The syntax for atoms is that of Prolog: variables start with upper case letters, atoms and terms are written in prefix notation, uninterpreted predicates and functors either start with lower case and contain alphanumerics and underscore, or are in 'single quotes'. The language also supports interpreted predicates and functors. These come in two varieties: defined predicates and functors, whose interpretation is specified by the TPTP language, and system predicates and functors, whose interpretation is ATP system specific. Interpreted predicates and functors are syntactically distinct from uninterpreted ones - they are = and !=, or start with a $, a ", or a digit. Non-variable symbols can be given a type globally, in the formula with role type. The defined types are $o - the Boolean type, $i - the type of individuals, $real - the type of reals, $rat - the type of rational, and $int - the type of integers. New types are introduced in formulae with the type role, based on $tType - the type of all types. Full details of the THF and TFF type systems are provided in [SS+12,BP13,SB10,KSR16], with the last providing an overview of the first three.
The defined predicates recognized so far are
System predicates and functors are used for interpreted predicates and functors that are available in particular ATP tools. System predicates and functors start with $$. The names are not controlled by the TPTP language, so they must be used with caution.
The connectives used to build non-atomic formulae are written using intuitive notations. The universal quantifier is !, the existential quantifier is ?, and the lambda binder is ^. Quantified formulae are written in the form Quantifier [Variables] : Formula. In the THF, TFF, and FOF languages, every variable in a Formula must be bound by a preceding quantification with adequate scope. Typed Variables are given their type by a :type suffix. The binary connectives are infix | for disjunction, infix & for conjunction, infix <=> for equivalence, infix => for implication, infix <= for reverse implication, infix <~> for non-equivalence (XOR), infix ~| for negated disjunction (NOR), infix ~& for negated conjunction (NAND), infix @ for application. The only unary connective is prefix ~ for negation. Negation has higher precedence than quantification, which in turn has higher precedence than the binary connectives. No precedence is specified between the binary connectives; brackets are used to ensure the correct association. The binary connectives are left associative.
The THF and TFF languages have conditional and let expressions.
The useful_info field of an annotated formula is optional, and if it is not used then the source field becomes optional. The source field is used to record where the annotated formula came from, and is most commonly a file record or an inference record. A file record stores the name of the file from which the annotated formula was read, and optionally the name of the annotated formula as it occurs in that file - this might be different from the name of the annotated formula itself, e.g., if an ATP systems reads an annotated formula, renames it, and then prints it out. An inference record stores information about an inferred formula. The useful_info field of an annotated formula is a list of arbitrary useful information formatted as Prolog terms, as required for user applications.
An example of a THF formula section, extracted from the problem file LCL633^1.p, is shown below. An example of a TFF formula section, extracted from the problem file DAT013=1.p, is shown below that. Example THF formulae
. An example of a FOF formula section, extracted from the problem file GRP194+1.p, is shown below that. An example of a clause section, extracted from the problem file GRP039-7.p, is shown below that.
The following interpreted predicates and interpreted functions are defined.
Each symbol is ad-hoc polymorphic over the numeric types (with one
exception - $quotient is not defined for $int).
All arguments must have the same numeric type.
All the functions, except for the coercion functions $to_int and
$to_rat, have the same range type as their arguments.
For example,
$sum can be used with the type signatures
($int * $int) > $int,
($rat * $rat) > $rat,
and
($real * $real) > $real.
The coercion function $to_int always has a $int result,
and the coercion function $to_rat always has a $rat result.
All the predicates have a $o result.
For example,
$less can be used with the type signatures
($int * $int) > $o,
($rat * $rat) > $o,
and
($real * $real) > $o.
%------------------------------------------------------------------------------
%----Signature
thf(a,type,(
a: $tType )).
thf(p,type,(
p: ( a > $i > $o ) > $i > $o )).
thf(g,type,(
g: a > $i > $o )).
thf(e,type,(
e: ( a > $i > $o ) > a > $i > $o )).
thf(r,type,(
r: $i > $i > $o )).
%----Axioms
thf(positiveness,axiom,(
! [X: a > $i > $o] :
( mvalid
@ ( mimpl @ ( mnot @ ( p @ X ) )
@ ( p
@ ^ [Z: a] :
( mnot @ ( X @ Z ) ) ) ) ) )).
thf(g,definition,
( g
= ( ^ [Z: a,W: $i] :
! [X: a > $i > $o] :
( mimpl @ ( p @ X ) @ ( X @ Z ) @ W ) ) )).
thf(e,definition,
( e
= ( ^ [X: a > $i > $o,Z: a,P: $i] :
! [Y: a > $i > $o] :
( mimpl @ ( Y @ Z )
@ ( mbox @ r
@ ^ [Q: $i] :
! [W: a] :
( mimpl @ ( X @ W ) @ ( Y @ W ) @ Q ) )
@ P ) ) )).
%----Conjecture
thf(thm,conjecture,
( mvalid
@ ^ [W: $i] :
! [Z: a] :
( mimpl @ ( g @ Z ) @ ( e @ g @ Z ) @ W ) )).
%------------------------------------------------------------------------------
%------------------------------------------------------------------------------
tff(list_type,type,(
list: $tType )).
tff(nil_type,type,(
nil: list )).
tff(mycons_type,type,(
mycons: ( $int * list ) > list )).
tff(sorted_type,type,(
fib_sorted: list > $o )).
tff(empty_fib_sorted,axiom,(
fib_sorted(nil) )).
tff(single_is_fib_sorted,axiom,(
! [X: $int] : fib_sorted(mycons(X,nil)) )).
tff(double_is_fib_sorted_if_ordered,axiom,(
! [X: $int,Y: $int] :
( $less(X,Y)
=> fib_sorted(mycons(X,mycons(Y,nil))) ) )).
tff(recursive_fib_sort,axiom,(
! [X: $int,Y: $int,Z: $int,R: list] :
( ( $less(X,Y)
& $greatereq(Z,$sum(X,Y))
& fib_sorted(mycons(Y,mycons(Z,R))) )
=> fib_sorted(mycons(X,mycons(Y,mycons(Z,R)))) ) )).
tff(check_list,conjecture,(
fib_sorted(mycons(1,mycons(2,mycons(4,mycons(7,mycons(100,nil)))))) )).
%------------------------------------------------------------------------------
%--------------------------------------------------------------------------
%----Definition of a homomorphism
fof(homomorphism1,axiom,
( ! [X] :
( group_member(X,f)
=> group_member(phi(X),h) ) )).
fof(homomorphism2,axiom,
( ! [X,Y] :
( ( group_member(X,f)
& group_member(Y,f) )
=> multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) ) )).
fof(surjective,axiom,
( ! [X] :
( group_member(X,h)
=> ? [Y] :
( group_member(Y,f)
& phi(Y) = X ) ) )).
%----Definition of left zero
fof(left_zero,axiom,
( ! [G,X] :
( left_zero(G,X)
<=> ( group_member(X,G)
& ! [Y] :
( group_member(Y,G)
=> multiply(G,X,Y) = X ) ) ) )).
%----The conjecture
fof(left_zero_for_f,hypothesis,
( left_zero(f,f_left_zero) )).
fof(prove_left_zero_h,conjecture,
( left_zero(h,phi(f_left_zero)) )).
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%----Redundant two axioms
cnf(right_identity,axiom,
( multiply(X,identity) = X )).
cnf(right_inverse,axiom,
( multiply(X,inverse(X)) = identity )).
... some clauses omitted here for brevity
cnf(property_of_O2,axiom,
( subgroup_member(X)
| subgroup_member(Y)
| multiply(X,element_in_O2(X,Y)) = Y )).
%----Denial of theorem
cnf(b_in_O2,negated_conjecture,
( subgroup_member(b) )).
cnf(b_times_a_inverse_is_c,negated_conjecture,
( multiply(b,inverse(a)) = c )).
cnf(a_times_c_is_d,negated_conjecture,
( multiply(a,c) = d )).
cnf(prove_d_in_O2,negated_conjecture,
( ~ subgroup_member(d) )).
%--------------------------------------------------------------------------
The Arithmetic System
Arithmetic must be done in the context of the THF or TFF types logics, which
support the predefined atomic numeric types
$int, $rat, and $real.
Using THF or TFF enforces semantics that separate
$int from $rat from $real from $i from $o.
The numbers are unbounded, and the reals of infinite precision (rather
than some specific implementation such as 32 bit 2's complement integer, or
IEEE floating point).
Systems that implement limited arithmetic must halt in an
SZS error state if they hit overflow.
| Symbol | Operation | Comments, examples - |
|---|---|---|
| $int | The type of integers | 123, -123<integer> ::- (<signed_integer>|<unsigned_integer>) <signed_integer> ::- <sign><unsigned_integer> <unsigned_integer> ::- <decimal> <decimal> ::- (<zero_numeric>|<positive_decimal>) <positive_decimal> ::- <non_zero_numeric><numeric>* <sign> ::: [+-] <zero_numeric> ::: [0] <non_zero_numeric> ::: [1-9] <numeric> ::: [0-9] |
| $rat | The type of rationals | 123/456, -123/456, +123/456
<rational> ::- (<signed_rational>|<unsigned_rational>) <signed_rational> ::- <sign><unsigned_rational> <unsigned_rational> ::- <decimal><slash><positive_decimal> <slash> ::: [/] |
| $real | The type of reals | 123.456, -123.456, 123.456E789, 123.456e789, -123.456E789, 123.456E-789, -123.456E-789 <real> ::- (<signed_real>|<unsigned_real>) <signed_real> ::- <sign><unsigned_real> <unsigned_real> ::- (<decimal_fraction>|<decimal_exponent>) <decimal_exponent> ::- (<decimal>|<decimal_fraction>)<exponent><decimal> <decimal_fraction> ::- <decimal><dot_decimal> <dot_decimal> ::- <dot><numeric><numeric>* <dot> ::: [.] <exponent> ::: [Ee] |
| = (infix) | Comparison of two numbers. | The numbers must be the same atomic type (see the type system). |
| $less/2 | Less-than comparison of two numbers. | $less, $lesseq, $greater,
and $greatereq are related by ! [X,Y] : ( $lesseq(X,Y) <=> ( $less(X,Y) | X = Y ) ) ! [X,Y] : ( $greater(X,Y) <=> $less(Y,X) ) ! [X,Y] : ( $greatereq(X,Y) <=> $lesseq(Y,X) ) i.e, only $less and equality need to be implemented to get all four relational operators. |
| $lesseq/2 | Less-than-or-equal-to comparison of two numbers. | |
| $greater/2 | Greater-than comparison of two numbers. | |
| $greatereq/2 | Greater-than-or-equal-to comparison of two numbers. | |
| $uminus/1 | Unary minus of a number. | $uminus, $sum, and $difference are
related by ! [X,Y] : $difference(X,Y) = $sum(X,$uminus(Y)) i.e, only $uminus and $sum need to be implemented to get all three additive operators. |
| $sum/2 | Sum of two numbers. | |
| $difference/2 | Difference between two numbers. | |
| $product/2 | Product of two numbers. | |
| $quotient/2 | Exact quotient of two $rat or $real numbers. | For non-zero divisors, the result can be computed.
For zero divisors the result is not specified.
In practice, if an ATP system does not "know" that the divisor
is non-zero, it should simply not evaluate the $quotient.
Users should always guard their use of $quotient using
inequality, e.g., ! [X: $real] : ( X != 0.0 => p($quotient(5.0,X)) ) |
| $quotient_e/2, $quotient_t/2, $quotient_f/2 | Integral quotient of two numbers. | The three variants use different rounding to an integral result:
|
| $remainder_e/2, $remainder_t/2, $remainder_f/2 | Remainder after integral division of two numbers. | For τ ∈ {$int,$rat,
$real}, ρ ∈ {e,
t,f}, $quotient_ρ and
$remainder_ρ are related by ! [N:τ,D:τ] : $sum($product($quotient_ρ(N,D),D),$remainder_ρ(N,D)) = N For zero divisors the result is not specified. |
| $floor/1 | Floor of a number. | The largest integral value (in the type of the argument) not greater than the argument. |
| $ceiling/1 | Ceiling of a number. | The smallest integral value (in the type of the argument) not less than the argument. |
| $truncate/1 | Truncation of a number. | The nearest integral value (in the type of the argument) with magnitude not greater than the absolute value of the argument. |
| $round/1 | Rounding of a number. | The nearest integral value (in the type of the argument) to the argument. If the argument is halfway between two integral values, the nearest even integral value to the argument. |
| $is_int/1 | Test for coincidence with an $int value. | |
| $is_rat/1 | Test for coincidence with a $rat value. | |
| $to_int/1 | Coercion of a number to $int. | The largest $int not greater than the argument. If applied to an argument of type $int this is the identity function. |
| $to_rat/1 | Coercion of a number to $rat. | This function is not fully specified.
If applied to a $int the result is the argument over
1.
If applied to a $rat this is the identity function.
If applied to a $real that is (known to be) rational the
result is the $rat value.
For other reals the result is not specified.
In practice, if an ATP system does not "know" that the argument
is rational, it should simply not evaluate the $to_rat.
Users should always guard their use of $to_rat using
$is_rat, e.g., ! [X: $real] : ( $is_rat(X) => p($to_rat(X)) ) |
| $to_real/1 | Coercion of a number to $real. |
The extent to which ATP systems are able to work with the arithmetic
predicates and functions can vary, from a simple ability to evaluate ground
terms, e.g., $sum(2,3) can be evaluated to 5, through
an ability to instantiate variables in equations involving such functions,
e.g., $product(2,$uminus(X)) = $uminus($sum(X,2))
can instantiate X to 2, to extensive algebraic manipulation
capability.
The syntax does not axiomatize arithmetic theory, but may be used to write
axioms of the theory.
In NXF the non-classical connectives are applied in a mixed
higher-order-applied/first-order-functional style, with the connectives applied using @
to a ()ed list of arguments.
In NHF the non-classical connectives are applied using @ in usual higher-order style with
curried function applications.
There are also short form unary connectives for unparameterised {$box} and
{$dia}, applied directly like negation: [.] and <.>, e.g.,
{$box} @ (p) can be
written [.] p.
Short forms and long forms can be used together, e.g., it’s OK to use {$necessary} and
[.] in the same problem or formula.
Full specification of the connectives and their use in formulae is in the BNF
starting at <nxf_atom>
and
<thf_defined_atomic>.
Semantics Specification
A semantic specification changes the meaning of things such as the boolean type $o,
universal quantification with !, etc - their existing meaning in classical logic should
not be confused with the meaning in the declared logic.
For plain $modal and all the *_modal logics the properties that may be
specified are $domains, $designation, $terms, and $modalities.
For $temporal_instant the properties are the $domains, $designation,
and $terms of the modal logic, $modalities with different possible values,
and another property $time.
The formulae of a problem can be either local (true in the current world) or global (true in all
worlds).
By default, formulae with the roles hypothesis and conjecture are local, and
all others are global.
These defaults can be overridden by adding a subrole, e.g., axiom-$local,
conjecture-$global.
The Non-classical Logics
The TPTP recognizes several normal modal logics, and one temporal logic.
Each logic has a define name:
Connectives
The non-classical connectives of NTF have the form {$name}.
For the logics recognized by the TPTP the connectives are:
(System names may also be used for user-defined connectives, e.g.,
{$$canadian_conditional}, thus allowing use of the TPTP syntax when experimenting with
logics that have not yet been formalized in the TPTP.)
A connective can be parameterized to reflect more complex non-classical connectives.
The form is
{$name(param1,param2,param2)}.
If the connective is indexed the index is given as the first parameter prefixed with a #,
e.g., {$knows(#manuel)} @ (nothing)},
so that the connective is {$knows(#manuel)} (and not the connective {$knows}
applied to the index #manuel).
All other parameters are key-value assignments, e.g., to list the agents of common knowledge the
form might be {$common($agents:=[alice,bob,claire])}.
An annotated formula with the role logic is used to specify the semantics of formulae.
The semantic specification typically comes first in a file.
A semantic specification consists of the defined name of the logic followed by
== and a list of properties value assignments.
Each specification is the property name, followed by == and either a value or a tuple
of specification details.
If the first element of a list of details is a value, that is the default value for all cases
that are not specified in the rest of the list.
Each detail after the optional default value is the name of a relevant part of the vocabulary of
the problem, followed by == and a value for that named part.
The BNF grammar is here.
The grammar is not very restrictive on purpose, to enable working with other logics as well.
It is possible to create a lot of nonsense specifications, and to say the same thing in different
meaningful ways.
A tool to check the sanity of a specification is available.
Sys ∈ { K,KB,K4,K5,K45,KB5,D,DB,D4,D5,D45,M,B,S4,S5,S5U }
Axi ∈ { K,M,B,D,4,5,CD,BoxM,C4,C }
Axi ∈ { K,M,B,D,4,5 },
Tmi ∈ { +,- },
Pi
∈ { $reflexivity, $irreflexivity, $transitivity, $asymmetry, $anti_symmetry,
$linearity, $forward_linearity, $backward_linearity, $beginning, $end, $no_beginning,
$no_end, $density, $forward_discreteness, $backward_discreteness }