TPTP Format for Derivations

Introduction

A derivation is a directed acyclic graph (DAG) whose leaf nodes are formulae from the input, whose interior nodes are formulae inferred from parent formulae, and whose root nodes are the final derived formulae. For example, a proof of a FOF theorem from some axioms, by refutation of the CNF of the axioms and negated conjecture, is a derivation whose leaf nodes are the FOF axioms and conjecture, whose internal nodes are formed from the process of clausification and then from inferences performed on the clauses, and whose root node is the false formula.

The information required to record a derivation is, minimally, the leaf formulae, and each inferred formula with references to its parent formulae. More detailed information that might be recorded and useful includes: the role of each formula; the name of the inference rule used in each inference step; sufficient details of each inference step to deterministically reproduce the inference; the semantic relationships of inferred formulae with respect to their parents.

Acceptable Derivations

The TPTP format requires certain features of a derivation so be deemed acceptable, e.g., as required by the CADE ATP System Competition (CASC). Required features are expressed as "must" in the documentation below, and marked ❗). Other features that are desirable are expressed as "should" in the documentation below. ATP systems might usefully offer a flag to indicate whether just an acceptable derivation is required, or a really nice derivation.

Proofs should not use a TPTP language that is more expressive than that of the problem.
Proofs must be syntactically correct in TPTP format.❗ This is checked using TPTP4X, available in SystemB4TSTP.
Proofs must be structurally correct:❗
- Annotated formulae must be uniquely named.
- Proofs must be acyclic.
- Proofs must show only relevant inference steps, i.e., in the DAG from leaves to a root.
- Proofs must have formulae from the problem as leaves, and end at the conjecture (for axiomatic proofs) or $false formulae (for proofs by contradiction, e.g., CNF refutations and closed tableau).
- Proofs that negate the conjecture must correctly annotate the step as status(cth) and have a a single parent with the role conjecture.
- Inference steps must list exactly those parents that are used in the inference.
- Proofs that make assumptions must propagate and eventually discharge the assumptions.
Proof structure is checked using GDV, available in SystemB4TSTP. To check only the structure of the derivation change GDV's "Command" to run_GDV %s 30 -d -u -l (i.e., add -d -u -l on the end).
Translations from one form to another, e.g., FOF to CNF, must be adequately documented.❗
Inference steps should not list parents that are not used in the inference, but can list parents that are not necessary for the inference. For example, in the inference of p(a) from the parents p(a) and a = a, the equality is not necessary but might be used in an (unnecessary) equality substitution step. In contrast, p(b) would probably not be used as a parent in that inference step. A genuine effort must be made to avoid listing unused and unnecessary parents.❗
Inference steps must be reasonably fine-grained. TO BE DECIDED. The maximal number of parents in any inference is considered in this regard.
New symbols introduced in derivations, e.g., to provide definitions for complex terms and formulae, in Skolemization and Herbrandization, in splitting, etc. must be documented in a new_symbols() record.❗ The new symbols listed in a new_symbols() record must be unique and globally new.❗
Proofs should graft in subproofs from external systems.
Steps with particular requirements
- Definitions that introduce a new symbol must be recorded like this.❗
- Tautologies that are introduced must be recorded like this.❗
- Theory axioms that are introduced and have to be accepted on faith, e.g., arithmetic axioms, must be recorded like this.❗
- Each Skolemization step of one existentially quantified variable must be performed and recorded in a separate annotated formula like this.❗

Specifying a TPTP Format Derivation

The top level building blocks of TPTP derivations are annotated formulae. An annotated formula has the form:

language(name,role,formula,source,useful info). A derivation is written as a list of annotated formulae.

Language: The languages currently supported are thf, tff, fof, and cnf. Proofs should not use a TPTP language that is more expressive than that of the problem.
Name: The name is a word starting lower case, e.g., original_f1, or a single quoted word, e.g., 'A crazy $ name', or a non-negative integer.
Role: The role is typically one of axiom for axiom leaves, definition for definition axiom leaves, conjecture for FOF conjecture leaves, negated_conjecture for negated conjectures, or plain for inferred formulae. A full list of the posssible roles is in the TPTP syntax.
Logical formula: The logical formula use the TPTP syntax.
Source: Each leaf formula has a file record or an introduced record, and each inferred formula has an inference record. Details below.
Useful information: The useful information is a []ed list of 'Prolog-like' terms.

The source and useful information are optional.

Sources:

File record: A file record contains the problem file name and the corresponding annotated formulae name in the problem file. Example:
```
     fof(john,axiom,
         human(john),
         file('CreatedEqual.p',john) ).
```
Introduced record: An introduced record contains the introduction rule name, a list of useful introduction information, and a list of any parent formulae. Example:
```
     fof(64,definition,
         ( sP1 <=> ! [X8] : ~ has_job(X8,boxer) ),
         introduced(definition,[new_symbols(definition,[sP1])],[])).
```
- Definitions that introduce a new symbol must be recorded like this.
- Tautologies that are introduced must be recorded like this.
- Theory axioms that are introduced and have to be accepted on faith, e.g., arithmetic axioms, must be recorded like this.
Inference record: An inference record contains the inference rule name, a list of useful inference information, and a list of its parent formulae. Parent formulae can be annotated formulae names and nested inference records. Example:
```
     cnf(c_0_20,negated_conjecture,
         sK0 = john,
         inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_16,c_0_17])]) ).
```
Common types of useful inference information are:
- The semantic relationship of the formula to its parents as an SZS ontology value in a status record. The most common ones are:
  - status(thm) when the inferred formula is a theorem (logical consequence) of its parents. This used for most inference rules.
  - status(cth) when the negation of the inferred formula is a theorem (logical consequence) of its parents. This used for, e.g., the step negating the conjecture for a proof by refutation.
  - status(esa) when the inferred formula is equisatisfiable with its single parent. This used for, e.g., Skolemization steps.
- Special information about recognized types of inferences, e.g., Skolemization.
- Details of new symbols introduced in the inference (when might this happen?), following the conventions for new symbol names.

Definitions

If a derivation defines a new symbol, the definition annotated formula must have the following form:

The annotated formula must have the definition role.

The formula must be binary, with an equality, equivalence, implication, or reverse implication connective.

The defined symbol must be on the left hand side of the connective, possibly under a negation.

Variable arguments of the new symbol must be universally quantified.

The source must be an introduced() record, with the type definition.

The useful information must include a new_symbols() record with the type definition, and the new symbol in the second argument list.

The parents of the introduced() record can be anything.

See the conventions for new symbol names.

(In problems definitions are axiom-like and do not need an introduced() record.)

Examples

tff(square,definition,
    ! [I: $int] :
      ( square(I) = $product(I,I) ),
    introduced(definition,[new_symbols(definition,[square])],[])).

fof(pugalist,definition,
    ( no_fighters <=> ! [X] : ~ has_job(X,boxer) ),
    introduced(definition,[new_symbols(definition,[no_fighters])],[])).

tff(bad_cat,definition,
    ! [C: cat] :
      ( ~ friendly(C) <=> ( horrible(C) & bad_pet(C) ) ),
    introduced(definition,[new_symbols(definition,[friendly])],[])).

Tautologies

If a derivation introduces a tautology, the annotated formula must have the following form:

The annotated formula must have the axiom role.
The source must be an introduced() record, with the type tautology.
The useful information and parents of the introduced() record can be anything.

Examples

tff(soliloquy,axiom,
    ! [X: alive] :
      ( to_be(X) | ~ to_be(X) ),
    introduced(tautology,[that_is_the_question],[hamlet])).

Theory Axioms

If a derivation introduces a theory axiom that has to be accepted on faith, e.g., arithmetic axioms, the annotated formula must have the following form:

The annotated formula must have the axiom role.
The source must be an introduced() record, with the type theory.
The useful information and parents of the introduced() record can be anything.

Examples

tff(square_root_2,axiom,
    ~ ? [X: $rat] :
        $product(X,X) = 2,
    introduced(theory,[],[hippasus_of_metapontum])).

Skolemization

Each Skolemization step of one existentially quantified variable should be performed and recorded in a separate annotated formula. So far two variants have emerged, each of which has particular output requirements.

Variables in the formula to be Skolemized must be uniquely (re)named.
Skolem symbols must be new, regardless of arity
New Skolem symbols must be recorded in a new_symbols() record, in the annotated formula where the new Skolem symbols first appear.
Skolem terms must have all universally quantified variables that are in scope as arguments.
The existentially quantified variables and the replacement Skolem terms must be recorded in skolemize() records, in the annotated formula where the existentially quantified variables are replaced.
If ε-terms are used, they must be defined before their use.

Plain example, single steps:

%----Starting formula
fof(marriage,plain,
    ! [Marriage] :
    ? [Bride] :
    ! [Parent] :
    ? [Groom] :
      ( in_love(Groom,Bride) 
     => give_hand_in_marriage(Parent,Bride,Groom,Marriage) ) ).

%----Skolemize Bride
fof(bride,plain,
    ! [Marriage] :
    ! [Parent] :
    ? [Groom] :
      ( in_love(Groom,sK0(Marriage))
     => give_hand_in_marriage(Parent,sK0(Marriage),Groom,Marriage) ),
    inference(skolemize,[status(esa),new_symbols(skolem,[sK0]),skolemize(Bride,sK0(Marriage))],[marriage]) ).

%----Skolemize Groom
fof(groom,plain,
    ! [Marriage] :
    ! [Parent] :
      ( in_love(sK1(Marriage,Parent),sK0(Marriage))
     => give_hand_in_marriage(Parent,sK0(Marriage),sK1(Marriage,Parent),Marriage) ),
    inference(skolemize,[status(esa),new_symbols(skolem,[sK1]),skolemize(Groom,sK1(Marriage,Parent))],[bride]) ).

"Hidden" existential quantification:
It is possible, but not nice, to record Skolemization steps with the existential quantification hidden, e.g., under a negated universal quantification.

fof(bride,plain,
    ! [Marriage] :
    ! [Parent] :
    ~ ! [Groom] :
        ~ ( in_love(Groom,sK0(Marriage))
         => give_hand_in_marriage(Parent,sK0(Marriage),Groom,Marriage) ) ).

%----Skolemize Groom
fof(groom,plain,
    ! [Marriage] :
    ! [Parent] :
      ~ ~ ( in_love(sK1(Marriage,Parent),sK0(Marriage))
         => give_hand_in_marriage(Parent,sK0(Marriage),sK1(Marriage,Parent),Marriage) ),
    inference(skolemize,[status(esa),new_symbols(skolem,[sK1]),skolemize(Groom,sK1(Marriage,Parent))],[bride]) ).

Plain example, multiple steps:
It is possible, but not nice, to record multiple Skolemization steps in one annotated formula.

%----Starting formula
fof(marriage,plain,
    ! [Marriage] :
    ? [Bride] :
    ! [Parent] :
    ? [Groom] :
      ( in_love(Groom,Bride) 
     => give_hand_in_marriage(Parent,Bride,Groom,Marriage) ) ).

%----Skolemize Bride and Groom
fof(groom,plain,
    ! [Marriage] :
    ! [Parent] :
      ( in_love(sK1(Marriage,Parent),sK0(Marriage))
     => give_hand_in_marriage(Parent,sK0(Marriage),sK1(Marriage,Parent),Marriage) ),
    inference(skolemize,[status(esa),new_symbols(skolem,[sk0,sK1]),skolemize(Bride,sK0(Marriage),skolemize(Groom,sK1(Marriage,Parent))],[marriage]) ).

Using ε-terms:
For people who justify Skolemization with ε-terms.

%----Starting formula
fof(marriage,plain,
    ! [Marriage] :
    ? [Bride] :
    ! [Parent] :
    ? [Groom] :
      ( in_love(Groom,Bride) 
     => give_hand_in_marriage(Parent,Bride,Groom,Marriage) ) ).

%----New symbol sK0 recorded here in the definition of the ε term. 
tff(sK0_defn,definition,
    ! [Marriage: $i] :
    ! [Parent: $i] :
      ( sK0(Marriage)
      = ( # [Bride: $i] :
          ? [Groom: $i] : 
            ( in_love(Groom,Bride)
           => give_hand_in_marriage(Parent,sK0(Marriage),Groom,Marriage) ) ),
    introduced(definition,[new_symbols(skolem,[sK0])],[marriage]) ).

%----Skolemize Bride
fof(bride,plain,
    ! [Marriage] :
    ! [Parent] :
    ? [Groom] :
      ( in_love(Groom,sK0(Marriage))
     => give_hand_in_marriage(Parent,sK0(Marriage),Groom,Marriage) ),
    inference(skolemize,[status(esa),skolemize(Bride,sK0(Marriage))],[marriage,sK0_defn]) ).

tff(sK1_defn,definition,
    ! [Marriage: $i] :
    ! [Parent: $i] :
      ( sK1(Marriage,Parent)
      = ( # [Groom: $i] : 
            ( in_love(Groom,sK0(Marriage))
           => give_hand_in_marriage(Parent,sK0(Marriage),sK1(Marriage,Parent),Marriage) ) ) ),
    introduced(definition,[new_symbols(skolem,[sK1])],[bride]) ).

%----Skolemize Groom
fof(groom,plain,
    ! [Marriage] :
    ! [Parent] :
      ( in_love(sK1(Marriage,Parent),sK0(Marriage))
     => give_hand_in_marriage(Parent,sK0(Marriage),sK1(Marriage,Parent),Marriage) ),
    inference(skolemize,[status(esa),skolemize(Bride,sK0(Marriage),skolemize(Groom,sK1(Marriage,Parent))],[bride,sK1_defn]) ).

New Symbol Names

These conventions provide guidelines for reasonable naming of such symbols, and specifiy how the new symbols must be introduced in inference() and introduced() records.

New non-variable symbol names are recommended to be of the form sKN, where K is an uppercase letter indicating the type of new symbol, and N is a non-negative integer used to make the basic symbol name unique.
- The conventions for K are
  - K for Skolems
  - F for defined functions
  - G for function symbols with other uses
  - P for defined predicates
  - Q for predicate symbols with other uses, e.g., for splitting
- The index N should be global across all symbol types, or local to each symbol type.
- Examples: sK1, sF99, sP127.
New symbol names can optionally be extended by an underscore then one or more alphanumeric and underscore characters. For examples, sK1_iProver, sF99_Splitting, sP127_3, SV99_ForallIntro. The names should remain unique without the extension, using the N suffix.
The introduction of new non-variable symbols must be recorded in a new_symbols() record in the second argument of an inference() or introduced() record. The format is new_symbols(Type,[Symbols]). The Type indicates the type of new symbol, with possible values skolem, definition, and others. Symbols is a comma separated list of new symbols of that type. An inference() or introduced() record can have multiple new_symbols() records. For examples:
```
fof(bride,plain,
    ! [Marriage] :
    ? [Groom] :
      in_love(Groom,sK0(Marriage)),
    inference(skolemize,[status(esa),new_symbols(skolem,[sK0]),skolemize(Bride,sK0(Marriage))],[marriage]) ).
    
fof(pugalist,definition,
    ( no_fighters <=> ! [X] : ~ has_job(X,boxer) ),
    introduced(definition,[new_symbols(definition,[no_fighters])],[])). 
```
ATP systems should check for clashes whenever they generate a new symbol, and whenever they read input after any new symbols have been generated.

If you are interested in the history and motivations behind this convention, read the source of this web page.

Example Derivation

Consider the toy FOF problem in the problem quick guide. Here is a derivation recording a proof by refutation of the CNF (adapted from the one produced by the ATP system E):

%------------------------------------------------------------------------------
fof(someone_not_john,conjecture,
    ? [X3] :
      ( human(X3)
      & created_equal(X3,john)
      & X3 != john ),
    file('CreatedEqual.p',someone_not_john) ).

fof(all_created_equal,axiom,
    ! [X1,X2] :
      ( ( human(X1)
        & human(X2) )
     => created_equal(X1,X2) ),
    file('CreatedEqual.p',all_created_equal) ).

fof(someone_got_an_a,axiom,
    ? [X3] :
      ( human(X3)
      & grade(X3) = a ),
    file('CreatedEqual.p',someone_got_an_a) ).

fof(john,axiom,
    human(john),
    file('CreatedEqual.p',john) ).

fof(distinct_grades,axiom,
    a != f,
    file('CreatedEqual.p',distinct_grades) ).

fof(john_failed,axiom,
    grade(john) = f,
    file('CreatedEqual.p',john_failed) ).

fof(c_0_6,negated_conjecture,
    ~ ? [X3] :
        ( human(X3)
        & created_equal(X3,john)
        & X3 != john ),
    inference(fof_simplification,[status(thm)],[inference(assume_negation,[status(cth)],[someone_not_john])]) ).

fof(c_0_7,plain,
    ! [X6,X7] :
      ( ~ human(X6)
      | ~ human(X7)
      | created_equal(X6,X7) ),
    inference(fof_nnf,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[all_created_equal])])]) ).

fof(c_0_8,negated_conjecture,
    ! [X5] :
      ( ~ human(X5)
      | ~ created_equal(X5,john)
      | X5 = john ),
    inference(fof_nnf,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_6])])]) ).

%----If E believed in epsilon terms ...
% tff(sK0_defn,definition,
%     ( sK0
%     = ( # [X3: $i] :
%           ( human(X3)
%           & ( grade(X3) = a ) ) ) ),
%     introduced(definition,[new_symbols(skolem,[sK0])],[someone_got_an_a]) ).

fof(c_0_9,plain,
    ( human(sK0)
    & grade(sK0) = a ),
    inference(skolemize,[status(esa),new_symbols(skolem,[sK0]),skolemize(X3,sK0)],[someone_got_an_a]) ).

cnf(c_0_10,plain,
    ( created_equal(X1,X2)
    | ~ human(X1)
    | ~ human(X2) ),
    inference(split_conjunct,[status(thm)],[c_0_7]) ).

cnf(c_0_11,plain,
    human(john),
    inference(split_conjunct,[status(thm)],[john]) ).

cnf(c_0_12,negated_conjecture,
    ( X1 = john
    | ~ human(X1)
    | ~ created_equal(X1,john) ),
    inference(split_conjunct,[status(thm)],[c_0_8]) ).

cnf(c_0_13,plain,
    human(sK0),
    inference(split_conjunct,[status(thm)],[c_0_9]) ).

cnf(c_0_14,plain,
    ( created_equal(X1,john)
    | ~ human(X1) ),
    inference(spm,[status(thm)],[c_0_10,c_0_11]) ).

fof(c_0_15,plain,
    a != f,
    inference(fof_simplification,[status(thm)],[distinct_grades]) ).

cnf(c_0_16,negated_conjecture,
    ( sK0 = john
    | ~ created_equal(sK0,john) ),
    inference(spm,[status(thm)],[c_0_12,c_0_13]) ).

cnf(c_0_17,plain,
    created_equal(sK0,john),
    inference(spm,[status(thm)],[c_0_14,c_0_13]) ).

fof(c_0_18,plain,
    a != f,
    inference(fof_nnf,[status(thm)],[c_0_15]) ).

cnf(c_0_19,plain,
    grade(sK0) = a,
    inference(split_conjunct,[status(thm)],[c_0_9]) ).

cnf(c_0_20,negated_conjecture,
    sK0 = john,
    inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_16,c_0_17])]) ).

cnf(c_0_21,plain,
    grade(john) = f,
    inference(split_conjunct,[status(thm)],[john_failed]) ).

cnf(c_0_22,plain,
    a != f,
    inference(split_conjunct,[status(thm)],[c_0_18]) ).

cnf(c_0_23,plain,
    $false,
    inference(sr,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[c_0_19,c_0_20]),c_0_21]),c_0_22]),
    [proof] ).
%------------------------------------------------------------------------------

Checking a Derivation

To check the syntax of a derivation you can run it through tptp4X, available in SystemOnTSTP. Select the "TSTP formulae" option and paste the formulae into the text box. Select "TPTP4X" as the system, and ensure that the "Output mode" is "System". Click "ProcessSolution". If the syntax is faulty you'll get an error massage.

You can download and install TPTP4X on your own Linux computer, from Github. You must get the JJParser submodule too, i.e.,
git clone --recurse-submodules https://github.com/TPTPWorld/TPTP4X.

You can verify a derivation using GDV, also available in SystemOnTSTP. Select the "TSTP formulae" option and paste the formulae into the text box. Select "GDV" as the system, and ensure that the "Output mode" is "System". Click "ProcessSolution". It might take a while for output to appear. If the derivation is dubious or faulty, you'll get an WARNING/ERROR messages.

You can download and install GDV on your own Linux computer, from Github. You must get the JJParser submodule too, i.e.,
git clone --recurse-submodules https://github.com/TPTPWorld/GDV.git.

For more information about GDV, you can read ...

@Article{Sut06,
    Author       = "Sutcliffe, G.",
    Year         = "2006",
    Title        = "{Semantic Derivation Verification}",
    Journal      = "International Journal on Artificial Intelligence Tools",
    Volume       = "15",
    Number       = "6",
    Pages        = "1053-1070"
}