Sequential method

A prover of logical formulas. Core of application is Sequential Method algorithm.

How it works

Application is divided in several independent modules:

Parser parse input string into AST.
1. Tokenizer split input expression into lexemes.
2. Infix-to-Postfix converter converts infix list of lexemes to RPN(Reverse Polish Notation).
3. Abstract Syntax Tree generator converts list of tokens(token = lexeme + additional information) to tree.
Sequential method logic of application.
Wrapper wrap results of Sequential method to unique data structure for mapping to JSON.

Now let's move to the heart of application.

Algorithm

Here are description of one algorithm iteration.

If all leaves are closed, finish with positive verdict.
If all leaves are atomic, finish with negative verdict.
For each non-atomic, non-closed leaf
- Expand leaf.
- Simplify result leaves.
Goto step 1

There is situation when algorithm will go to INFINITE LOOP. In this case we can use König's infinity lemma to finish with negative result.

Sequential formulas

mark	notation	branches
₊!	₊!A, Σ	_-A, Σ
_-!	_-!A, Σ	₊A, Σ
₊v	₊(A v B), Σ	₊A, Σ	₊B, Σ
_-v	_-(A v B), Σ	_-A, _-B, Σ
₊&	₊(A & B), Σ	₊A, ₊B, Σ
_-&	_-(A & B), Σ	_-A, Σ	_-B, Σ
₊->	₊(A -> B), Σ	_-A, Σ	₊B, Σ
_-->	_-(A -> B), Σ	₊A, _-B, Σ
₊∃	₊∃xA, Σ	₊A_x[y], Σ
_-∃	_-∃xA, Σ	_-A_x[z₁], ..., _-A_x[z_m], Σ, _-∃xA
₊∀	₊∀xA, Σ	₊A_x[z₁], ..., ₊A_x[z_m], Σ, ₊∀xA
_-∀	_-∀xA, Σ	_-A_x[y], Σ

Where

y denotes new unique name.
z₁, ..., z_m denotes names used in current sequence.
A_x[y] denotes renomination of predicate from x to y.

Let's look at examples

Example 1

First example demonstrates how exactly sequences is expanding when implication and disjunction are in charge.

  P[x] = P[x] | Q[x]
          |
          v
-(P[x] -> P[x] | Q[x])
          |
          v
+P[x], -(P[x] | Q[x])
          |
          v
 -P[x], -Q[x], +P[x]
          |
          v
          X

As you can see sequential tree has only one branch, and it's closed. According to algorithm above, expression is truthful. It's quite simple example, so let's move to something more interesting.

Example 2

Here is more complicated example with quantifiers. It's also truthful, but now we have two different closed branches.

                       #xP[x] -> Q[x] = P[x] -> #xQ[x]
                                      |
                                      v
                    -((#xP[x] -> Q[x]) -> P[x] -> #xQ[x])
                                      |
                                      v
                    +(#xP[x] -> Q[x]), -(P[x] -> #xQ[x])
                   /                                   \
                  /                                     \
                 v                                       v
    -#xP[x], -(P[x] -> #xQ[x])                 +Q[x], -(P[x] -> #xQ[x])
                |                                         |
                v                                         v
-P[x], -(P[x] -> #xQ[x]), -#xP[x]                +P[x], -#xQ[x], +Q[x]
                |                                         |
                v                                         v
 +P[x], -#xQ[x], -P[x], -#xP[x]               -Q[x], +P[x], +Q[x], -#xQ[x]
                |                                         |
                v                                         v
                X                                         X

As you can see all two branches closed at the same time. So expression is truthful and no counter examples exists.

Example 3

This example show you how to deal with unclosed sequences, and how to get a counter example.

                 P[x] -> #xQ[x] = #xP[x] -> Q[x]
                                |
                                v
               -(P[x] -> #xQ[x] -> #xP[x] -> Q[x])
                                |
                                v
               +(P[x] -> #xQ[x]), -(#xP[x] -> Q[x])
              /                                   \
             /                                     \
            v                                       v
-P[x], -(#xP[x] -> Q[x])                 +#xQ[x], -(#xP[x] -> Q[x])
           |                                         |
           v                                         v
 +#xP[x], -Q[x], -P[x]                    +Q[z], -(#xP[x] -> Q[x])
           |                                         |
           v                                         v
  +P[y], -Q[x], -P[x]                      +#xP[x], -Q[x], +Q[z]
                                                     |
                                                     v
                                            +P[w], -Q[x], +Q[z]

Left branch

name	delta	values
	x -> a,	P[a] := False,
A	y -> b	P[b] := True,
		Q[a] := False

Let's try to build interpretation on this counter example. There are many different options, but we can stop on:

x := integer
P[x] := x == 1
Q[x] := x != x                // always false

Right branch

name	delta	values
	x -> a,	Q[a] := False,
B	z -> b,	Q[b] := True,
	w -> c	P[c] := True

In this case we can use next interpretation:

x := integer
P[x] := (x + x) % 2 == 0      // always true
Q[x] := x % 2 == 1

Summary

We have got two unclosed branches. Each one produces unique counter example. Each counter example can give us many different interpretations. We have already built some of them. You can manually check these interpretations, to ensure that expression is false. Just evaluate input expression with these interpretations.

Let's stop at this example and move forward to syntax overview.

Language specification

This is language grammar in Backus-Naur-Form.

<expression>                ::= <formula> "=" <formula>
<formula>                   ::= [ "(" ]
                                <predicate>
                              | <logical-operation>
                              | <quantifier>
                                [ ")" ]
<predicate>                 ::= <predicate-name> "[" <variable-arguments> "]"
<predicate-name>            ::= <capital-letter> { <letter> }
<predicate-arguments>       ::= <variable-name> { "," <variable-name> }
<variable-name>             ::= <letter> { <letter> }

<logical-operation>         ::= <unary-operation>
                              | <binary-operation>
<binary-operation>          ::= <formula> <binary-operation-keyword> <formula>
<binary-operation-keyword>  ::= "&"
                              | "|"
                              | "->"
<unary-operation>           ::= "!" <formula>

<quantifier>                ::= <quantifier-keyword> <predicate-name> <formula>
<quantifier-keyword>        ::= "#"        Stands for "exists"
                              | "@"        Stands for "for all"

<letter>                    ::= "a"
                              | "b"
                              | ...
                              | "z"
<capital-letter>            ::= "A"
                              | "B"
                              | ...
                              | "Z"

If your expression does not fit this grammar then application will rise an parser exception and inform you with error field of response.

Valid expressions

Some valid examples of expressions:

P[x] = Q[x]
P[x] = P[x] | Q[x]
#xP[x] -> Q[x] = P[x] -> #xQ[x]
#x@yP[x, y] = @y#xP[x, y]
P[x] -> #xQ[x] = #xP[x] -> Q[x]
P[x] | Q[x] | R[x] | T[x] | S[x] = @xP[x]
@xP[x] = Q[x]

API

There are only one service available, named check. To use it your query should contains field named expr with expression you want to check. Both get and post methods are supported.

API Usage

Simple request check?expr=P[x]=P[x]|Q[x] will give you response listed below.

{
   "tree":{
      "root":{
         "formulas":[
            {
               "formula":"P[x] -> Q[x]",
               "value":false
            }
         ],
         "children":[
            {
               "formulas":[
                  {
                     "formula":"P[x]",
                     "value":true
                  },
                  {
                     "formula":"Q[x]",
                     "value":false
                  }
               ],
               "children":[],
               "closed":false
            }
         ],
         "closed":false
      }
   },
   "verity":false,
   "examples":[
      {
         "name":"E",
         "delta":{
            "x":"m"
         },
         "example":{
            "P":{
               "[m]":true
            },
            "Q":{
               "[m]":false
            }
         }
      }
   ],
   "error":null
}

Now let's look closer to response structure.

Response structure

	name	type	description
response	tree	{root: node}	sequential tree with root node
	verity	Boolean	null, if infinite loop
	examples	[example]	list of examples
	error	string	error message from server
node	formulas	[formula]	list of formulas
	children	[node]	list of child nodes
	closed	boolean	is sequence closed
formula	formula	string	actual formula
formula	value	boolean	logical value
example	name	string	name of example
	delta	{x -> a}	map of new variable names
	example	{P[x] -> val}	map of predicates/variables/values

Installation

Requirements

Java Runtime Environment with Java SE8 support
Maven 3 to build

Building

You can build app using Maven with ease. Just type mvn clean package. This will generate JAR file target/sequential-method-1.0-SNAPSHOT.jar.

Running

If you are using Maven, you can run application without building using mvn spring-boot:run.

Or if you have pre-built JAR you can run it by typing:

java -jar target/sequential-method-1.0-SNAPSHOT.jar

Docker

You can run this application using prebuild docker image like this docker run -it -p 8080:8080 lionell/math-logic. Alternatively you can use Docker Compose docker-compose up -d. This will start service locally and expose port 8080.

Used materials

Here is a list of materials used in app:

D3.js visualizing sequential tree
HTML5 UP design of main page
Spring framework
Maven app architecture
Docker isolated environment

Contributions

Here I want to say thanks to my friend Fetiorin who helped me with this app.

He did a great job on the client-side.

Licence

MIT

License

lionell/sequential-method

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