FormalSystems 0.1.3
Play with formal systems from "Gödel Escher Bach", Douglas Hofstadter
This is a Python implementation of Douglas Hofstadter formal systems, from his book Gödel, Escher, Bach: An Eternal Golden Braid (commonly GEB).
In fact, you may define your own formal systems using a quite simple syntax, close to free text. Examples for MIU, pg, fg and NDP formal systems from the book are implemented in directory definitions.
A main Python script gives you possibilities to play with the formal system, including:
 axiom schema support (iteration, decision procedure)
 theorem step by step generation (using different algorithms)
 theorem derivation
Formal system definition
Examples
The MIU system may be define with:
axioms:  MI rules:  x is .*, xI => xIU  x is .*, Mx => Mxx  x is .*, y is .*, xIIIy => xUy  x y .* , xUUy => xy
The underlying syntax is YAML (see raw format). You can define one or several axioms, or even an infinite number of axioms using a schema, as in the pg formal system:
axioms:  x is +, xpgx rules:  x y z are +, xpygz => xpygz
Syntax
Axiom definitions should be formatted like this ([] means this is optional):
[def_1, [def_2, ...]] expr
Where:
 def_i is an optional definition of wildcard, using a regular expression, for example:
 “.*” may be anything including the empty string
 “+” is a string composed of ““
The definitions are written using char [is] regexp or char1 char2 [are] regexp if different wildcards have the same definition. Note that you should use only one character for wildcard definition.
 expr is the axiom expression
Rules for theorem production should be formatted like this:
[def_1, [def_2, ...]] cond_1 [and cond_2 [and ...]] => th_1 [and th_2 [and ...]]
Where:
 def_i is the same as before
 cond_i is a required theorem, in order to produce new theorems (separated by and if several conditions)
 th_i is a produced theorems with the rule
Installation
Install with:
$ python setup.py install user
A script should be put in ~/.local/bin, make sure this path is in your $PATH:
$ export PATH=$PATH:~/.local/bin
Tests
If installation is successful, run the tests with:
$ cd tests
$ python test_formalsystems.py v
Main script
After installation, you should have the main script FormalSystemsMain.py deployed somewhere where you $PATH points to, under the name FormalSystems. If it is not the case, you can always execute the script directly, assuming the dependencies are properly installed (just pyyaml and LEPL).
Usage of the main script is fully documented in help argument.
You may generate theorems step by step if the number of axioms is finite:
$ FormalSystems definitions/MIU.yaml iteration 3 > Finite number of axioms, using step algorithm STEP 1: MI P (1) x is .*, xI => xIU for MI gives MIU P (2) x is .*, Mx => Mxx for MI gives MII . (3) x is .*, y is .*, xIIIy => xUy for MI . (4) x y .* , xUUy => xy for MI STEP 2: MIU/MII P (1) x is .*, xI => xIU for MII gives MIIU . (1) x is .*, xI => xIU for MIU P (2) x is .*, Mx => Mxx for MII gives MIIII P (2) x is .*, Mx => Mxx for MIU gives MIUIU . (3) x is .*, y is .*, xIIIy => xUy for MII . (3) x is .*, y is .*, xIIIy => xUy for MIU . (4) x y .* , xUUy => xy for MII . (4) x y .* , xUUy => xy for MIU STEP 3: MIIU/MIIII/MIUIU
Or using a bucket where axioms are thrown and theorems computed iteratively if the number of axioms is infinite:
$ FormalSystems definitions/pg.yaml iteration 4 > Infinite number of axioms, using bucket algorithm [Adding pg to bucket] === BUCKET 1: pg P (1) x y z are +, xpygz => xpygz for pg gives pg [Adding pg to bucket] === BUCKET 2: pg/pg P (1) x y z are +, xpygz => xpygz for pg gives pg P (1) x y z are +, xpygz => xpygz for pg gives pg [Adding pg to bucket] === BUCKET 3: pg/pg/pg P (1) x y z are +, xpygz => xpygz for pg gives pg P (1) x y z are +, xpygz => xpygz for pg gives pg P (1) x y z are +, xpygz => xpygz for pg gives pg [Adding pg to bucket] === BUCKET 4: pg/pg/pg/pg
Options are available to display theorem derivation as well:
$ FormalSystems definitions/NDP.yaml quiet derivation P > Infinite number of axioms, using bucket algorithm > Rule with several parents, using recursivity === BUCKET 1: NDP === BUCKET 2: NDP/SD/P === BUCKET 3: NDP/SD/NDP === BUCKET 4: NDP/NDP/SD/P/NDP === BUCKET 5: NDP/NDP/NDP/SD/SD/SD/NDP === BUCKET 6: NDP/NDP/NDP/NDP/SD/NDP === BUCKET 7: NDP/NDP/NDP/NDP/NDP/SD/SD/SD/NDP === BUCKET 8: NDP/NDP/NDP/NDP/NDP/NDP/SD/SD/SD/SD/NDP === BUCKET 9: NDP/NDP/NDP/NDP/NDP/NDP/NDP/SD/SD/SD/SD/P/NDP === Theorem P found, derivation: [1 ] Axiom gives NDP [2 ] (1) x y are +, xNDPy => xNDPxy for NDP gives NDP [3 ] Axiom gives NDP [3 ] (1) x y are +, xNDPy => xNDPxy for NDP gives NDP [4 ] Axiom gives NDP [4 ] (1) x y are +, xNDPy => xNDPxy for NDP gives NDP [4 ] (2) z is +, NDPz => zSD for NDP gives SD [5 ] (1) x y are +, xNDPy => xNDPxy for NDP gives NDP [5 ] (3) x z are +, zSDx and xNDPz => zSDx for SD and NDP gives SD [6 ] (3) x z are +, zSDx and xNDPz => zSDx for SD and NDP gives SD [7 ] (4) z is +, zSDz => Pz for SD gives P
Python API
Some tests using doctests:
>>> from formalsystems.formalsystems import FormalSystem, Theorem
MIU formal system:
>>> fs = FormalSystem() >>> fs.read_formal_system('./definitions/MIU.yaml') >>> r = fs.apply_rules_step(fs.iterate_over_schema(), step=4, verbose=False) STEP 1: MI STEP 2: MIU/MII STEP 3: MIIU/MIIII/MIUIU STEP 4: MIIIIU/MIIIIIIII/MIIUIIU/MIUIUIUIU/MIU/MUI >>> print [str(a) for a in fs.iterate_over_schema()] ['MI']
pg formal system:
>>> fs = FormalSystem() >>> fs.read_formal_system('./definitions/pg.yaml') >>> r = fs.apply_rules_bucket_till(fs.iterate_over_schema(), max_turns=4, verbose=False) === BUCKET 1: pg === BUCKET 2: pg/pg === BUCKET 3: pg/pg/pg === BUCKET 4: pg/pg/pg/pg >>> r = fs.apply_rules_bucket_till(fs.iterate_over_schema(), min_len=9, verbose=False) === BUCKET 1: pg === BUCKET 2: pg/pg === BUCKET 3: pg/pg/pg
NDP formal system:
>>> fs = FormalSystem() >>> fs.read_formal_system('./definitions/NDP.yaml') >>> r = fs.apply_rules_bucket_till(fs.iterate_over_schema(), max_turns=2, full=True, verbose=False) === BUCKET 1: NDP === BUCKET 2: NDP/SD/P
Successful derivation:
>>> fs = FormalSystem() >>> fs.read_formal_system('./definitions/NDP.yaml') >>> r = fs.derivation_asc(fs.iterate_over_schema(), Theorem('P'), full=True, max_turns=10) <BLANKLINE> ... === Theorem P found, derivation: ...
Failed derivation:
>>> fs = FormalSystem() >>> fs.read_formal_system('./definitions/MIU.yaml') >>> r = fs.derivation_step(fs.iterate_over_schema(), Theorem('MIUIU'), step=5) <BLANKLINE> ... === Theorem MIUIU found, derivation: ... >>> r = fs.derivation_step(fs.iterate_over_schema(), Theorem('MU'), step=5) <BLANKLINE> ... === Theorem MU not found
File  Type  Py Version  Uploaded on  Size  

FormalSystems0.1.3.tar.gz (md5)  Source  20160705  14KB  
 Author: Alex Prengère
 Home Page: http://alexprengere.github.com/FormalSystems

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