%----v9.0.0.0 (TPTP version.internal development number)
%----v9.0.0.1 Added <tff_quantifier>. Added <hash> (for epsilon terms) to <thf_quantifier> and
%---- <tff_quantifier>. Changed <tff_quantified_formula> to use <tff_quantifier>.
%----v9.0.0.2 Removed axiom_of_choice as an <intro_type>
%---- Renamed <inference_parents> to <parents>
%---- <inference_record> :== inference(<inference_rule>,<useful_info>,<parents>)
%---- Aligned introduced and creator with inferred
%---- <internal_source> :== introduced(<intro_type>,<useful_info>,<parents>)
%---- <creator_source> :== creator(<creator_name>,<useful_info>,<parents>)
%----v9.0.0.3 Moved <hash> to <fof_quantifier> so it's available in all languages.
%----v9.0.0.4 Removed ()s around <thf_defined_infix> and <thf_infix_unary>
%---- Removed ()s around <tff_defined_infix> and <tff_infix_unary>
%----v9.0.0.5 Fixed typo in <ntf_domain_type>
%----v9.0.0.6 Converted <source> and it's descendents to grammar rules. That required adding
%---- a grammar rule for <intro_type>.
%---- Changed <file_name> to <atomic_word>, to allow non-quoted filenames.
%--------------------------------------------------------------------------------------------------
%----README ... this header provides important meta- and usage information
%----
%----Intended uses of the various parts of the TPTP syntax are explained in the TPTP technical
%----manual, linked from www.tptp.org.
%----
%----Four kinds of separators are used, to indicate different types of rules:
%---- ::= is used for regular grammar rules, for syntactic parsing.
%---- :== is used for semantic grammar rules. These define specific values that make semantic
%---- sense when more general syntactic rules apply.
%---- ::- is used for rules that produce tokens.
%---- ::: is used for rules that define character classes used in the construction of tokens.
%----
%----White space may occur between any two tokens. White space is not specified in the grammar, but
%----there are some restrictions to ensure that the grammar is compatible with standard Prolog: a
%----<TPTP_file> should be readable with read/1.
%----
%----The syntax of comments is defined by the <comment> rule. Comments may occur between any two
%----tokens, but do not act as white space. Comments will normally be discarded at the lexical
%----level, but may be processed by systems that understand them (e.g., if the system comment
%----convention is followed).
%----
%----Multiple languages are defined. Depending on your need, you can implement just the one(s) you
%----need. The common rules for atoms, terms, etc, come after the definitions of the languages, and
%----mostly all needed for all the languages.
%----Top of Page-----------------------------------------------------------------------------------
%----Files. Empty file is OK.
<TPTP_file> ::= <TPTP_input>*
<TPTP_input> ::= <annotated_formula> | <include>
%----Formula records
<annotated_formula> ::= <thf_annotated> | <tff_annotated> | <tcf_annotated> | <fof_annotated> |
<cnf_annotated> | <tpi_annotated>
%----Future languages may include ... english | efof | tfof | mathml | ...
<tpi_annotated> ::= tpi(<name>,<formula_role>,<tpi_formula><annotations>).
<tpi_formula> ::= <fof_formula>
<thf_annotated> ::= thf(<name>,<formula_role>,<thf_formula><annotations>).
<tff_annotated> ::= tff(<name>,<formula_role>,<tff_formula><annotations>).
<tcf_annotated> ::= tcf(<name>,<formula_role>,<tcf_formula><annotations>).
<fof_annotated> ::= fof(<name>,<formula_role>,<fof_formula><annotations>).
<cnf_annotated> ::= cnf(<name>,<formula_role>,<cnf_formula><annotations>).
<annotations> ::= ,<source><optional_info> | <null>
%----In derivations the annotated formulae names must be unique, so that parent references (see
%----<inference_record>) are unambiguous.
%----Types for problems.
%----Note: The previous <source_type> from ...
%---- <formula_role> ::= <user_role>-<source>
%----... is now gone. Parsers may choose to be tolerant of it for backwards compatibility.
<formula_role> ::= <lower_word> | <lower_word>-<general_term>
<formula_role> :== axiom | hypothesis | definition | assumption | lemma | theorem |
corollary | conjecture | negated_conjecture | plain | type |
interpretation | fi_domain | fi_functors | fi_predicates | unknown
%----"axiom"s are accepted, without proof. There is no guarantee that the axioms of a problem are
%----consistent. "hypothesis"s are assumed to be true for a particular problem, and are used like
%----"axiom"s. "definition"s are intended to define symbols. They are either universally quantified
%----equations, or universally quantified equivalences with an atomic lefthand side. They can be
%----treated like "axiom"s. "assumption"s can be used like axioms, but must be discharged before a
%----derivation is complete. "lemma"s and "theorem"s have been proven from the "axiom"s. They can
%----be used like "axiom"s in problems, and a problem containing a non-redundant "lemma" or
%----"theorem" is ill-formed. They can also appear in derivations. "theorem"s are more important
%----than "lemma"s from the user perspective. "conjecture"s are to be proven from the
%----"axiom"(-like) formulae. A problem is solved only when all "conjecture"s are proven.
%----"negated_conjecture"s are formed from negation of a "conjecture" (usually in a FOF to CNF
%----conversion). "plain"s have no specified user semantics. "interpretation"s record all aspects
%----of an interpretation. "fi_domain", "fi_functors", and "fi_predicates" are are thge old way of
%----recording the domain, interpretation of functors, and interpretation of predicates, for a
%----finite interpretation. "type" defines the type globally for one symbol; treat as $true.
%----"unknown"s have unknown role, and this is an error situation.
%----Top of Page-----------------------------------------------------------------------------------
%----THF formulae.
<thf_formula> ::= <thf_logic_formula> | <thf_atom_typing> | <thf_subtype>
<thf_logic_formula> ::= <thf_unitary_formula> | <thf_unary_formula> | <thf_binary_formula> |
<thf_defined_infix> | <thf_definition> | <thf_sequent>
<thf_binary_formula> ::= <thf_binary_nonassoc> | <thf_binary_assoc> | <thf_binary_type>
%----There's no precedence among binary connectives
<thf_binary_nonassoc> ::= <thf_unit_formula> <nonassoc_connective> <thf_unit_formula>
<thf_binary_assoc> ::= <thf_or_formula> | <thf_and_formula> | <thf_apply_formula>
<thf_or_formula> ::= <thf_unit_formula> <vline> <thf_unit_formula> |
<thf_or_formula> <vline> <thf_unit_formula>
<thf_and_formula> ::= <thf_unit_formula> & <thf_unit_formula> |
<thf_and_formula> & <thf_unit_formula>
%----@ (denoting apply) is left-associative and lambda is right-associative.
%----^ [X] : ^ [Y] : f @ g (where f is a <thf_apply_formula> and g is a <thf_unitary_formula>)
%----should be parsed as: (^ [X] : (^ [Y] : f)) @ g. That is, g is not in the scope of either
%----lambda.
<thf_apply_formula> ::= <thf_unit_formula> @ <thf_unit_formula> |
<thf_apply_formula> @ <thf_unit_formula>
<thf_unit_formula> ::= <thf_unitary_formula> | <thf_unary_formula> | <thf_defined_infix>
<thf_preunit_formula> ::= <thf_unitary_formula> | <thf_prefix_unary>
<thf_unitary_formula> ::= <thf_quantified_formula> | <thf_atomic_formula> | <variable> |
(<thf_logic_formula>)
%----All variables must be quantified
<thf_quantified_formula> ::= <thf_quantification> <thf_unit_formula>
<thf_quantification> ::= <thf_quantifier> [<thf_variable_list>] :
<thf_variable_list> ::= <thf_typed_variable> | <thf_typed_variable>,<thf_variable_list>
<thf_typed_variable> ::= <variable> : <thf_top_level_type>
<thf_unary_formula> ::= <thf_prefix_unary> | <thf_infix_unary>
<thf_prefix_unary> ::= <thf_unary_connective> <thf_preunit_formula>
<thf_infix_unary> ::= <thf_unitary_term> <infix_inequality> <thf_unitary_term>
<thf_atomic_formula> ::= <thf_plain_atomic> | <thf_defined_atomic> | <thf_system_atomic> |
<thf_fof_function>
<thf_plain_atomic> ::= <constant> | <thf_tuple>
%----<thf_plain_atomic> includes <thf_tuple> because tuples can be formulae
%----in logic definitions
<thf_defined_atomic> ::= <defined_constant> | <thf_defined_term> | (<thf_conn_term>) |
<nhf_long_connective> | <thf_let>
% <thf_conditional>
%----<thf_conditional> is omitted from <thf_defined_atomic> because $ite is
%----read simply as a <thf_apply_formula>
<thf_defined_term> ::= <defined_term> | <th1_defined_term>
%----The ()s are really optional. I have reformated the TPTP to include them so, e.g.,
%----! [X:foo] a = X is formatted as ! [X:foo] (a = X) to save the lives of parsers that would
%----parse it as (! [X:foo] a) = X and throw an error.
<thf_defined_infix> ::= <thf_unitary_term> <defined_infix_pred> <thf_unitary_term>
% <thf_defined_infix> ::= <thf_unitary_term> <defined_infix_pred> <thf_unitary_term>
%----Defined terms can't be formulae. See TFF. FIX HERE.
<thf_system_atomic> ::= <system_constant>
%----<thf_conditional> is written and read as a <thf_apply_formula>
% <thf_conditional> ::= $ite(<thf_logic_formula>,<thf_logic_formula>, <thf_logic_formula>)
<thf_let> ::= $let(<thf_let_types>,<thf_let_defns>, <thf_logic_formula>)
<thf_let_types> ::= <thf_atom_typing> | [<thf_atom_typing_list>]
<thf_atom_typing_list> ::= <thf_atom_typing> | <thf_atom_typing>,<thf_atom_typing_list>
<thf_let_defns> ::= <thf_let_defn> | [<thf_let_defn_list>]
<thf_let_defn> ::= <thf_logic_formula> <assignment> <thf_logic_formula>
<thf_let_defn_list> ::= <thf_let_defn> | <thf_let_defn>,<thf_let_defn_list>
<thf_unitary_term> ::= <thf_atomic_formula> | <variable> | (<thf_logic_formula>)
<thf_conn_term> ::= <nonassoc_connective> | <assoc_connective> | <infix_equality> |
<infix_inequality> | <thf_unary_connective>
%----Note that syntactically this allows (p @ =), but for = the first argument must be known to
%----infer the type of =, so that's not allowed, i.e., only (= @ p).
<thf_tuple> ::= [] | [<thf_formula_list>]
%----Allows first-order style in THF.
<thf_fof_function> ::= <functor>(<thf_arguments>) | <defined_functor>(<thf_arguments>) |
<system_functor>(<thf_arguments>)
%----Arguments recurse back up to formulae (this is the THF world here)
<thf_arguments> ::= <thf_formula_list>
<thf_formula_list> ::= <thf_logic_formula> | <thf_logic_formula>,<thf_formula_list>
%----<thf_top_level_type> appears after ":", where a type is being specified
%----for a term or variable. <thf_unitary_type> includes <thf_unitary_formula>,
%----so the syntax is very loose, but trying to be more specific about
%----<thf_unitary_type>s (ala the semantic rule) leads to reduce/reduce
%----conflicts.
<thf_atom_typing> ::= <untyped_atom> : <thf_top_level_type> | (<thf_atom_typing>)
<thf_top_level_type> ::= <thf_unitary_type> | <thf_mapping_type> | <thf_apply_type>
%----Removed along with adding <thf_binary_type> to <thf_binary_formula>, for
%----TH1 polymorphic types with binary after quantification.
%---- | (<thf_binary_type>)
<thf_unitary_type> ::= <thf_unitary_formula>
<thf_unitary_type> :== <thf_atomic_type> | <th1_quantified_type>
<thf_atomic_type> :== <type_constant> | <defined_type> | <variable> | <thf_mapping_type> |
(<thf_atomic_type>)
<th1_quantified_type> :== !> [<thf_variable_list>] : <thf_unitary_type>
<thf_apply_type> ::= <thf_apply_formula>
<thf_binary_type> ::= <thf_mapping_type> | <thf_xprod_type> | <thf_union_type>
%----Mapping is right-associative: o > o > o means o > (o > o).
<thf_mapping_type> ::= <thf_unitary_type> <arrow> <thf_unitary_type> |
<thf_unitary_type> <arrow> <thf_mapping_type>
%----Xproduct is left-associative: o * o * o means (o * o) * o. Xproduct
%----can be replaced by tuple types.
<thf_xprod_type> ::= <thf_unitary_type> <star> <thf_unitary_type> |
<thf_xprod_type> <star> <thf_unitary_type>
%----Union is left-associative: o + o + o means (o + o) + o.
<thf_union_type> ::= <thf_unitary_type> <plus> <thf_unitary_type> |
<thf_union_type> <plus> <thf_unitary_type>
%----Tuple types, e.g., [a,b,c], are allowed (by the loose syntax) as tuples.
<thf_subtype> ::= <untyped_atom> <subtype_sign> <atom>
%----These are also used for NHF logic definitions
<thf_definition> ::= <thf_atomic_formula> <identical> <thf_logic_formula>
<thf_sequent> ::= <thf_tuple> <gentzen_arrow> <thf_tuple>
%----Top of Page-----------------------------------------------------------------------------------
%----TFF formulae.
<tff_formula> ::= <tff_logic_formula> | <tff_atom_typing> | <tff_subtype>
<tff_logic_formula> ::= <tff_unitary_formula> | <tff_unary_formula> | <tff_binary_formula> |
<tff_defined_infix> | <txf_definition> | <txf_sequent>
%----<tff_defined_infix> up here to avoid confusion in a = b | p, for TXF.
%----For plain TFF it can be in <tff_defined_atomic>
<tff_binary_formula> ::= <tff_binary_nonassoc> | <tff_binary_assoc>
<tff_binary_nonassoc> ::= <tff_unit_formula> <nonassoc_connective> <tff_unit_formula>
<tff_binary_assoc> ::= <tff_or_formula> | <tff_and_formula>
<tff_or_formula> ::= <tff_unit_formula> <vline> <tff_unit_formula> |
<tff_or_formula> <vline> <tff_unit_formula>
<tff_and_formula> ::= <tff_unit_formula> & <tff_unit_formula> |
<tff_and_formula> & <tff_unit_formula>
<tff_unit_formula> ::= <tff_unitary_formula> | <tff_unary_formula> | <tff_defined_infix>
<tff_preunit_formula> ::= <tff_unitary_formula> | <tff_prefix_unary>
<tff_unitary_formula> ::= <tff_quantified_formula> | <tff_atomic_formula> |
<txf_unitary_formula> | (<tff_logic_formula>)
<txf_unitary_formula> ::= <variable>
<tff_quantified_formula> ::= <tff_quantifier> [<tff_variable_list>] : <tff_unit_formula>
%----Quantified formulae bind tightly, so cannot include infix formulae
<tff_variable_list> ::= <tff_variable> | <tff_variable>,<tff_variable_list>
<tff_variable> ::= <tff_typed_variable> | <variable>
<tff_typed_variable> ::= <variable> : <tff_atomic_type>
<tff_unary_formula> ::= <tff_prefix_unary> | <tff_infix_unary>
%FOR PLAIN TFF <fof_infix_unary>
<tff_prefix_unary> ::= <tff_unary_connective> <tff_preunit_formula>
<tff_infix_unary> ::= <tff_unitary_term> <infix_inequality> <tff_unitary_term>
%FOR PLAIN TFF <tff_atomic_formula> ::= <fof_atomic_formula>
<tff_atomic_formula> ::= <tff_plain_atomic> | <tff_defined_atomic> | <tff_system_atomic>
<tff_plain_atomic> ::= <constant> | <functor>(<tff_arguments>)
<tff_plain_atomic> :== <proposition> | <predicate>(<tff_arguments>)
<tff_defined_atomic> ::= <tff_defined_plain>
%---To avoid confusion in TXF a = b | p | <tff_defined_infix>
<tff_defined_plain> ::= <defined_constant> | <defined_functor>(<tff_arguments>) | <nxf_atom> |
<txf_let>
% <txf_conditional>
%----<txf_conditional> is omitted from <tff_defined_plain> because $ite is
%----read simply as a <tff_atomic_formula>
<tff_defined_plain> :== <defined_proposition> | <defined_predicate>(<tff_arguments>) |
<nxf_atom> | <txf_conditional> | <txf_let>
%----This is the only one that is strictly a formula, type $o. In TXF, if the
%----LHS/RHS is a formula that does not look like a term, then it must be ()ed
%----per <tff_unitary_term>. If you put an un()ed formula that looks like a term
%----it will be interpreted as a term.
<tff_defined_infix> ::= <tff_unitary_term> <defined_infix_pred> <tff_unitary_term>
<tff_system_atomic> ::= <system_constant> | <system_functor>(<tff_arguments>)
<tff_system_atomic> :== <system_proposition> | <system_predicate>(<tff_arguments>)
%----<txf_conditional> is written and read as a <tff_defined_atomic>
<txf_conditional> :== $ite(<tff_logic_formula>,<tff_term>,<tff_term>)
<txf_let> ::= $let(<txf_let_types>,<txf_let_defns>,<tff_term>)
<txf_let_types> ::= <tff_atom_typing> | [<tff_atom_typing_list>]
<tff_atom_typing_list> ::= <tff_atom_typing> | <tff_atom_typing>,<tff_atom_typing_list>
<txf_let_defns> ::= <txf_let_defn> | [<txf_let_defn_list>]
<txf_let_defn> ::= <txf_let_LHS> <assignment> <tff_term>
<txf_let_LHS> ::= <tff_plain_atomic> | <txf_tuple>
<txf_let_defn_list> ::= <txf_let_defn> | <txf_let_defn>,<txf_let_defn_list>
<nxf_atom> ::= <nxf_long_connective> @ (<tff_arguments>)
<tff_term> ::= <tff_logic_formula> | <defined_term> | <txf_tuple>
<tff_unitary_term> ::= <tff_atomic_formula> | <defined_term> | <txf_tuple> | <variable> |
(<tff_logic_formula>)
<txf_tuple> ::= [] | [<tff_arguments>]
<tff_arguments> ::= <tff_term> | <tff_term>,<tff_arguments>
%----<tff_atom_typing> can appear only at top level.
<tff_atom_typing> ::= <untyped_atom> : <tff_top_level_type> | (<tff_atom_typing>)
<tff_top_level_type> ::= <tff_atomic_type> | <tff_non_atomic_type>
<tff_non_atomic_type> ::= <tff_mapping_type> | <tf1_quantified_type> | (<tff_non_atomic_type>)
<tf1_quantified_type> ::= !> [<tff_variable_list>] : <tff_monotype>
<tff_monotype> ::= <tff_atomic_type> | (<tff_mapping_type>) | <tf1_quantified_type>
<tff_unitary_type> ::= <tff_atomic_type> | (<tff_xprod_type>)
<tff_atomic_type> ::= <type_constant> | <defined_type> | <variable> |
<type_functor>(<tff_type_arguments>) | (<tff_atomic_type>) |
<txf_tuple_type>
<tff_type_arguments> ::= <tff_atomic_type> | <tff_atomic_type>,<tff_type_arguments>
<tff_mapping_type> ::= <tff_unitary_type> <arrow> <tff_atomic_type>
<tff_xprod_type> ::= <tff_unitary_type> <star> <tff_atomic_type> |
<tff_xprod_type> <star> <tff_atomic_type>
%----For TXF only
<txf_tuple_type> ::= [<tff_type_list>]
<tff_type_list> ::= <tff_top_level_type> | <tff_top_level_type>,<tff_type_list>
<tff_subtype> ::= <untyped_atom> <subtype_sign> <atom>
%----These are also used for NXF logic definitions
<txf_definition> ::= <tff_atomic_formula> <identical> <tff_term>
<txf_sequent> ::= <txf_tuple> <gentzen_arrow> <txf_tuple>
%----Top of Page-----------------------------------------------------------------------------------
%----Typed non-classical here
%----Have to duplicate NHF and NXF because they lead to <thf_definition> and <txf_definition>
<nhf_long_connective> ::= {<ntf_connective_name>} | {<ntf_connective_name>(<nhf_parameter_list>)}
<nhf_parameter_list> ::= <nhf_parameter> | <nhf_parameter>,<nhf_parameter_list>
<nhf_parameter> ::= <ntf_index> | <nhf_key_pair>
<nhf_key_pair> ::= <thf_definition>
<nxf_long_connective> ::= {<ntf_connective_name>} | {<ntf_connective_name>(<nxf_parameter_list>)}
<nxf_parameter_list> ::= <nxf_parameter> | <nxf_parameter>,<nxf_parameter_list>
<nxf_parameter> ::= <ntf_index> | <nxf_key_pair>
<nxf_key_pair> ::= <txf_definition>
<ntf_connective_name> ::= <def_or_sys_constant>
<ntf_index> ::= <hash><tff_unitary_term>
<ntf_short_connective> ::= [.] | <less_sign>.<arrow> | {.} | (.)
%----Short connectives are unary operators, cannot be indexed
%---- | [<ntf_index>] | <less_sign><ntf_index><arrow> |
%---- {<ntf_index>}
%----NXF logic specifications. Captured by <txf_definition>
%----NHF logic specifications are captured by <thf_definition>
<ntf_semantics_spec> :== <ntf_logic_name> <identical> [<ntf_logic_spec_list>]
<ntf_logic_name> :== $modal | $alethic_modal | $deontic_modal | $epistemic_modal |
$doxastic_modal | $temporal_instant
<ntf_logic_spec_list> :== <ntf_logic_spec> | <ntf_logic_spec>,<ntf_logic_spec_list>
<ntf_logic_spec> :== <ntf_domains_spec> | <ntf_designation_spec> | <ntf_terms_spec> |
<ntf_modalities_spec> | <ntf_time_spec>
<ntf_domains_spec> :== $domains <identical> <ntf_domains_value>
<ntf_domains_value> :== <ntf_domain_type> | [<ntf_domain_type_list>]
<ntf_domain_type> :== $constant | $varying | $cumulative | $decreasing |
<tff_atomic_type> <identical> <ntf_domains_value>
<ntf_domains_type_list> :== <ntf_domain_type> | <ntf_domain_type>,<ntf_domain_type_list>
<ntf_designation_spec> :== $designation <identical> <ntf_designation_value>
<ntf_designation_value> :== <ntf_designation_type> | [<ntf_designation_type_list>]
<ntf_designation_type> :== $rigid | $flexible |
<tff_atomic_type> <identical> <ntf_designation_value>
<ntf_designation_type_list> :== <ntf_designation_type> |
<ntf_designation_type>,<ntf_designation_type_list>
<ntf_terms_spec> :== $terms <identical> <ntf_terms_value>
<ntf_terms_value> :== <ntf_terms_type> | [<ntf_terms_type_list>]
<ntf_terms_type> :== $local | $global | <tff_atomic_type> <identical> <ntf_terms_value>
<ntf_terms_type_list> :== <ntf_terms_type> | <ntf_terms_type>,<ntf_terms_type_list>
<ntf_modalities_spec> :== $modalities <identical> <ntf_modalities_value>
<ntf_modalities_value> :== <ntf_modalities_type> | [<ntf_modalities_type_list>]
<ntf_modalities_type> :== <ntf_modal_system> | <ntf_modal_axiom> |
<tff_atomic_type> <identical> <ntf_modalities_value>
<ntf_modalities_type_list> :== <ntf_modalities_type> |
<ntf_modalities_type>,<ntf_modalities_type_list>
<ntf_time_spec> :== $time <identical> <ntf_time_value>
<ntf_time_value> :== <ntf_time_type> | [<ntf_time_type_list>]
<ntf_time_type> :== $reflexivity | $irreflexivity | $transitivity | $asymmetry |
$anti_symmetry | $linearity | $forward_linearity | $backward_linearity |
$beginning | $end | $no_beginning | $no_end | $density |
$forward_discreteness | $backward_discreteness |
<tff_atomic_type> <identical> <ntf_time_value>
<ntf_time_type_list> :== <ntf_time_type> | <ntf_time_type>,<ntf_time_type_list>
<ntf_modal_system> :== $modal_system_K | $modal_system_M | $modal_system_B | $modal_system_D |
$modal_system_S4 | $modal_system_S5
<ntf_modal_axiom> :== $modal_axiom_K | $modal_axiom_M | $modal_axiom_B | $modal_axiom_D |
$modal_axiom_4 | $modal_axiom_5
%----Top of Page-----------------------------------------------------------------------------------
%----TCF formulae.
<tcf_formula> ::= <tcf_logic_formula> | <tff_atom_typing>
<tcf_logic_formula> ::= <tcf_quantified_formula> | <cnf_formula>
<tcf_quantified_formula> ::= ! [<tff_variable_list>] : <tcf_logic_formula>
%----Top of Page-----------------------------------------------------------------------------------
%----FOF formulae.
<fof_formula> ::= <fof_logic_formula> | <fof_sequent>
<fof_logic_formula> ::= <fof_binary_formula> | <fof_unary_formula> | <fof_unitary_formula>
%----Future answer variable ideas | <answer_formula>
<fof_binary_formula> ::= <fof_binary_nonassoc> | <fof_binary_assoc>
%----Only some binary connectives are associative
%----There's no precedence among binary connectives
<fof_binary_nonassoc> ::= <fof_unit_formula> <nonassoc_connective> <fof_unit_formula>
%----Associative connectives & and | are in <binary_assoc>
<fof_binary_assoc> ::= <fof_or_formula> | <fof_and_formula>
<fof_or_formula> ::= <fof_unit_formula> <vline> <fof_unit_formula> |
<fof_or_formula> <vline> <fof_unit_formula>
<fof_and_formula> ::= <fof_unit_formula> & <fof_unit_formula> |
<fof_and_formula> & <fof_unit_formula>
<fof_unary_formula> ::= <unary_connective> <fof_unit_formula> | <fof_infix_unary>
%----<fof_term> != <fof_term> is equivalent to ~ <fof_term> = <fof_term>
<fof_infix_unary> ::= <fof_term> <infix_inequality> <fof_term>
%----<fof_unitary_formula> are in ()s or do not have a connective
<fof_unit_formula> ::= <fof_unitary_formula> | <fof_unary_formula>
<fof_unitary_formula> ::= <fof_quantified_formula> | <fof_atomic_formula> | (<fof_logic_formula>)
%----All variables must be quantified
<fof_quantified_formula> ::= <fof_quantifier> [<fof_variable_list>] : <fof_unit_formula>
<fof_variable_list> ::= <variable> | <variable>,<fof_variable_list>
<fof_atomic_formula> ::= <fof_plain_atomic_formula> | <fof_defined_atomic_formula> |
<fof_system_atomic_formula>
<fof_plain_atomic_formula> ::= <fof_plain_term>
<fof_plain_atomic_formula> :== <proposition> | <predicate>(<fof_arguments>)
<fof_defined_atomic_formula> ::= <fof_defined_plain_formula> | <fof_defined_infix_formula>
<fof_defined_plain_formula> ::= <fof_defined_plain_term>
<fof_defined_plain_formula> :== <defined_proposition> | <defined_predicate>(<fof_arguments>)
<fof_defined_infix_formula> ::= <fof_term> <defined_infix_pred> <fof_term>
%----System terms have system specific interpretations
<fof_system_atomic_formula> ::= <fof_system_term>
%----<fof_system_atomic_formula>s are used for evaluable predicates that are
%----available in particular tools. The predicate names are not controlled by
%----the TPTP syntax, so use with due care. Same for <fof_system_term>s.
%----FOF terms.
<fof_plain_term> ::= <constant> | <functor>(<fof_arguments>)
%----Defined terms have TPTP specific interpretations
<fof_defined_term> ::= <defined_term> | <fof_defined_atomic_term>
<fof_defined_atomic_term> ::= <fof_defined_plain_term>
%----None yet | <defined_infix_term>
%----None yet <defined_infix_term> ::= <fof_term> <defined_infix_func> <fof_term>
%----None yet <defined_infix_func> ::=
<fof_defined_plain_term> ::= <defined_constant> | <defined_functor>(<fof_arguments>)
%----System terms have system specific interpretations
<fof_system_term> ::= <system_constant> | <system_functor>(<fof_arguments>)
%----Arguments recurse back to terms (this is the FOF world here)
<fof_arguments> ::= <fof_term> | <fof_term>,<fof_arguments>
%----These are terms used as arguments. Not the entry point for terms because
%----<fof_plain_term> is also used as <fof_plain_atomic_formula>. The <tff_
%----options are for only TFF, but are here because <fof_plain_atomic_formula>
%----is used in <fof_atomic_formula>, which is also used as
%----<tff_atomic_formula>.
<fof_term> ::= <fof_function_term> | <variable>
<fof_function_term> ::= <fof_plain_term> | <fof_defined_term> | <fof_system_term>
%----Top of Page-----------------------------------------------------------------------------------
%----This section is the FOFX syntax. Not yet in use.
<fof_sequent> ::= <fof_formula_tuple> <gentzen_arrow> <fof_formula_tuple> |
(<fof_sequent>)
<fof_formula_tuple> ::= [] | [<fof_formula_tuple_list>]
<fof_formula_tuple_list> ::= <fof_logic_formula> | <fof_logic_formula>,<fof_formula_tuple_list>
%----Top of Page-----------------------------------------------------------------------------------
%----CNF formulae (variables implicitly universally quantified)
<cnf_formula> ::= <cnf_disjunction> | ( <cnf_formula> )
<cnf_disjunction> ::= <cnf_literal> | <cnf_disjunction> <vline> <cnf_literal>
<cnf_literal> ::= <fof_atomic_formula> | ~ <fof_atomic_formula> |
~ (<fof_atomic_formula>) | <fof_infix_unary>
%----Top of Page-----------------------------------------------------------------------------------
%----Connectives - THF
<thf_quantifier> ::= <fof_quantifier> | <th0_quantifier> | <th1_quantifier>
<thf_unary_connective> ::= <unary_connective> | <ntf_short_connective>
%----TH0 quantifiers are also available in TH1
<th1_quantifier> ::= !> | ?*
<th0_quantifier> ::= ^ | @+ | @-
%----Connectives - THF and TFF
<subtype_sign> ::= <<
%----Connectives - TFF
<tff_unary_connective> ::= <unary_connective> | <ntf_short_connective>
<tff_quantifier> ::= <fof_quantifier>
%----Connectives - FOF
<fof_quantifier> ::= ! | ? | <hash>
<nonassoc_connective> ::= <=> | => | <= | <~> | ~<vline> | ~&
<assoc_connective> ::= <vline> | &
<unary_connective> ::= ~
%----The seqent arrow
<gentzen_arrow> ::= -->
<assignment> ::= :=
<identical> ::= ==
%----Types for THF and TFF
<type_constant> ::= <type_functor>
<type_functor> ::= <atomic_word>
<defined_type> ::= <atomic_defined_word>
<defined_type> :== $oType | $o | $iType | $i | $tType | $real | $rat | $int
%----$oType/$o is the Boolean type, i.e., the type of $true and $false.
%----$iType/$i is non-empty type of individuals, which may be finite or
%----infinite. $tType is the type of all types. $real is the type of <real>s.
%----$rat is the type of <rational>s. $int is the type of <signed_integer>s
%----and <unsigned_integer>s.
<system_type> :== <atomic_system_word>
%----For all language types
<atom> ::= <untyped_atom> | <defined_constant>
<untyped_atom> ::= <constant> | <system_constant>
<proposition> :== <predicate>
<predicate> :== <atomic_word>
<defined_proposition> :== <defined_predicate>
<defined_proposition> :== $true | $false
<defined_predicate> :== <atomic_defined_word>
<defined_predicate> :== $distinct |
$less | $lesseq | $greater | $greatereq | $is_int | $is_rat |
$box | $dia
%----$distinct is part of the TFF, TXF, THF, NXF, and NHF syntax. $distinct takes one or more
%----constants of the same type as arguments, and indicates that the arguments are pairwise !=.
%----$distinct can be used only as a fact in an axiom-like annotated formula (e.g., not in a
%----conjecture), and not under any connective.
<defined_infix_pred> ::= <infix_equality>
<system_proposition> :== <system_predicate>
<system_predicate> :== <atomic_system_word>
<infix_equality> ::= =
<infix_inequality> ::= !=
<constant> ::= <functor>
<functor> ::= <atomic_word>
<defined_constant> ::= <defined_functor>
<defined_functor> ::= <atomic_defined_word>
<defined_functor> :== $uminus | $sum | $difference | $product |
$quotient | $quotient_e | $quotient_t | $quotient_f |
$remainder_e | $remainder_t | $remainder_f |
$floor | $ceiling | $truncate | $round |
$to_int | $to_rat | $to_real
<system_constant> ::= <system_functor>
<system_functor> ::= <atomic_system_word>
<def_or_sys_constant> ::= <defined_constant> | <system_constant>
<th1_defined_term> ::= !! | ?? | @@+ | @@- | @=
<defined_term> ::= <number> | <distinct_object>
<variable> ::= <upper_word>
%----Top of Page-----------------------------------------------------------------------------------
%----Formula sources
%----Expanded semantic rules for IDV. It was <source> ::= <general_term>
<source> ::= <dag_source> | <internal_source> | <external_source> | unknown |
[<sources>]
%----Alternative sources are recorded like this, thus allowing representation
%----of alternative derivations with shared parts.
<sources> ::= <source> | <source>,<sources>
%----Only a <dag_source> can be a <name>, i.e., derived formulae can be
%----identified by a <name> or an <inference_record>
<dag_source> ::= <name> | <inference_record>
<inference_record> ::= inference(<inference_rule>,<useful_info>,<parents>)
<inference_rule> ::= <atomic_word>
%----Examples are deduction | modus_tollens | modus_ponens | rewrite | resolution |
%---- paramodulation | factorization | cnf_conversion | cnf_refutation | ...
<internal_source> ::= introduced(<intro_type>,<useful_info>,<parents>)
<intro_type> ::= <atomic_word>
<intro_type> :== definition | tautology | assumption
%----This should be used to record the symbol being defined, or the function
%----for the axiom of choice
<external_source> ::= <file_source> | <theory> | <creator_source>
<file_source> ::= file(<file_name><file_info>)
<file_info> ::= ,<name> | <null>
<theory> ::= theory(<theory_name><optional_info>)
<theory_name> ::= <atomic_word>
<theory_name> :== equality | ac
%----More theory names may be added in the future. The <optional_info> is
%----used to store, e.g., which axioms of equality have been implicitly used,
%----e.g., theory(equality,[rst]). Standard format still to be decided.
<creator_source> ::= creator(<creator_name>,<useful_info>,<parents>)
<creator_name> ::= <atomic_word>
%----<parents> can be empty in cases when there is a justification for a tautologous theorem. In
%----cases when a tautology is introduced as a leaf, e.g., for splitting, then use an
%----<internal_source>.
<parents> ::= [] | [<parent_list>]
<parent_list> ::= <parent_info> | <parent_info>,<parent_list>
<parent_info> ::= <source><parent_details>
<parent_details> ::= :<general_list> | <null>
%----Useful info fields
<optional_info> ::= ,<useful_info> | <null>
<useful_info> ::= <general_list>
<useful_info> :== [] | [<info_items>]
<info_items> :== <info_item> | <info_item>,<info_items>
<info_item> :== <formula_item> | <inference_item> | <general_function>
%----Useful info for formula records
<formula_item> :== <description_item> | <iquote_item>
<description_item> :== description(<atomic_word>)
<iquote_item> :== iquote(<atomic_word>)
%----<iquote_item>s are used for recording exactly what the system output about
%----the inference step. In the future it is planned to encode this information
%----in standardized forms as <parent_details> in each <inference_record>.
%----Useful info for inference records
<inference_item> :== <inference_status> | <assumptions_record> | <new_symbol_record> |
<refutation>
<inference_status> :== status(<status_value>) | <inference_info>
%----These are the success status values from the SZS ontology. The most
%----commonly used values are:
%---- thm - Every model of the parent formulae is a model of the inferred formula. Regular logical
%---- consequences.
%---- cth - Every model of the parent formulae is a model of the negation of the inferred formula.
%---- Used for negation of conjectures in FOF to CNF conversion.
%---- esa - There exists a model of the parent formulae iff there exists a model of the inferred
%---- formula. Used for Skolemization steps.
%----For the full hierarchy see the SZSOntology file distributed with the TPTP.
<status_value> :== suc | unp | sap | esa | sat | fsa | thm | eqv | tac | wec | eth | tau |
wtc | wth | cax | sca | tca | wca | cup | csp | ecs | csa | cth | ceq |
unc | wcc | ect | fun | uns | wuc | wct | scc | uca | noc
%----<inference_info> is used to record standard information associated with an arbitrary inference
%----rule. The <inference_rule> is the same as the <inference_rule> of the <inference_record>. The
%----<atomic_word> indicates the information being recorded in the <general_list>. The
%----<atomic_word> are (loosely) set by TPTP conventions, and include esplit, sr_split, and
%----discharge.
<inference_info> :== <inference_rule>(<atomic_word>,<general_list>)
%----An <assumptions_record> lists the names of assumptions upon which this
%----inferred formula depends. These must be discharged in a completed proof.
<assumptions_record> :== assumptions([<name_list>])
%----A <refutation> record names a file in which the inference recorded here
%----is recorded as a proof by refutation.
<refutation> :== refutation(<file_source>)
%----A <new_symbol_record> provides information about a newly introduced symbol.
<new_symbol_record> :== new_symbols(<atomic_word>,[<new_symbol_list>])
<new_symbol_list> :== <principal_symbol> | <principal_symbol>,<new_symbol_list>
%----Principal symbols are predicates, functions, variables
<principal_symbol> :== <functor> | <variable>
%----Include directives
<include> ::= include(<file_name><include_optionals>).
<include_optionals> ::= <null> | ,<formula_selection> | ,<formula_selection>,<space_name>
<formula_selection> ::= [<name_list>] | <star>
<name_list> ::= <name> | <name>,<name_list>
<space_name> ::= <name>
%----Non-logical data
<general_term> ::= <general_data> | <general_data>:<general_term> | <general_list>
<general_data> ::= <atomic_word> | <general_function> | <variable> | <number> |
<distinct_object> | <formula_data>
<general_function> ::= <atomic_word>(<general_terms>)
%----A <general_data> bind() term is used to record a variable binding in an
%----inference, as an element of the <parent_details> list.
<general_data> :== bind(<variable>,<formula_data>) | bind_type(<variable>,<bound_type>)
<bound_type> :== $thf(<thf_top_level_type>) | $tff(<tff_top_level_type>)
<formula_data> ::= $thf(<thf_formula>) | $tff(<tff_formula>) | $fof(<fof_formula>) |
$cnf(<cnf_formula>) | $fot(<fof_term>)
<general_list> ::= [] | [<general_terms>]
<general_terms> ::= <general_term> | <general_term>,<general_terms>
%----General purpose
<name> ::= <atomic_word> | <integer>
%----Integer names are expected to be unsigned
<atomic_word> ::= <lower_word> | <single_quoted>
%----<single_quoted>s are the enclosed <atomic_word> without the quotes. Therefore the <lower_word>
%----<atomic_word> cat and the <single_quoted> <atomic_word> 'cat' are the same, but <numbers>s and
%----<variable>s are not <lower_word>s, so 123' and 123, and 'X' and X, are different. Quotes can
%----be removed from a <single_quoted> <atomic_word>, and is recommended if doing so produces a
%----<lower_word> <atomic_word>.
<atomic_defined_word> ::= <dollar_word>
<atomic_system_word> ::= <dollar_dollar_word>
<number> ::= <integer> | <rational> | <real>
%----Numbers are always interpreted as themselves, and are thus implicitly
%----distinct if they have different values, e.g., 1 != 2 is an implicit axiom.
%----All numbers are base 10 at the moment.
<file_name> ::= <atomic_word>
<null> ::=
%----Top of Page-----------------------------------------------------------------------------------
%----Rules from here on down are for defining tokens (terminal symbols) of the grammar, assuming
%----they will be recognized by a lexical scanner.
%----A ::- rule defines a token, a ::: rule defines a macro that is not a token. Usual regexp
%----notation is used. Single characters are always placed in []s to disable any special meanings
%----(for uniformity this is done to all characters, not only those with special meanings).
%----These are tokens that appear in the syntax rules above. No rules defined here because they
%----appear explicitly in the syntax rules, except that <vline>, <star>, <plus> denote "|", "*",
%----"+", respectively.
%----Keywords: fof cnf thf tff include
%----Punctuation: ( ) , . [ ] :
%----Operators: ! ? ~ & | <=> => <= <~> ~| ~& * +
%----Predicates: = != $true $false
%----For lex/yacc there cannot be spaces on either side of the | here
<comment> ::- <comment_line>|<comment_block>
<comment_line> ::- [%]<printable_char>*
<comment_block> ::: [/][*]<not_star_slash>[*][*]*[/]
<not_star_slash> ::: ([^*]*[*][*]*[^/*])*[^*]*
%----Defined comments are a convention used for annotations that are used as additional input for
%----systems. They look like comments, but start with %$ or /*$. A wily user of the syntax can
%----notice the $ and extract information from the "comment" and pass that on as input to the
%----system. They are analogous to pragmas in programming languages. To extract these separately
%----from regular comments, the rules are:
%---- <defined_comment> ::- <def_comment_line>|<def_comment_block>
%---- <def_comment_line> ::: [%]<dollar><printable_char>*
%---- <def_comment_block> ::: [/][*]<dollar><not_star_slash>[*][*]*[/]
%----A string that matches both <defined_comment> and <comment> should be recognized as
%----<defined_comment>, so put these before <comment>. Defined comments that are in use include:
%---- TO BE ANNOUNCED
%----System comments are a convention used for annotations that may used as additional input to a
%----specific system. They look like comments, but start with %$$ or /*$$. A wily user of the
%----syntax can notice the $$ and extract information from the "comment" and pass that on as input
%----to the system. The specific system for which the information is intended should be identified
%----after the $$, e.g., /*$$Otter 3.3: Demodulator */ To extract these separately from regular
%----comments, the rules are:
%---- <system_comment> ::- <sys_comment_line>|<sys_comment_block>
%---- <sys_comment_line> ::: [%]<dollar><dollar><printable_char>*
%---- <sys_comment_block> ::: [/][*]<dollar><dollar><not_star_slash>[*][*]*[/]
%----A string that matches both <system_comment> and <defined_comment> should
%----be recognized as <system_comment>, so put these before <defined_comment>.
<single_quoted> ::- <single_quote><sq_char><sq_char>*<single_quote>
%----<single_quoted>s contain visible characters. \ is the escape character for ' and \, i.e.,
%----\' is not the end of the <single_quoted>. The token does not include the outer quotes, e.g.,
%----'cat' and cat are the same. See <atomic_word> for information about stripping the quotes.
<distinct_object> ::- <double_quote><do_char>*<double_quote>
%---Space and visible characters upto ~, except " and \ distinct_object>s contain visible
%----characters. \ is the escape character for " and \, i.e., \" is not the end of the
%----<distinct_object>. <distinct_object>s are different from (but may be equal to) other tokens,
%----e.g., "cat" is different from 'cat' and cat. Distinct objects are always interpreted as
%----themselves, so if they are different they are unequal, e.g., "Apple" != "Microsoft" is
%----implicit.
<dollar_word> ::- <dollar><alpha_numeric>*
<dollar_dollar_word> ::- <dollar><dollar><alpha_numeric>*
<upper_word> ::- <upper_alpha><alpha_numeric>*
<lower_word> ::- <lower_alpha><alpha_numeric>*
%----Tokens used in syntax, and cannot be character classes
<vline> ::- [|]
<star> ::- [*]
<plus> ::- [+]
<arrow> ::- [>]
<less_sign> ::- [<]
<hash> ::- [#]
%----Numbers. Signs are made part of the same token here.
<real> ::- (<signed_real>|<unsigned_real>)
<signed_real> ::- <sign><unsigned_real>
<unsigned_real> ::- (<decimal_fraction>|<decimal_exponent>)
<rational> ::- (<signed_rational>|<unsigned_rational>)
<signed_rational> ::- <sign><unsigned_rational>
<unsigned_rational> ::- <decimal><slash><positive_decimal>
<integer> ::- (<signed_integer>|<unsigned_integer>)
<signed_integer> ::- <sign><unsigned_integer>
<unsigned_integer> ::- <decimal>
<decimal> ::- (<zero_numeric>|<positive_decimal>)
<positive_decimal> ::- <non_zero_numeric><numeric>*
<decimal_exponent> ::- (<decimal>|<decimal_fraction>)<exponent><exp_integer>
<decimal_fraction> ::- <decimal><dot_decimal>
<dot_decimal> ::- <dot><numeric><numeric>*
<exp_integer> ::- (<signed_exp_integer>|<unsigned_exp_integer>)
<signed_exp_integer> ::- <sign><unsigned_exp_integer>
<unsigned_exp_integer> ::- <numeric><numeric>*
<slash> ::- <slash_char>
<slosh> ::- <slosh_char>
%----Character classes
<percentage_sign> ::: [%]
<double_quote> ::: ["]
<do_char> ::: ([\40-\41\43-\133\135-\176]|[\\]["\\])
<single_quote> ::: [']
%---Space and visible characters upto ~, except ' and \
<sq_char> ::: ([\40-\46\50-\133\135-\176]|[\\]['\\])
<sign> ::: [+-]
<dot> ::: [.]
<exponent> ::: [Ee]
<slash_char> ::: [/]
<slosh_char> ::: [\\]
%% <bar> ::: [|]
<zero_numeric> ::: [0]
<non_zero_numeric> ::: [1-9]
<numeric> ::: [0-9]
<lower_alpha> ::: [a-z]
<upper_alpha> ::: [A-Z]
<alpha_numeric> ::: (<lower_alpha>|<upper_alpha>|<numeric>|[_])
<dollar> ::: [$]
<printable_char> ::: .
%----<printable_char> is any printable ASCII character, codes 32 (space) to 126 (tilde).
%----<printable_char> does not include tabs, newlines, bells, etc. The use of . does not not
%----exclude tab, so this is a bit loose.
<viewable_char> ::: [.\n]
%----Top of Page-----------------------------------------------------------------------------------