Crate pocket_prover
source · [−]Expand description
Pocket-Prover
A fast, brute force, automatic theorem prover for first order logic
- For generic automated theorem proving, see monotonic_solver
- For a debuggable SAT solver, see debug_sat
extern crate pocket_prover;
use pocket_prover::*;
fn main() {
println!("Socrates is mortal: {}", prove!(&mut |man, mortal, socrates| {
// Using `imply` because we want to prove an inference rule.
imply(
// Premises.
and(
// All men are mortal.
imply(man, mortal),
// Socrates is a man.
imply(socrates, man),
),
// Conclusion.
imply(socrates, mortal)
)
}));
}
Motivation
The motivation is to provide the analogue of a “pocket calculator”, but for logic, therefore called a “pocket prover”.
This library uses an approach that is simple to implement from scratch in a low level language.
This is useful in cases like:
- Study logic without the hurdle of doing manual proofs
- Checking your understanding of logic
- Verify that logicians are wizards and not lizards
- Due to a series of unfortunate events, you got only 24 hours to learn logic and just need the gist of it
- Memorizing source code for situations like The Martian
- A tiny mistake and the whole planet blows up (e.g. final decisions before the AI singularity and you need to press the right buttons)
In addition this library can be used to create extensible logical systems.
For more information, see the Prove
trait.
Implementation
This library uses brute-force to check proofs, instead of relying on axioms of logic.
64bit CPUs are capable of checking logical proofs of 6 arguments (booleans)
in O(1), because proofs can be interpreted as tautologies (true for all input)
and 2^6 = 64
.
This is done by replacing bool
with u64
and organizing inputs
using bit patterns that simulate a truth table of 6 arguments.
To extend to 10 arguments, T
and F
are used to alternate the 4 extra arguments.
To extend to N arguments, recursive calls are used down to less than 10 arguments.
Path Semantical Logic
Notice! Path Semantical Logic is at early stage of research.
This library has experimental support for a subset of Path Semantical Logic. Implementation is based on paper Faster Brute Force Proofs.
Path Semantical Logic separates propositions into levels, such that an equality between two propositions in level N+1, propagates into equality between uniquely associated propositions in level N.
For example, if f
has level 1 and x
has level 0,
then imply(f, x)
associates x
uniquely with f
,
such that the core axiom of Path Semantics
is satisfied.
This library has currently only support for level 1 and 0.
These functions are prefixed with path1_
.
The macros count!
and prove!
will automatically expand
to path1_count!
and path1_prove!
.
Each function takes two arguments, consisting of tuples of propositions, e.g. (f, g), (x, y)
.
Arbitrary number of arguments is supported.
extern crate pocket_prover;
use pocket_prover::*;
fn main() {
println!("=== Path Semantical Logic ===");
println!("The notation `f(x)` means `x` is uniquely associated with `f`.");
println!("For more information, see the section 'Path Semantical Logic' in documentation.");
println!("");
print!("(f(x), g(y), h(z), f=g ⊻ f=h) => (x=y ∨ x=z): ");
println!("{}\n", prove!(&mut |(f, g, h), (x, y, z)| {
imply(
and!(
imply(f, x),
imply(g, y),
imply(h, z),
xor(eq(f, g), eq(f, h))
),
or(eq(x, y), eq(x, z))
)
}));
print!("(f(x), g(y), f=g => h, h(z)) => (x=y => z): ");
println!("{}\n", prove!(&mut |(f, g, h), (x, y, z)| {
imply(
and!(
imply(f, x),
imply(g, y),
imply(eq(f, g), h),
imply(h, z)
),
imply(eq(x, y), z)
)
}));
}
Path Semantical Quality
Pocket-Prover has a model of Path Semantical Quality that resembles quantum logic.
To write x ~~ y
you use q(x, y)
or qual(x, y)
.
q(x, y)
is the same as and!(eq(x, y), qubit(x), qubit(y))
.
q(x, x)
is the same as qubit(x)
.
A qubit is a kind of “superposition”.
One can also think about it as introducing a new argument qubit(x)
that depends on x
.
Since qubits can collide with other propositions,
one must repeat measurements e.g. using measure
to get classical states.
However, sometimes one might wish to amplify quantum states, using amplify
or amp
.
To use quality with path semantics, one should use ps_core
.
Path Semantical Logic is designed for equality, not quality.
use pocket_prover::*;
fn main() {
println!("Path semantics: {}", measure(1, || prove!(&mut |a, b, c, d| {
imply(
and!(
ps_core(a, b, c, d),
imply(a, c),
imply(b, d)
),
imply(qual(a, b), qual(c, d))
)
})));
}
Re-exports
Modules
Helper utilities for extracting data from proofs.
Macros
An AND relation of variable arguments.
Evaluates an expression for all bit configurations.
Generates a “{}{}{}…” format for bits.
Path Semantical Logic: A contractible “family of types”.
Counts the number of solutions of a variable argument boolean function.
An IMPLY chain of variable arguments.
An OR relation of variable arguments.
Path Semantical Logic: Counts the number of solutions of a variable argument boolean function.
Path Semantical Logic: Returns true
if proposition is correct, false
otherwise.
Prints a truth table with result of a boolean expression.
Prints a truth table extracted from a theory, assigning each case a bit and automatically flip expression properly.
Returns true
if proposition is correct, false
otherwise.
Helper macro for counting size of a tuple.
Helper macro for binding to a tuple pattern.
An XOR relation of variable arguments.
Constants
The False proposition.
Used to alternate higher than 6 arguments, set to 0
.
0xaaaa_aaaa_aaaa_aaaa
0xcccc_cccc_cccc_cccc
0xf0f0_f0f0_f0f0_f0f0
0xff00_ff00_ff00_ff00
0xffff_0000_ffff_0000
0xffff_ffff_0000_0000
The True proposition.
Used to alternate higher than 6 arguments, set to 1
.
Traits
Implemented by base logical systems.
Used to construct logical systems.
Implemented by logical systems to define core rules.
Implemented by types to use with all
and any
.
Implemented by logical systems to extend existing ones.
Implemented by observables.
Implemented by provable systems of logic.
Functions
Enumerates the type, checking that all outputs are true.
Amplify a “wavefunction” of a proposition using its qubit transform.
Returns true
if all arguments are true
.
An AND relation of 3 argument.
An AND relation of 4 arguments.
An AND relation of 5 arguments.
An AND relation of 6 arguments.
An AND relation of 7 arguments.
An AND relation of 8 arguments.
An AND relation of 9 arguments.
An AND relation of 10 arguments.
An AND relation of variable number of arguments.
Enumerates the type, checking that at least one output is true.
Path semantical aquality a ~¬~ b
.
Path semantical contravariant quality a ¬~~ b
.
Counts the number of solutions of a 1-argument boolean function.
Counts the number of solutions of a 2-argument boolean function.
Counts the number of solutions of a 3-argument boolean function.
Counts the number of solutions of a 4-argument boolean function.
Counts the number of solutions of a 5-argument boolean function.
Counts the number of solutions of a 6-argument boolean function.
Counts the number of solutions of a 7-argument boolean function.
Counts the number of solutions of an 8-argument boolean function.
Counts the number of solutions of a 9-argument boolean function.
Counts the number of solutions of a 10-argument boolean function.
Counts the number of solutions of an n-argument boolean function.
Returns true
if arguments are equal.
Ignores argument, always returning false
.
Ignores both arguments, returning false
for all inputs.
Ignores all 3 arguments, returning false
for all inputs.
Ignores all 4 arguments, returning false
for all inputs.
Ignores all 5 arguments, returning false
for all inputs.
Ignores all 6 arguments, returning false
for all inputs.
Ignores all 7 arguments, returning false
for all inputs.
Ignores all 8 arguments, returning false
for all inputs.
Ignores all 9 arguments, returning false
for all inputs.
Ignores all 10 arguments, returning false
for all inputs.
Aligns equality of qubits up to some homotopy level.
Returns argument.
First argument implies the second.
An IMPLY chain of 3 arguments.
An IMPLY chain of 4 arguments.
An IMPLY chain of 5 arguments.
An IMPLY chain of 6 arguments.
An IMPLY chain of 7 arguments.
An IMPLY chain of 8 arguments.
An IMPLY chain of 9 arguments.
An IMPLY chain of 10 arguments.
An IMPLY chain of variable number of arguments.
Defines a groupoid relation from x
to a
and b
.
Defines an n-groupoid relation from x
to a
and b
.
Defines a homotopy level n
relation from x
to a
and b
.
Defines a proposition relation of proposition x
to potential proofs a
and b
.
Defines a set relation from a set x
to potential members a
and b
.
Measures result repeatedly.
If input is true
, returns false
and vice versa.
Returns true
if at least one argument is true
.
An OR relation of 3 arguments.
An OR relation of 4 arguments.
An OR relation of 5 arguments.
An OR relation of 6 arguments.
An OR relation of 7 arguments.
An OR relation of 8 arguments.
An OR relation of 9 arguments.
An OR relation of 10 arguments.
An OR relation of variable number of arguments.
Path Semantical Logic: Counts the number of solutions of a 1-argument boolean function,
Path Semantical Logic: Counts the number of solutions of a 2-argument boolean function,
Path Semantical Logic: Counts the number of solutions of a 3-argument boolean function,
Path Semantical Logic: Counts the number of solutions of a 4-argument boolean function,
Path Semantical Logic: Counts the number of solutions of a 5-argument boolean function,
Path Semantical Logic: Counts the number of solutions of a 6-argument boolean function,
Path Semantical Logic: Counts the number of solutions of a 7-argument boolean function,
Path Semantical Logic: Counts the number of solutions of a 8-argument boolean function,
Path Semantical Logic: Counts the number of solutions of a 9-argument boolean function,
Path Semantical Logic: Counts the number of solutions of a 10-argument boolean function,
Path Semantical Logic: Counts the number of solutions of a n-argument boolean function,
Path Semantical Logic: Counts the number of solutions of a n+m-argument boolean function,
Path Semantical Logic: Computes number of cases.
Path Semantical Logic: Returns true
if proposition is correct, false
otherwise.
Path Semantical Logic: Returns true
if proposition is correct, false
otherwise.
Path Semantical Logic: Returns true
if proposition is correct, false
otherwise.
Path Semantical Logic: Returns true
if proposition is correct, false
otherwise.
Path Semantical Logic: Returns true
if proposition is correct, false
otherwise.
Path Semantical Logic: Returns true
if proposition is correct, false
otherwise.
Path Semantical Logic: Returns true
if proposition is correct, false
otherwise.
Path Semantical Logic: Returns true
if proposition is correct, false
otherwise.
Path Semantical Logic: Returns true
if proposition is correct, false
otherwise.
Path Semantical Logic: Returns true
if proposition is correct, false
otherwise.
Path Semantical Logic: Returns true
if proposition is correct, false
otherwise.
Path Semantical Logic: Returns true
if proposition is correct, false
otherwise.
Returns T
if a
is true
, F
otherwise.
In logical terminology this corresponds to a proposition.
Returns true
if proposition is correct, false
otherwise.
Returns true
if proposition is correct, false
otherwise.
Returns true
if proposition is correct, false
otherwise.
Returns true
if proposition is correct, false
otherwise.
Returns true
if proposition is correct, false
otherwise.
Returns true
if proposition is correct, false
otherwise.
Returns true
if proposition is correct, false
otherwise.
Returns true
if proposition is correct, false
otherwise.
Returns true
if proposition is correct, false
otherwise.
Returns true
if proposition is correct, false
otherwise.
Returns true
if proposition is correct, false
otherwise.
Assumes the path semantical acore axiom.
Assumes the path semantical core axiom.
Path semantical quality a ~~ b
.
Prepares a qubit using a proposition as seed.
Ignores argument, always returning true
.
Ignores both arguments, returning true
for all inputs.
Ignores all 3 arguments, returning true
for all inputs.
Ignores all 4 arguments, returning true
for all inputs.
Ignores all 5 arguments, returning true
for all inputs.
Ignores all 6 arguments, returning true
for all inputs.
Ignores all 7 arguments, returning true
for all inputs.
Ignores all 8 arguments, returning true
for all inputs.
Ignores all 9 arguments, returning true
for all inputs.
Ignores all 10 arguments, returning true
for all inputs.
Assumes univalence axiom for some homotopy level.
Returns true
if only one argument is true
.
An XOR relation of 3 arguments.
An XOR relation of 4 arguments.
An XOR relation of 5 arguments.
An XOR relation of 6 arguments.
An XOR relation of 7 arguments.
An XOR relation of 8 arguments.
An XOR relation of 9 arguments.
An XOR relation of 10 arguments.
An XOR relation of variable number of arguments.