OTTER 2017-12-03

OTTER(Organized Techniques

for Theorem-proving and Effective Research)

Heuristics computing 2017/12/03 9:25 – 11:00

Keio University, SFC

■Organized Techniques for Theorem-proving and Effective Research ■Otter is designed to prove theorems stated in first-order logic with equality

OTTER

Install# apt-get install otter# wget http://www.mcs.anl.gov/research/projects/AR/otter/dist33/otter-3.3f.tar.gz# tar zxvf otter-3.3f.tar.gz; cd otter-3.3f; ./configure; make; make install

Argonne National Laboratory the largest national laboratory by size and scope in the Midwest. Argonne was initially formed to carry out Enrico Fermi's work on nuclear reactors as part of the Manhattan Project.

Larry Wos is a mathematician, a researcher in the Mathematics and Computer Science Division of Argonne National Laboratories.

Prolog(1972)

C(1973)

LISP(1953)

1950 200019801970 1990 2010

JAVA(1995)

Ocaml(04)

ML(1973)Type Inference

Isabelle(1986)OTTER(96)

Haskell(1986)

gcc(1985)

Google MapReduce(04)

scala(03)

Coq(1984) Proverif(2003)

Python(1991)

SQL86(86)

Shift-Reduce

Imperative

Proof Assistant

Higher order

Automated reasoning

lamda

Unification

Compiler

Database

Mainframe Object orientedProgramming Data mining BigData(AI2)Algorithm(A1)

PopField(1982)

BoltzmanMachine(85)

BackProp(1986)

SOM(81) SVM(91)Deep Learning(2012)

B: Imperative

Thermodynamics(熱力学)

Proof Assistant Kernel

Autoencoder

Boosting(04)

2015

GraphComputation

Pig(09)

pthread

Redis(09)

multicore

Proof Assistantwith heuristics

SledgeHummer(03)

GPU

A: declarative

C: reasoning

D: heuristics

Deep learning

λ calculus

turing machine

π calculus

optimization

resolution

propagation

recursion

OTTER and Four computing paradigmsWorld war II

Natural dedution

Puzzles and programs

Truth tellers and liars

Tower of Hanoi

Two inverter puzzle

Ferrying problem15 puzzle

N-coins balance puzzle

#apt-get install otter git#git clone https://github.com/RuoAndo/otter-book

Self-replicatingrobots

Tarski parallelaxiom

StrategiesRestriction Strategies

Set of Support

Weighting

Direction Strategies

Ratio

Resonance

Look-ahead Strategies

Hot List

Redundancy control

Subsumption

DemodulationParamodulation

Dynamic Hot List

Subtautology

Resonance restriction

Level SaturationRecursive Tail

Hints

First-in Last-outProof checking filter

http://www.mcs.anl.gov/research/projects/AR/strategies.html

Set of Support

Imperative

Declarative

Automated reasoning

Loop

Lambda / Monad

Primitive recursive function ?

State Machine

Recursion

TAPE

CLOCK

ATOM

LIST

RECURSION

HEATRESOANCE

Leibniz

Spinoza

?

Example 1: Truth tellers and liars

http://jinavie.tumblr.com/post/38901128124/truth-and-lies#.WiOmGEpl91s

On the fabled Island of Knights and Knaves, we meet two people, A and B, one of whom is a knight and one a knave. The knight always tells the truth, the knave always lies.A says: "C is a knave."B says: "A is a knight."

Who is the knight, who the knave, and who the spy?

%assign(max_proofs,-1).set(hyper_res).set(print_lists_at_end).assign(stats_level,0).

list(usable).-P(T(x)) | -P(Says(x,y)) | P(y).-P(L(x)) | -P(Says(x,y)) | -P(y).P(T(x)) | P(L(x)).-P(T(x)) | -P(L(x)).end_of_list.

list(sos).P(Says(A, L(A))).end_of_list.

https://github.com/RuoAndo/otter-book/tree/master/takefuji-lab/2017

configuration rule fact

Usable Set+

+++

+-

-

lightest

+

-

EMPTY

Usable Set

SoS Set

SoS Set

--

prg2-4.out.lined-----> EMPTY CLAUSE at 0.00 sec --> 8 [hyper,7,4,6] $F.

Size of Usable set (nonliear)

Main Loop

A

B

C

C ⊃ BB ⊃ A

There exists A[i] where B∪￢A → Ф (empty)

A’

Clause Sets and representation

SoS (Set of Support)

Demodulators

Usable

Passive

Term Rewriting

Resolutionand paramodulation

conflict

a b

a b c

a b

c

IF A THEN B : -A | B

IF A & B THEN B : -A | -B | C

IF A THEN B or C : -A | B | C

Example 2: Truth tellers, liars and spyOn the fabled Island of Knights and Knaves, we meet three people, A, B, and C, one of whom is a knight, one a knave, and one a spy. The knight always tells the truth, the knave always lies, and the spy can either lie or tell the truth.A says: "C is a knave."B says: "A is a knight."C says: "I am the spy."Who is the knight, who the knave, and who the spy?

set(hyper_res).list(usable).

-P(T(x)) | -P(Says(x,y)) | P(y).-P(L(x)) | -P(Says(x,y)) | -P(y).P(T(x)) | P(L(x)) | P(N(x)).-P(T(x)) | -P(L(x)).-P(T(x)) | -P(N(x)).-P(L(x)) | -P(N(x)).

-P(T(x)) | P(L(x)) | P(N(x)).-P(L(x)) | P(T(x)) | P(N(x)).

end_of_list.

list(sos).P(Says(A,L(C))).P(Says(B,T(A))).P(Says(C,N(C))).

end_of_list.

Always true (恒真）

Always false（恒偽）

Satisfiable（証明可能） Unsatisfiable（証明不可能）

私は正直者だと言っているスパイ

私は嘘つきだと言っているスパイ

https://github.com/RuoAndo/otter-book/tree/master/takefuji-lab/2017

Generalization: consistency is non-deterministic

(∃y) (x) ～ Dem (x, y) ⊃ (x) ～ Dem(x, sub(n, A, n))

Dem(x,y) : yの言っていることをxが確かめる。Island is consistent if there are only truth tellers and liars.

島に正直者と嘘つきしかいない場合、確かめる事のできない言明がある。その言明とは、「私は嘘好きである」というものである。

(∃y) (x) ～ Dem (x, y)

Dem(x, sub(n,A,n)): 自分について言っていること(sub(n,A,n)をxが確かめることができる。If the island is consistent, no one in the island know if there are some spy.

「私は嘘つきである」という言明を確かめることができない場合、誰も島にスパイがいることを発見できない。

しかし、OTTERは発見できる。

15 puzzle

set(para_into).

list(usable).EQUAL(l(hole,l(n(x),y)),l(n(x),l(hole,y))).EQUAL(l(hole,l(x,l(y,l(z,l(u,l(n(w),v)))))),l(n(w), l(x,l(y,l(z,l(u,l(hole,v))))))).

-STATE(l(n(1),l(n(2),l(n(3),l(n(4),l(end,l(n(5), l(n(6),l(n(7),l(n(8),l(end,l(n(9),l(n(10),l(n(11), l(n(12),l(end,l(n(13),l(n(14),l(n(15), l(hole,end)))))))))))))))))))).end_of_list.

list(sos).STATE(l(n(1),l(n(6),l(n(2),l(n(4),l(end,l(n(5), l(hole,l(n(3),l(n(8),l(end,l(n(9),l(n(10),l(n(7), l(n(11),l(end,l(n(13),l(n(14),l(n(15), l(n(12),end)))))))))))))))))))).end_of_list.

Complexity of 15 puzzle

1 6 2 45 3 89 10 7 1113 14 15 12

Prg4-8.in : SCORE 437

1 6 2 45 3 89 10 7 1113 14 15 12

Prg4-8-7.in : SCORE 891

11 2 6 94 14 3 51 12 7 1513 10 6

Prg4-8-2.in : SCORE ????

https://github.com/RuoAndo/otter-book/tree/master/15puzzle# cd otter-book# git pull

Paramodulation Usable List: list(usable)

SoS List: list(list)

OTTER 2017-12-03

Education

Transcript of OTTER 2017-12-03