TPTP Documents File: SZSOntology
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The SZS success ontology provides status values to describe what is known or
has been successfully established about the relationship between the axioms and
conjecture in logical data. The SZS no-success ontology provides status values
to describe why a success ontology value has not been established. The SZS
dataform ontology provides status values to describe the nature of logical data.
All status values are expressed as "OneWord" to make system output parsing
simple, and also have a three letter mnemonic.
Commonly Used Ontology Values
-----------------------------
The ontologies are very fine grained ontology, which have more status values
and dataforms than are commonly used by ATP systems. Suitable subsets for
practical purposes are as follows:
+ For Success
- FOF problems with a conjecture - report Theorem or CounterSatisfiable
- FOF problems without a conjecture - report Satisfiable or Unsatisfiable
- CNF problems - report Satisfiable or Unsatisfiable
+ For No-success
- System stopped due to CPU limit - report Timeout
- System gave up due to incompleteness - report GaveUp
- System stopped due to an error - report Error
+ For Dataforms
- A generic proof - report Proof
- A CNF refutation - report CNFRefutation
- A generic model - report Model
- A finite model - report FiniteModel
- An infinite model - report InfiniteModel
- A saturation - report Saturation
Success ontology values are also used in TPTP format proofs to record the
relationship between the parents and inferred formula of each inference step.
Commonly used values are:
- The inferred formula is a theorem of the parents (logical consequences,
e.g., resolvants, etc.) - report Theorem
- The inferred and parent formulae are equisatisfiable (e.g., Skolemization)
- report EquiSatisfiable
- The negation of the inferred formula is a theorem of the parents (e.g.,
negating the conjecture in a proof by refutation) - report CounterTheorem
Standard Presentation of Ontology Values
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The solution status should be reported in a line starting "% SZS status"
(the leading '%' makes the line into a TPTP language comment). For examples:
% SZS status Unsatisfiable for SYN075+1
% SZS status GaveUp for SYN075+1
A success or no-success ontology value should be presented as early as
possible, at least before any data output to justify the value. The justifying
data should be delimited by lines starting "% SZS output start" and
"% SZS output end". For example:
% SZS output start CNFRefutation for SYN075-1
...
% SZS output end CNFRefutation for SYN075-1
All "SZS" lines lines can optionally have software specific information
appended, separated by a :. For examples:
% SZS status GaveUp for SYN075+1 : Could not complete CNF conversion
% SZS output end CNFRefutation for SYN075-1 : Completed in CNF conversion
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The SZS Success Ontology
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The ontology assumes that the input is a 2-tuple of the form <Ax, C>, where
Ax is a set (conjunction) of axioms and C is a single conjecture formula. This
is a common standard usage of ATP systems. If the input is not of the form
<Ax, C>, it is treated as a conjecture formula (even if it is a "set of axioms"
from the user view point, e.g., a set of formulae all with the TPTP role
"axiom"), and the 2-tuple is <TRUE, C>. The ontology values can also be
interpreted in terms of the formula F, of the form Ax => C. The ontology values
are based on the possible relationships between the sets of models of Ax and C.
In the figure below many of the "OneWord" status values are abbreviated - see
the list below for the official full "OneWord"s. The lines in the ontology can
be followed up the hierarchy as isa links, e.g., an ETH isa EQV isa (SAT and
a THM).
Success
SUC
_____________________________|_____________________________
| | | |
UNP SAP CSP CUP
from____|__________/ | | \__________|___from
CSA | | | | SAT
EquSat | | EquCtrSat
to ESA | | ECS to
CUP____| | | |____UNP
Sat'ble Theorem CtrThm CtrSat
SAT THM CTH CSA
| \_________.|._______________________________.|._________/ |
| | | | | |
| FinSat | NoConq | FinUns |
| FSA |_________ NOC _________| FUN |
_|____________|_ \_____________/ _|____________|_
| | | | | | ^ |
Eqvlnt TautC WeakC CtraAx WeakCC UnsCon |CtrEqu
EQV TAC WEC CAX WCC UNC | CQE
__|___ _|__ ___|__ ___|___ __|___ __|_ |___|__
| | / \ | | | | | | / \ | |
Equiv Taut- Weaker Weaker SatCon SatCCo Weaker Weaker Unsat Equiv
Thm ology TautCo TautCo CtraAx CtraAx CtrThm UnsCon -able CtrTh
ETH TAU WTC WTH SCA SCC WCT WUC UNS ECT
__|__ _|___
| \ / |
TauCon WCon UnsCon
CtraAx CtraAx CtraAx
TCA WCA UCA
+ Success (SUC):
The logical data has been processed successfully.
+ UnsatisfiabilityPreserving (UNP):
If there does not exist a model of Ax then there does not exist a model of
C, i.e., if Ax is unsatisfiable then C is unsatisfiable.
+ SatisfiabilityPreserving (SAP):
If there exists a model of Ax then there exists a model of C, i.e., if Ax
is satisfiable then C is satisfiable.
- F is satisfiable.
+ EquiSatisfiable (ESA):
There exists a model of Ax iff there exists a model of C, i.e., Ax is
(un)satisfiable iff C is (un)satisfiable.
+ Satisfiable (SAT):
Some interpretations are models of Ax, and
some models of Ax are models of C.
- F is satisfiable, and ~F is not valid.
- Possible dataforms are Models of Ax | C.
+ FinitelySatisfiable (FSA):
Some finite interpretations are finite models of Ax, and
some finite models of Ax are finite models of C.
- F is satisfiable, and ~F is not valid.
- Possible dataforms are FiniteModels of Ax | C.
+ Theorem (THM):
All models of Ax are models of C.
- F is valid, and C is a theorem of Ax.
- Possible dataforms are Proofs of C from Ax.
+ Equivalent (EQV):
Some interpretations are models of Ax,
all models of Ax are models of C, and
all models of C are models of Ax.
- F is valid, C is a theorem of Ax, and Ax is a theorem of C.
- Possible dataforms are Proofs of C from Ax and of Ax from C.
+ TautologousConclusion (TAC):
Some interpretations are models of Ax, and
all interpretations are models of C.
- F is valid, and C is a tautology.
- Possible dataforms are Proofs of C.
+ WeakerConclusion (WEC):
Some interpretations are models of Ax,
all models of Ax are models of C, and
some models of C are not models of Ax.
- See Theorem and Satisfiable.
+ EquivalentTheorem (ETH):
Some, but not all, interpretations are models of Ax,
all models of Ax are models of C, and
all models of C are models of Ax.
- See Equivalent.
+ Tautology (TAU):
All interpretations are models of Ax, and
all interpretations are models of C.
- F is valid, ~F is unsatisfiable, and C is a tautology.
- Possible dataforms are Proofs of Ax and of C.
+ WeakerTautologousConclusion (WTC):
Some, but not all, interpretations are models of Ax, and
all interpretations are models of C.
- F is valid, and C is a tautology.
- See TautologousConclusion and WeakerConclusion.
+ WeakerTheorem (WTH):
Some interpretations are models of Ax,
all models of Ax are models of C,
some models of C are not models of Ax, and
some interpretations are not models of C.
- See Theorem and Satisfiable.
+ ContradictoryAxioms (CAX):
No interpretations are models of Ax.
- F is valid, and anything is a theorem of Ax.
- Possible dataforms are Refutations of Ax.
+ SatisfiableConclusionContradictoryAxioms (SCA):
No interpretations are models of Ax, and
some interpretations are models of C.
- See ContradictoryAxioms.
+ TautologousConclusionContradictoryAxioms (TCA):
No interpretations are models of Ax, and
all interpretations are models of C.
- See TautologousConclusion and SatisfiableConclusionContradictoryAxioms.
+ WeakerConclusionContradictoryAxioms (WCA):
No interpretations are models of Ax, and
some, but not all, interpretations are models of C.
- See SatisfiableConclusionContradictoryAxioms and
SatisfiableCounterConclusionContradictoryAxioms.
+ CounterUnsatisfiabilityPreserving (CUP):
If there does not exist a model of Ax then there does not exist a model of
~C, i.e., if Ax is unsatisfiable then ~C is unsatisfiable.
+ CounterSatisfiabilityPreserving (CSP):
If there exists a model of Ax then there exists a model of ~C, i.e., if Ax
is satisfiable then ~C is satisfiable.
+ EquiCounterSatisfiable (ECS):
There exists a model of Ax iff there exists a model of ~C, i.e., Ax is
(un)satisfiable iff ~C is (un)satisfiable.
+ CounterSatisfiable (CSA):
Some interpretations are models of Ax, and
some models of Ax are models of ~C.
- F is not valid, ~F is satisfiable, and C is not a theorem of Ax.
- Possible dataforms are Models of Ax | ~C.
+ CounterTheorem (CTH):
All models of Ax are models of ~C.
- F is not valid, and ~C is a theorem of Ax.
- Possible dataforms are Proofs of ~C from Ax.
+ CounterEquivalent (CEQ):
Some interpretations are models of Ax,
all models of Ax are models of ~C, and
all models of ~C are models of Ax
(i.e., all interpretations are models of Ax xor of C).
- F is not valid, and ~C is a theorem of Ax.
- Possible dataforms are Proofs of ~C from Ax and of Ax from ~C.
+ UnsatisfiableConclusion (UNC):
Some interpretations are models of Ax, and
all interpretations are models of ~C
(i.e., no interpretations are models of C).
- F is not valid, and ~C is a tautology.
- Possible dataforms are Proofs of ~C.
+ WeakerCounterConclusion (WCC):
Some interpretations are models of Ax, and
all models of Ax are models of ~C, and
some models of ~C are not models of Ax.
- See CounterTheorem and CounterSatisfiable.
+ EquivalentCounterTheorem (ECT):
Some, but not all, interpretations are models of Ax,
all models of Ax are models of ~C, and
all models of ~C are models of Ax.
- See CounterEquivalent.
+ FinitelyUnsatisfiable (FUN):
All finite interpretations are finite models of Ax, and
all finite interpretations are finite models of ~C
(i.e., no finite interpretations are finite models of C).
+ Unsatisfiable (UNS):
All interpretations are models of Ax, and
all interpretations are models of ~C.
(i.e., no interpretations are models of C).
- F is unsatisfiable, ~F is valid, and ~C is a tautology.
- Possible dataforms are Proofs of Ax and of C, and Refutations of F.
+ WeakerUnsatisfiableConclusion (WUC):
Some, but not all, interpretations are models of Ax, and
all interpretations are models of ~C.
- See Unsatisfiable and WeakerCounterConclusion.
+ WeakerCounterTheorem (WCT):
Some interpretations are models of Ax,
all models of Ax are models of ~C,
some models of ~C are not models of Ax, and
some interpretations are not models of ~C.
- See CounterSatisfiable.
+ SatisfiableCounterConclusionContradictoryAxioms (SCC):
No interpretations are models of Ax, and
some interpretations are models of ~C.
- See ContradictoryAxioms.
+ UnsatisfiableConclusionContradictoryAxioms (UCA):
No interpretations are models of Ax, and
all interpretations are models of ~C
(i.e., no interpretations are models of C).
- See UnsatisfiableConclusion and
- SatisfiableCounterConclusionContradictoryAxioms.
+ NoConsequence (NOC):
Some interpretations are models of Ax,
some models of Ax are models of C, and
some models of Ax are models of ~C.
- F is not valid, F is satisfiable, ~F is not valid, ~F is satisfiable, and
C is not a theorem of Ax.
- Possible dataforms are pairs of models, one Model of Ax | C and one Model
of Ax | ~C.
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The NoSuccess Ontology
----------------------
In order to understand and make productive use of a lack of success, it is
necessary to precisely specify the reason for and nature of the lack of
success. The SZS no-success ontology provides status values for describing the
reasons. Note that no-success is not the same as failure: failure means that
the software has completed its attempt to process the logical data and could
not establish a success ontology value. In contrast, no-success might be
because the software is still running, or that it has not yet even started
processing the logical data. In the figure below many of the "OneWord" status
values are abbreviated - see the list below for the official full "OneWord"s.
NoSuccess
NOS
____________________|___________________
| | |
Open Unknown Assumed
OPN UNK ASS(UNK,SUC)
_________________|_________________
| | |
Stopped InProgress NotTried
STP INP NTT
____________________|________________ ____|____
| | | | |
Error Forced GaveUp | NotTriedYet
ERR FOR GUP | NTY
____|____ ____|____ _________|__________ |
| | | | | | | | |
OSError InputEr User ResourceOut Incompl | Inappro
OSE INE USR RSO INC | IAP
____|____ ___|___ v
| | | | to
SynError SemError Timeout MemyOut ERR
SYE SEE TMO MMO
|
TypeError
TYE
+ NoSuccess (NOS):
The logical data has not been processed successfully (yet).
+ Open (OPN):
A success value has never been established.
+ Unknown (UNK):
Success value unknown, and no assumption has been made.
+ Assumed (ASS(U,S)):
The success ontology value S has been assumed because the actual value is
unknown for the no-success ontology reason U. U is taken from the
subontology starting at Unknown in the no-success ontology.
+ Stopped (STP):
Software attempted to process the data, and stopped without a success status.
+ Error (ERR):
Software stopped due to an error.
+ OSError (OSE):
Software stopped due to an operating system error.
+ InputError (INE):
Software stopped due to an input error.
+ SyntaxError (SYE):
Software stopped due to an input syntax error.
+ SemanticError (SEE):
Software stopped due to an input semantic error.
+ TypeError (TYE):
Software stopped due to an input type error (for typed logical data).
+ Forced (FOR):
Software was forced to stop by an external force.
+ User (USR):
Software was forced to stop by the user.
+ ResourceOut (RSO):
Software stopped because some resource ran out.
+ Timeout (TMO):
Software stopped because the CPU time limit ran out.
+ MemoryOut (MMO):
Software stopped because the memory limit ran out.
+ GaveUp (GUP):
Software gave up of its own accord.
+ Incomplete (INC):
Software gave up because it's incomplete.
+ Inappropriate (IAP):
Software gave up because it cannot process this type of data.
+ InProgress (INP):
Software is still running.
+ NotTried (NTT):
Software has not tried to process the data.
+ NotTriedYet (NTY):
Software has not tried to process the data yet, but might in the future.
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The Dataform Ontology
---------------------
The dataform ontology provides suitable values for describing the form of
logical data. The ontology values are commonly used to describe data provided
to justify a success ontology value, e.g., if an ATP system reports the success
ontology value Theorem it might output a proof to justify that. In the figure
below many of the "OneWord" status values are abbreviated - see the list below
for the official full "OneWord"s.
LogicalData
LDa
_________________________|________________________
| | |
None Solution NotSoln
Non Sol NSo
___________________|__________________ ___|____
| | | | |
Proof Interpretation ListFrm Assure IncoPrf
Prf Int Lof Ass IPr
___|___ _|_____________ |________________ |
| | | | | | | | IncoRef
Derivn Refutn DomInt Model | LiTHF LiFOF LiCNF IRf
Der Ref Din Mod | Lth Lfo Lcn |
| ______|___________ _|__ | InCNFRe
CNFRef | | | \ | \ | ICf
CRf FinInt InfInt HerInt DomMod Saturn
FIn IIn HIn DMo Sat
| __________________|
| / ^ / ^ /
FinMod InfMod HerMod
FMo IMo HMo
+ LogicalData (LDa):
Logical data.
+ Solution (Sln):
A solution.
+ Proof (Prf):
A proof.
+ Derivation (Der):
A derivation (inference steps ending in the theorem, in the Hilbert style).
+ Refutation (Ref):
A refutation (starting with Ax U ~C and ending in FALSE).
+ CNFRefutation (CRf):
A refutation in clause normal form, including, for FOF Ax or C, the
translation from FOF to CNF (without the FOF to CNF translation it's an
IncompleteRefutation).
+ Interpretation (Int):
An interpretation.
+ DomainInterpretation (DIn):
An interpretation expressed as a domain, a mapping for functions, and a
relation for predicates.
+ FiniteInterpretation (FIn):
A domain interpretation with a finite domain.
+ InfiniteInterpretation (IIn):
A domain interpretation with an infinite domain.
+ HerbrandInterpretation (HIn):
A Herbrand interpretation.
+ Model (Mod):
A model.
+ DomainModel (DMo):
A model expressed as a domain, a mapping for functions, and a relation for
predicates.
+ FiniteModel (FMo):
A domain model with a finite domain.
+ InfiniteModel (IMo):
A domain model with an infinite domain.
+ HerbrandModel (HMo):
A Herbrand model.
+ Saturation (Sat):
A model expressed as a saturating set of formulae.
+ ListOfFormulae (Lof):
A list of formulae.
+ ListOfTHF (Lth):
A list of THF formulae.
+ ListOfFOF (Lfo):
A list of FOF formulae.
+ ListOfCNF (Lcn):
A list of CNF formulae.
+ IncompleteProof (IPr):
A proof with part missing.
+ IncompleteRefutation (IRf):
A refutation with parts missing.
+ IncompleteCNFRefutation (ICf):
A CNF refutation with parts missing.
+ Assurance (Ass):
Only an assurance of the success ontology value.
+ None (Non):
Nothing.
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