Logictools

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What is this?

A toolkit for proving first order predicate logic formulas, answering questions and converting between different syntaxes for logic.

Think about logic as a generic way of writing rules: proving formulas and answering questions is a way of detecting what follows from the rules we wrote.

The toolkit is built around a high-performance reasoner gkc (see the paper) and runs in the browser using Wasm without any server involvement. For more advanced use you may want to download the command-line version.

Use it for learning logic and experimenting with a reasoner.

Contents:

Basic syntax
What does it mean?
Special symbols and additional constructions
Automated reasoning
What is the practical use?

Basic syntax

A good way to learn syntax and meaning is to go through the explanations in the simple examples. Nevertheless, the following presents a compact description.

Gkc supports several syntaxes: the simple syntax used in most examples, TPTP syntax and JSON-LD-LOGIC syntax: see the separate Json menu for the examples and explanations.

The simple syntax is an extension of the TPTP FOF and CNF syntaxes as described in the TPTP technical manual, combining them into one and extending with a list type and a few convenience predicates and constructions.

The simple syntax input is a list of logical formulas, each terminated with a period. The whole list is interpreted as a big conjunction (and) of formulas.

Each formula is either a literal (a positive or a negative atom) or is constructed from simpler formulas using logical operators like

& for conjunction (and),
| for disjunction (or),
- or ~ for negation (not),
=> for implication (if-then),
"for all values of variables holds ..." construction ! [variables] : formula (example ! [X,Y] : p(X,Y))
"exist values of variables so that ..." construction ? [variables] : formula.

Additional logical operators will be described later.

An atom is a predicate symbol applied to a number of term arguments like father(john,pete) and denoting a relation between the arguments. A negated atom has a minus sign prepended like -father(john,pete).

Equality is a special predicate written like gender(john)=male and meaning, as expected, that the term on the left side is equal to the term on the right side. Inequality is another special predicate written like gender(john)!=female

The terms denote objects. A simplest term is just a symbol/constant like john: think of it as a post-it note glued to some real thing or idea: a real object may have many such notes glued to it. A variable like X is also a term: think of it as placeholder which can be replaced by any other term.

Variables must start with an upper-case character, other symbols with a lower-case character. There are also other options for symbols, more about these below.

A variable which is not in the scope of a for all ! or exists quantifier ? is treated as if it is bound by the for all quantifier. In fact, explicit quantifiers can be skipped altogether: they are just a convenience construction.

Numbers like 12 or 3.45 can be also used as constant terms.

More complex terms can be constructed from simpler terms using functions like age(mother(john)) or $sum(X,$multiply(2,X)). A value of such a function is not always defined: it can be used just as a constructor for complex terms. A function may also be defined on some other terms, like $sum is defined on numbers, but not on lists. You may define functions yourself using equality like mother(john)=eve or age(mother(john))=age(mother(pete)).

Both line comments % and multi-line comments /* ... */ are available.

What does it mean?

Logic can be seen as a fairly general way of writing down facts and rules, while proving the formulas means detecting what follows from the rules.

Think about an atom like father(john,pete) as saying that there is a row with two cells john,pete in a two-column table father. In fact, the core ideas of relational database systems and SQL stem from this interpretation, although they branch off into specific directions. For example, in conventional logic notation all the elements in row columns are given positionally, one after another, while in SQL columns have names.

Each predicate like father should be seen as giving rise to two different tables: one for positive facts and another for negative facts: -father(john,mike) means that in the negative -father table there is a row john,mike. Importantly, the facts we state only say that there are these-and-these rows, but they do not say that there cannot be other rows, unknown to us. This inpreretation is called open world assumption contrasting the closed world assumption used in database systems and Prolog.

Logical formulas are then understood as rules for deriving new rows in the tables. For example. father(X,Y) => child(Y,X) means that if there is a row in the father table, the corresponding row with arguments exchanged must be put into the child table.

A rule like -father(X,X) can be seen as saying that the negative -father table contains rows like object_n,object_n for all objects. Since we could potentially have infinitely many objects (think numbers and terms constructed from functions) we cannot always physically fill in all the rows.

A contradiction occurs when a row in the positive table also occurs in the corresponding negative table. For example, saying father(john,john) would contradict the row john,john in the negative -father table, given the rule -father(X,X).

Finding a contradiction is used as one convenient way of finding whether some fact/row follows from our rules. Suppose we have two statements father(john,pete) and father(X,Y) => child(Y,X) and we want to prove an obvious fact that child(pete,john) follows from these two. Let us denote these three statements as a, b, c. Then we really want to prove that a & b => c is true, which is equivalent to the question of whether the negated form -(a & b => c) is contradictory, which is equivalent to the question of whether the conjunction a & b & -c is contradictory. Hence we could add the negated form -child(pete,john) to our two initial statements and check whether a contradiction can be found.

Finally, an important principle of logic as used here is that no common knowledge - or rather, no knowledge at all - is assumed. Everything we want to use must be written explicitly as logical facts and rules. The names of the predicates mean nothing in itself: a reasoner knows nothing about the father predicate except what is explicitly written in the problem statement. You could replace all the nice human-understandable names with meaningless names and the logical meaning of the problem statement would be exactly the same.

Special symbols and additional constructions

Ordinary symbols must start with lower-case character and must not contain any whitespace or non-alphanumeric characters like (, -, ~, =, " etc: however, underscore _ and dollar $ are allowed.

You can put whitespace or any symbol except a single quote into symbols by surrounding the symbol with single quote symbols like this: 'John Smith'. Internal quotes must be prefixed by a backslash \.

Additionally you can make a symbol variable by prefixing it with a question mark like this: ?smth.

Double quotes surrounding a symbol like "John Smith" have a special meaning: the symbol is treated as distinct (essentially, a string) meaning that it is known to be unequal to any syntactically different distinct symbols, numbers and lists. This has real effect for equality and string predicates, but not otherwise.

In addition to the typically used constructions &,|, -/~, =>, !, ? the syntax provides a number of additional infix logical constructions with two arguments. The full list is:

| for disjunction,
& for conjunction,
<=> for equivalence,
=> for implication,
<= for reverse implication,
<~> for non-equivalence (XOR),
~| for negated disjunction (NOR),
~& for negated conjunction (NAND),
@ for application, used mainly in the higher-order context in TPTP.

There are several predicates and function symbols with a pre-defined meaning: all these are prefixed with a dollar $.

$ans is used for finding the actual terms we look for as answers: see the answers example.

Arithmetic

The numbers and arithmetic functions and predicates are defined following the TPTP arithmetic system plus a few convenience operators for writing infix terms:

Type detection predicates $is_int, $is_real.
Comparison predicates $less, $lesseq, $greater, $greatereq.
Type conversion functions $to_int, $to_real.
Arithmetic functions on integers and reals: $sum, $difference, $product, $quotient, $quotient_e, $remainder_e, $remainder_t, $remainder_f, $floor, $ceiling, $uminus, $truncate, $round.
Note: these comparison predicates and arithmetic functions take exactly two arguments.
Example: $less($sum(1,$to_int(2.1)),$product(3,3)).
Additional convenience predicate is used: $is_number is true if and only if $is_int or $is_real is true.
Additional infix convenience functions +, -, *, / are used with the same meaning as $sum, $difference, $product and $quotient, respectively.
Example: $less(1+(1+2),(3*3))
Note: these convenience functions take also exactly two arguments.

NB! Do not use a variable or a non-numeric constant as a first element of the infix arithmetic expression like p(X*2), otherwise the whole expression will be parsed as a single variable X*2. No such restrictions apply for the prefix form.

Lists

You can use the list syntax like [] for the empty list and [a,b,c] for a three-element list. The bracket notation is syntactic sugar for the constant $nil and functional list-constructing term $list(a,$list(b,$list(c,$nil))), respectively.

Observe that $list(X0,X1) constructs a list by prepending X0 to the list X1, which is generally different from a two-element list [X0,X1].

Terms constructed using $list or $nil are interpreted as having a list type:

A list type object is inequal to any number or a distinct symbol.
Syntactically different list type objects A and B are unequal if at any position the corresponding elements of A and B are unequal typed values: numbers, lists or distinct symbols.

The following predicate and two functions are defined on lists:

$is_list(L) evaluates to true if A is a list and false is A is a number or a distinct symbol.
$first(L) returns the first element of the list.
$rest(L) returns the rest of the list, i.e. the result of removing the first element.

These functions can be applied to non-list arguments, where they are left as is and not evaluated.

Distinct symbols as strings

Symbols in double quotes like in "person" stand for for distinct symbols which can be viewed as a string type. A distinct symbol is not equal to any other syntactically different symbol and not equal to any numbers or lists.

Gkc defines a function and three predicates on distinct symbols:

$strlen(S) returns the integer length of a distinct symbol S as a string.
$substr(A,B) evaluates to true if a distinct symbol A is a substring of a distinct symbol B, and false otherwise, provided that A and B are distinct symbols.
$substrat(A,B,C) evaluates to true if a distinct symbol A is a substring of a distinct symbol B exactly at the integer position C (starting from 0), and false otherwise, provided that A and B are distinct symbols and C is an integer.
$is_distinct(A) evaluates to true if A is a distinct symbol and false if A is a number or a list.

Automated reasoning

Automated reasoners prove logical theorems, answer questions and (sometimes) can detect unprovability. Basically, they are highly optimized search systems often encompassing a large number of different search strategies.

Most of the reasoners are specialized to perform well on some specific kinds of problems or are extended with additional special capabilities. For example, the gkc prover used in this site is specialized for very large problems, although this capability cannot be used well in the confined browser environment. Hint: you may want to download the command-line version for more advanced use.

A large special category of provers focuses on propositional logic. See the Propositional logic menu for checking out their core ideas. The easiest way to find top level propositional solvers is to check the The international SAT Competition: you will see the competition results for various problem categories, can download competition problems, source code and descriptions of the provers. A good alternative choice is to check out minisat: an easily extensible and hackable state of the art solver which is a basis for several other state of the art solvers. You can run crypto-minisat in your browser, compiled from C to javascript using LLVM.

Good reading material for predicate logic provers:

Automated Theorem Proving Course Content by Geoff Sutcliffe is probably the best introductory material
Implementation tutorial by Stephan Schulz
A detailed theoretical coursebook by Marek Kosta and Christoph Weidenbach
Handbook of Practical Logic and Automated Reasoning contains code examples in ML in addition to an excellent text
Handbook of Automated_Reasoning for in-depth papers (here on Amazon).

For experimenting with general-purpose predicate logic solvers check out:

The TPTP Problem Library for Automated Theorem Proving by Geoff Sutcliffe and Christian Suttner.
Online solvers on TPTP
The CADE ATP System Competition
PyRes A well-documented simple implementation in Python for illustrating the basic machinery.

Several specialized categories of predicate logic solvers exist in addition to the general-purpose solvers mentioned above. Some of the categories are:

Interactive theorem provers like Lean, HOL, Mizar, coq.
SMT (satisfiability modulo theories) solvers used for formal verification like Z3 and PVS
Answer set based solvers like DLV and Smodels, see the competition
Description logic solvers, see the list of implementations
OWL (web ontology language) solvers used in the context of semantic web, see the list of implementations

What is the practical use?

For the generic reasoners, mostly niche applications so far. The eighties-hype of expert systems and symbolic A.I. with little practical value lead to the famous A.I. winter. This said, the propositional and SMT provers are seriously used for verification, test generation and configuration management.

The main obstacles for wider use: (a) it is very hard, or practically impossible (although theoretically possible) to hand-encode most of human expert knowledge in logical rules, (b) proof search without intelligent guidance is really hard, (c) basic predicate logic without probabilities, default reasoning etc etc is extremely impractical for encoding commonsense knowledge.

There is hope for a new rise of reasoners as parts of a machine-learning-and-reasoning hybrid systems performing common sense reasoning. Check out a preprint proposing several mechanisms in this direction.

Solve a predicate logic reasoning task:

Result:

What is this?

Basic syntax

What does it mean?

Special symbols and additional constructions

Arithmetic

Lists

Distinct symbols as strings

Automated reasoning

What is the practical use?

Example 1 for basics

Example 2 for answers

Example 3 for rules

Example 4 for indefinite answers

Example 5 for more rules

Example 6 for equalities and functions

Example 7 for equality

Example 8 for multiple answers

Example algebra

Example with an unprovable problem

Example with a hard problem

Blocks world

Example 13 for various characters

Example 14 for TPTP syntax with steam

Steam using simple formulas

Example 16 for the strategy

Example 17 for arithmetic

Example 18 for lists

Example 19 for distinct symbols

Example 20 from natural language

Example 21 from organization theory

Example 22 from boolean algebra

Example 23 from medicine

Example 24: Dreadbury Mansion

Example 25 from group theory

Example 26 from set theory

Solve a predicate logic reasoning task: help

Result:

What is this?

Basic syntax

What does it mean?

Special symbols and additional constructions

Arithmetic

Lists

Distinct symbols as strings

Automated reasoning

What is the practical use?

Example 1 for basics

Example 2 for answers

Example 3 for rules

Example 4 for indefinite answers

Example 5 for more rules

Example 6 for equalities and functions

Example 7 for equality

Example 8 for multiple answers

Example algebra

Example with an unprovable problem

Example with a hard problem

Blocks world

Example 13 for various characters

Example 14 for TPTP syntax with steam

Steam using simple formulas

Example 16 for the strategy

Example 17 for arithmetic

Example 18 for lists

Example 19 for distinct symbols

Example 20 from natural language

Example 21 from organization theory

Example 22 from boolean algebra

Example 23 from medicine

Example 24: Dreadbury Mansion

Example 25 from group theory

Example 26 from set theory

Solve a predicate logic reasoning task: