sizes, because cut-elimination is at least exponential;
I the decision cost corresponds to the computation cost,i.e.it can behuge;
I this ideafails for classical logic, because cut-elimination is non-con uent. ←−In Gentzen!
One big problem is that semantics usually does not takesizeinto consideration (only results matter).
Another big problem (in AG's opinion) is that we should design
languages starting fromhowwe compute with them, not fromwhatwe compute.Gentzen was mostly interested in the `what' and his view has not been suf ciently challenged.