Resolution Principle and Fuzzy Logic

Fuzzy Predicate Logic with Evaluated Syntax (FPL) (Novak, V.) is a well-studied and wide-used logic capable of expressing vagueness. It has a lot of applications based on robust theoretical background. It also requires an efficient formal proof theory. However the most widely applied resolution principle (Dukic, N.) brings syntactically several obstacles mainly arising from normal form transformation. FPL is associating with even harder problems when trying to use the resolution principle. Solutions to these obstacles based on the non-clausal resolution (Bachmair, L.) were already proposed in (Habiballa, H.).


Introduction
Fuzzy Predicate Logic with Evaluated Syntax (FPL) (Novák, V.) is a well-studied and wide-used logic capable of expressing vagueness.It has a lot of applications based on robust theoretical background.It also requires an efficient formal proof theory.However the most widely applied resolution principle (Dukić, N.) brings syntactically several obstacles mainly arising from normal form transformation.FPL is associating with even harder problems when trying to use the resolution principle.Solutions to these obstacles based on the non-clausal resolution (Bachmair, L.) were already proposed in (Habiballa, H.).
In this article it would be presented a natural integration of these two formal logical systems into fully functioning inference system with effective proof search strategies.It leads to the refutational resolution theorem prover for FPL (RRTP FPL ).Another issue addressed in the paper concerns to the efficiency of presented inference strategies developed originally for the proving system.It is showed their perspectives in combination with standard proof-search strategies.The main problem for the fuzzy logic theorem proving lies in the large amount of possible proofs with different degrees and there is presented an algorithm (Detection of Consequent Formulas -DCF) solving this problem.The algorithm is based on detection of such redundant formulas (proofs) with different degrees.
The article presents the method which is the main point of the work on any automated prover.There is a lot of strategies which makes proofs more efficient when we use refutational proving.We consider well-known strategies -orderings, filtration strategy, set of support etc.One of the most effective strategies is the elimination of consequent formulas.It means the check if a resolvent is not a logical consequence of a formula in set of axioms or a previous resolvent.If such a condition holds it is reasonable to not include the resolvent into the set of resolvents, because if the refutation can be deduced from it, then so it can be deduced from the original resolvent, which it implies of.

First-order logic
For the purposes of (RRTP FPL ) it will be used generalized principle of resolution, which is defined in the research report (Bachmair, L.).There is a propositional form of the rule defined at first and further it is lifted into first-order logic.It is introduced the propositional form of the general resolution.

General resolution -propositional version
where the propositional logic formulas F and F ′ are the premises of inference and G is an occurrence of a subformula of both F and F ′ .The expression F[G/⊥] ∨ F ′ [G/⊤] is the resolvent of the premises on G. Every occurrence of G is replaced by false in the first formula and by true in the second one.It is also called F the positive, F' the negative premise, and G the resolved subformula.
The proof of the soundness of the rule is similar to clausal resolution rule proof.Suppose the Interpretation I in which both premises are valid.In I, G is either true or false.If G (¬G) is true in I, so is Revised version of the paper which forms the core of the handbook (Bachmair, L.) is closely related with notion of selection functions and ordering constraints.By a selection functions it is meant a mapping S that assigns to each clause C a (possibly empty) multiset S(C) of negative literals in C. In other words, the function S selects (a possibly empty) negative subclause of C. We say that an atom A, or a literal ¬A, is selected by S if ¬A occurs in S(C).There are no selected atoms or literals if S(C) is empty.Lexicographic path ordering can be used as an usual ordering over a total precedence.But in this case the ordering is admissible if predicate symbols have higher precedence than logical symbols and the constants ⊤ and ⊥ are smaller than the other logical symbols.It means the ordering is following A ≻ ≡ ≻ ⊃ ≻¬≻∨≻∧≻⊤≻⊥ .The handbook also addresses another key issues for automated theorem proving -the efficiency of the proof search.This efficiency is closely related with the notion of redundancy.
If we want to generalize the notion of resolution and lift it into first-order case we have to define first the notion of selection function for general clauses.General clauses are multisets of arbitrary quantifier-free formulas, denoting the disjunction of their elements.Note that we can also work with a special case of such general clause with one element, which yields to a standard quantifier-free formula of first-order logic.A (general selection) function is a mapping S that assigns to each general clause C a (possibly empty) set C of non-empty sequences of (distinct) atoms in C such that either S(C) is empty or else, for all interpretations I in which C is false, there exists a sequence A 1 , ..., A k in S(C), all atoms of which are true in I.A sequence A 1 , ..., A k in S(C) is said to be selected (by S).
We have to define the notion of polarity for these reasons according to the handbook (Bachmair, L.).It is based on the following assumption that a subformula Even it should seem that determining of the polarity of any subformula is NP-complete (hard) problem, we can use syntactic criteria for this computation.In this case the complexity of the algorithm is linear (note that we base our theory on similar syntactic criteria below -structural notions definition).

F is positive in a clause C if it is an element of C.
Note that this proposition applies both to formulas and clauses and allows us to determine polarity of any subformula in a formula.It is safe to select any sequence of negative atoms in a general clause, since a negative atom cannot be false in an interpretation the clause is false.
With the notion of the polarity as a selection function there is possible to state another notion of General resolution based on orderings applied to clauses.

General ordered resolution with selection O
where (i and (iii) no clause C i contains a selected atom.
According to the (Bachmair, L.) an inference system based on this rule is refutationally complete.When trying to extend this into the first-order case we to use lifting lemma.2. a (axiom), 3. b (axiom) 4. ⊥∨¬ ⊤ ∨ ¬ b ∨ c (a is a negative atom in (1) -selected in (1) as negative premise, and (2) as positive premise respectively) ⇒¬b ∨ c 5. ⊥∨¬ ⊤ ∨ c (b is a negative atom in (4) -selected in (4) as negative premise, and (3) as positive premise respectively) ⇒ c

Lemma 1. Lifting lemma Let M be a set of clauses and K
In the example we used the notion of polarity as a selection function.For example in the line 4 we select the atom a upon negative polarity (according the proposition criteria 1, 3 and 2level ordered) in formula 1 (it means 1. is a negative premise).
Further we can observe the behavior of the rule within the frame of clausal form resolution. Consider following table showing various cases of resolution on clauses.4), (5) on a) ⇒ b ∧ d 7. ⊥∧d ∨⊤∧d (( 6), (6)  When trying to refine the general resolution rule for fuzzy predicate logic, it is important to devise a sound and complete unification algorithm.Standard unification algorithms require variables to be treated only as universally quantified ones.We will present a more general unification algorithm, which can deal with existentially quantified variables without the need for those variables be eliminated by skolemization.It should be stated that the following unification process does not allow an occurrence of the equivalence connective.It is needed to remove equivalence by rewrite rule:

Definition 1. Structural notions of a FOL formula
Let F be a formula of FOL then the structural mappings Sub (subformula), Sup (superformula), Pol (polarity) and Lev (level) are defined as follows: For mappings Sub and Sup reflexive and transitive closures Sub * and Sup * are defined recursively as follows: These structural mappings provide framework for assignment of quantifiers to variable occurrences.It is needed for the correct simulation of skolemization (the information about a variable quantification in the prenex form).Subformula and superformula mappings and its closures encapsulate essential hierarchical information of a formula structure.Level gives 58 Fuzzy Logic -Algorithms, Techniques and Implementations www.intechopen.comthe ordering with respect to the scope of variables (which is also essential for skolemization simulation -unification is restricted for existential variables).Polarity enables to decide the global meaning of a variable (e.g.globally an existential variable is universal if its quantification subformula has negative polarity).Sound unification requires further definitions on variable quantification.We will introduce notions of the corresponding quantifier for a variable occurrence, substitution mapping and significance mapping (we have to distinguish between original variables occurring in special axioms and newly introduced ones in the proof sequence).

Definition 2. Variable assignment, substitution and significance
Let F be a formula of FOL, G = p(t 1 , ..., t n ) ∈ Sub * (F) atom in F and α a variable occurring in t i .
Variable mappings Qnt(quantifier assignment), Sbt (variable substitution) and Sig(significance) are defined as follows: Note that with Qnt mapping (assignment of first name matching quantifier variable in a formula hierarchy from bottom) we are able to distinguish between variables of the same name and there is no need to rename any variable.Sbt mapping holds substituted terms in a quantifier and there is no need to rewrite all occurrences of a variable when working with this mapping within unification.It is also clear that if Qnt(α)=∅ then α is a free variable.These variables could be simply avoided by introducing new universal quantifiers to F. Significance mapping is important for differentiating between original formula universal variables and newly introduced ones during proof search (an existential variable can't be bounded with it).
Before we can introduce the standard unification algorithm, we should formulate the notion of global universal and global existential variable (it simulates conversion into prenex normal form).

Definition 3. Global quantification
Let F be a formula without free variables and α be a variable occurrence in a term of F.
Var ∀ (F) and Var ∃ (F) are sets of global universal and existential variables.

if t
are function symbols with arguments and f = g then unifier for (t ′ , u ′ ) exists iff unifier exists for every pair

In any other case unifier does not exist (fail-state). MGU(A)=σ for a set of atoms
With above defined notions it is simple to state the general resolution rule for FOL (without the equivalence connective).It conforms to the definition from (Bachmair, L.).
where σ = MGU(A) is the most general unifier (MGU) of the set of the atoms Note that with Qnt mapping we are able to distinguish variables not only by its name (which may not be unique) but also with this mapping (it is unique).Sig property enables to separate variables, which were not originally in the scope of an existential variable.When utilizing the rule it should be set the Sig mapping for every variable in axioms and negated goal to one.We present a very simple example of existential variable unification before we introduce the refutational theorem prover for FOL.

Example 3. Variable Unification Restriction
We would try to prove if ∀X∃Yp(X, Y) ⊢∃ Y∀Xp(X, Y)?We will use refutational proving and therefore we will construct a special axiom from the first formula and negation of the second formula: F 0 : ∀X∃Yp(X, Y).F 1 (¬query) : ¬∃Y∀Xp(X, Y).There are 2 trivial and 2 non-trivial combinations how to resolve F 0 and F 1 (combinations with the same formula as the positive and the negative premise could not lead to refutation since they are consistent): Trivial cases: R[F 1 &F 1 ] : ⊥∨⊤and R[F 0 &F 0 ] : ⊥∨⊤.Both of them lead to ⊤ and the atoms are simply unifiable since the variables are the same.Non-trivial cases:[F 1 &F 0 ] : no resolution is possible.Y ∈ Var ∀ (F 1 ) and Y ∈ Var ∃ (F 0 ) can't unify since VUR for (Y, Y) does not hold -there is a variable X ∈ Sup * (Qnt(Y))(over the scope), X ∈ Var ∀ (F 0 ), Sbt(X)=∅); the case with variable X is identical.
[F 0 &F 1 ] : no resolution is possible (the same reason as above).No refutation could be derived from F 0 and F 1 due to VUR.

Fuzzy predicate logic and refutational proof
The fuzzy predicate logic with evaluated syntax is a flexible and fully complete formalism, which will be used for the below presented extension (Novák, V.).In order to use an efficient form of the resolution principle we have to extend the standard notion of a proof (provability value and degree) with the notion of refutational proof (refutation degree).Propositonal version of the fuzzy resolution principle has been already presented in (Habiballa, H.). We suppose that set of truth values is Łukasiewicz algebra.Therefore we assume standard notions of conjunction, disjunction etc. to be bound with Łukasiewicz operators.

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Resolution Principle and Fuzzy Logic www.intechopen.com We will assume Łukasewicz algebra to be where [0, 1] is the interval of reals between 0 and 1, which are the smallest and greatest elements respectively.Basic and additional operations are defined as follows: , where ∧ is infimum operation.The following properties of L Ł will be used in the sequel: The syntax and semantics of fuzzy predicate logic is following: • terms t 1 , ..., t n are defined as in FOL • predicates with p 1 , ..., p m are syntactically equivalent to FOL ones.Instead of 0 we write ⊥ and instead of 1 we write ⊤, connectives -& (Łukasiewicz conjunction), ∇ (Łukasiewicz disjunction), ⇒ (implication), ¬ (negation), ∀X (universal quantifier),∃X (existential quantifier) and furthermore by F J we denote set of all formulas of fuzzy logic in language J • FPL formulas have the following semantic interpretations (D is the universe): Interpretation of terms is equivalent to FOL, D(p i (t i 1 , ..., • for every subformula defined above Sub, Sup, Pol, Lev, Qnt, Sbt, Sig and other derived properties defined for classical logic hold (where the classical FOL connective is presented the Łukasiewicz one has the same mapping value).
Graded fuzzy predicate calculus assigns grade to every axiom, in which the formula is valid.It will be written as a A where A is a formula and a is a syntactic evaluation.We use several standard notions defined in (Novák, V.) namely: inference rule, formal fuzzy theory with set of logical and special axioms, evaluated formal proof.

Definition 6. Inference rule
An n-ary inference rule r in the graded logical system is a scheme r : using which the evaluated formulas a 1 A 1 , ..., a n A n are assigned the evaluated formula r evl (a 1 , ..., a n ) r syn (A 1 , ..., A n ).The syntactic operation r syn is a partial n-ary operation on F J and the evaluation operation r evl is an n-ary lower semicontinous operation on L (i.e. it preserves arbitrary suprema in all variables).

Definition 7. Formal fuzzy theory
A formal fuzzy theory T in the language J is a triple where LAx ⊂ ∼ F J is a fuzzy set of logical axioms, SAx ⊂ ∼ F J is a fuzzy set of special axioms, and R is a set of sound inference rules.

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Fuzzy Logic -Algorithms, Techniques and Implementations www.intechopen.comDefinition 8. Evaluated proof, refutational proof and refutation degree An evaluated formal proof of a formula A from the fuzzy set X ⊂ ∼ F J is a finite sequence of evaluated formulas w := a 0 A 0 , a 1 A 1 , ..., a n A n such that A n := A and for each i ≤ n, either there exists an m-ary inference rule r such that a i A i := r evl (a i 1 , ..., a i m ) r syn (A i 1 , ..., A i m ), i 1 , ..., i m < nor a i A i := X(A i ) A i .
We will denote the value of the evaluated proof by Val(w)=a n .An evaluated refutational formal proof of a formula A from X is w, where additionally a 0 A 0 := 1 ¬A and A n := ⊥.Val(w)=a n is called refutation degree of A.

Definition 9. Provability and truth
Let T be a fuzzy theory and A ∈ F J a formula.We write T ⊢ a A and say that the formula A is a theorem in the degree a, or provable in the degree a in the fuzzy theory T.
T ⊢ a A iffa = {Val(w)| wisaproofofAfromLAx∪ SAx} (5) We write T |= a A and say that the formula A is true in the degree a in the fuzzy theory T.
The fuzzy modus ponens rule could be formulated: where from premise A holding in the degree a and premise A⇒B holding in the degree b we infer B holding in the degree a ⊗ b.
In classical logic r MP could be viewed as a special case of the resolution.The fuzzy resolution rule presented below is also able to simulate fuzzy r MP .From this fact the completeness of a system based on resolution can be deduced.It will only remain to prove the soundness.It is possible to introduce following notion of resolution w.r.t. the modus ponens.
Definition 11.General resolution for fuzzy predicate logic (GR FPL ) where σ = MGU(A) is the most general unifier (MGU) of the set of the atoms Proof.Before we solve the core of GR FPL we should prove that the unification algorithm preserves soundness.But it could be simply proved since in the classical FPL with the rule of Modus-Ponens (Novák, V.) from the axiom ⊢ (∀x)A⇒A[x/t] and ⊢ (∀x)A we can prove A[x/t].For r GR we may rewrite the values of the left and right parts of equation ( 10):

is called positive and F' is called negative premise, G represents an occurrence of an atom. The expression Fσ
It is sufficient to prove the equality for ⇒ since all other connectives could be defined by it.By induction on the complexity of formula |A|, defined as the number of occurrences of connectives, we can prove: Let premises F 1 and F 2 be atomic formulas.Since they must contain the same subformula then Induction step: Let premises F 1 and F 2 be complex formulas and let A and B are subformulas of F 1 , C and D are subformulas of F 2 and G is an atom where generally F 1 =( A⇒B) and Since they must contain the same subformula and for A, B, C, D the induction presupposition hold it remain to analyze the following cases: We have rewritten the expression into Łukasiewicz interpretation.Now we will try to rewrite the right side of the inequality, which has to be proven.
The left and right side of the equation ( 10) are equal and therefore for this case holds. 2.
By induction we have proven that the inequality holds and the r R is sound.The induction of the case where only one of the premises has greater complexity is included in the above solved induction step.
There is a proof: All the schemes of the type A j ∇A , j > i could be simplified by sound simplification rules and the formula ¬A∇A may be removed.
The proof w ′′ := a 0 A 0 , ..., a i A i , a i+2 A i+2 ∇A, ..., a A is a correct proof of A in the degree a since the formulas are either axioms or results of application of resolution.
Theorem 1. Completeness for fuzzy logic with r GR , r s∇ , r s⇒ instead of r MP Formal fuzzy theory, where r MP is replaced with r GR ,r s∇ ,r s⇒ , is complete i.e. for every A from the set of formulas T ⊢ a A iff T |= a A.
Proof.The left to right implication (soundness of such formal theory) could be easily done from the soundness of the resolution rule.Conversely it is sufficient to prove that the rule r MP can be replaced by r GR , r s∇ , r s⇒ .Indeed, let w be a proof: Then we can replace it by the proof: Using the last sequence we can easily make a proof with r MP also with the proposed r R and simplification rules.Since usual formal theory with r MP is complete as it is proved in (Novák, V.), every fuzzy formal theory with these rules is also complete.Note that the non-ground case (requiring unification) could be simulated in the same way like in the proof of soundness.

Implementation and efficiency
The author also currently implements the non-clausal theorem prover into fuzzy logic as an extension of previous prover for FOL (GEneralized Resolution Deductive System -GERDS) (Habiballa, H.). Experiments concerning prospective inference strategies can be performed with this extension.The prover called Fuzzy Predicate Logic GEneralized Resolution Deductive System (Fig. 1) -FPLGERDS provides standard interface for input (knowledge base and goals) and output (proof sequence and results of fuzzy inference, statistics).

Fig. 1. Fuzzy Predicate Logic GEneralized Resolution Deductive System
There are already several efficient strategies proposed by author (mainly Detection of Consequent Formulas (DCF) adopted for the usage also in FPL).With these strategies the proving engine can be implemented in real-life applications since the complexity of theorem proving in FPL is dimensionally harder than in FOL (the need to search for all possible proofs -we try to find the best refutation degree).The DCF idea is to forbid the addition of a resolvent which is a logical consequence of any previously added resolvent.For refutational theorem proving it is a sound and complete strategy and it is emiprically very effective.Completeness of such a strategy is also straight-forward in FOL: DCF could be implemented by the same procedures like General Resolution (we may utilize self-resolution).Self-resolution has the same positive and negative premise and needs to We have proved that R new is a logical consequence of R old .
In FPL we have to enrich the DCF procedure by the limitation on the provability degree.if We will now show some efficiency results concerning many-valued logic both for Fuzzy Predicate Logic.We have used the above mentioned application FPLGERDS and originally developed DCF strategy for FPL.It is clear that inference in RRTP FPL and RRTP FDL on general knowledge bases is a problem solved in exponential time.Nevertheless as we would like to demonstrate the need to search for every possible proof (in contrast to the two-valued logic) will not necessarily in particular cases lead to the inefficient theory.We have devised knowledge bases (KB) on the following typical problems related to the use of fuzzy logic.
We have performed experimental measurements concerning efficiency of the presented non-clausal resolution principle and also DCF technique.These measurements were done using the FPLGERDS application (Habiballa, H.). Special testing knowledge bases were prepared and several types of inference were tested on a PC with standard Intel Pentium 4 processor as described below.

Fuzzy predicate Logic redundancy-based inefficient knowledge bases
As it was shown above in the theorem proving example the problem of proof search is quite different in FPL and FDL in comparison with the two-valued logic.We have to search for the best refutation degree using refutational theorem proving in order to make sensible conclusions from the inference process.It means we cannot accept the first successful proof, but we have to check "all possible proofs" or we have to be sure that every omitted proof is worse that some another one.The presented DCF and DCF Kill technique belong to the third sort of proof search strategies, i.e. they omit proofs that are really worse than some another (see the explication above).Proofs and formulas causing this could be called redundant proofs and redundant formulas.Fuzzy logic makes this redundancy dimensionally harder since we could produce not only equivalent formulas but also equivalent formulas of different evaluation degree.

Example 4. Redundant knowledge base
Consider the following knowledge base (fragment): ..., Searching for the best proof of a goal will produce a lot of logically equivalent formulas with different degrees.These resolvents make the inference process inefficient and one of the essential demands to the presented refutational theorem prover is a reasonable inference strategy with acceptable time complexity.
We have compared efficiency of the standard breadth-first search, linear search and modified linear search (starting from every formula in knowledge base) and also combinations with DCF and DCF-kill technique (Habiballa, H.). We have prepared knowledge bases of the size 120, 240, 360, 480 and 600 formulas.It has been compared the time and space efficiency on the criterion of 2 redundancy levels.This level represents the number of redundant formulas to which the formula is equivalent (including the original formula).For example the level 5 means the knowledge base contain 5 equivalent redundant formulas for every formula (including the formula itself).The basic possible state space search techniques and DCF heuristics and their combinations are presented in the following tables.
We use standard state space search algorithms in the FPLGERDS application -Breadth-first and Linear search.Breadth-first method searches for every possible resolvent from the formulas of the level 0 (goal and special axioms).These resolvents form formulas of the level 1 and we try to combine them with all formulas of the same and lower level and continue by the same procedure until no other non-redundant resolvent could be found.Linear search performs depth-first search procedure, where every produced resolvent is used as one of the premises in succeeding step of inference.The first produced resolvents arises from the goal formula.Modified linear search method posses the same procedure as linear one, but it starts from goal and also from all the special axioms.
DCF methods for reduction of resolvent space are basically three.formulas on an atom.This procedure is also used for the implementation of the theorem.
A "virtual tree" is created from candidate and former resolvent (axiom) connected by negated implication.Then it remains to perform self-resolution on such formula until a logical value is obtained.Let us compare the efficiency of standard strategies and the above-defined one.We have built-up 9 combinations of inference strategies from the mentioned proof search and DCF heuristics.They have different computational strength, i.e. their completeness is different for various classes of formulas.Fully complete (as described above) for general formulas of FPL and FDL are only breadth-first search combinations.Linear search strategies are not complete even for two-valued logic and horn clauses.Modified linear search has generally bad completeness results when an infinite loop is present in proofs, but for guarded knowledge bases it can assure completeness preserving better space efficiency than breadth-first search.
We tested presented inference strategies on sample knowledge bases with redundancy level 5 with 20, 40, 60, 80 and 100 groups of mutually redundant formulas (total number of formulas in knowledge base is 120, 240, 360, 480 and 600).At first we have tested their time efficiency for inference process.As it could be observed from figure 2, the best results have LDK and LDC strategies.For simple guarded knowledge bases (not leading to an infinite loop in proof search and where the goal itself assures the best refutation degree) these two methods are very efficient.DCF strategies significantly reduces the proof search even in comparison with LT strategy (standard), therefore the usage of any non-trivial DCF heuristics is significant.Next important result concludes from the comparison of BDK and MDK, MDC strategies.We can conclude that MDK and MDC strategies are relatively comparable to BDK and moreover BDK preserves completeness for general knowledge bases.
Space complexity is even more significantly affected by the DCF heuristics.There is an interesting comparison of trivial and non-trivial DCF heuristics in figure 3.Even BDK strategy brings significant reduction of resolvents amount, while LDK, LDC, MDK, MDC strategies have minimal necessary amount of kept resolvents during inference process.The second examined redundancy level 10 shows also important comparison for increasing redundancy in knowledge bases.Tested knowledge bases contained 10, 20, 30, 40 and 50 groups of 10 equivalent formulas (the total number of formulas was 110, 220, 330, 440 and 550 formulas).Time efficiency results shows that higher redundancy level causes expected increase in the necessary time for the best proof search (figure 4).The approximate increase is double, while the proportion shows good results for MDK, MDC and LDK, LDC (linear search based) strategies.This property also holds for space complexity as shown in figure 5. Performed experiments shows the significance of originally developed DCF strategies in combination with standard breadth-first search (important for general knowledge bases -BDK).We also outlined high efficiency for linear search based strategies (mainly LDK).Even this strategy is not fully complete and could be used only for guarded fragment of FDL, this problem is already known in classical (two-valued) logic programming and automated theorem proving.We also use these highly efficient linear search strategies, even they are not complete.

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Logic -Algorithms, Techniques and Implementations www.intechopen.comresolve all possible combinations of an atom.It uses the following scheme:R old ⊢ R new ⇔¬(R old → R new ) ⊢⊥Even the usage of this teachnique is a semidecidable problem, we can use time or step limitation of the algorithm and it will not affect the completeness of the RRTP FOL .Example: R new = p(a), R old = ∀x(p(x)), ¬(∀x(p(x)) → p(a)) MGU: Sbt(x)=a, Res = ¬(⊥→⊥) ∨¬(⊤→⊤) ⇒⊥

Fig. 4 .
Fig. 4. Time complexity for redundancy level 10 (seconds) Let G = p(t 1 , ..., t n ) and G ′ = r(u 1 , ..., u n ) be atoms.Most general unifier (substitution mapping) MGU(G, G') = σ is obtained by following atom and term unification steps or the algorithm returns fail-state for unification.For the purposes of the algorithm we define the Variable Unification Restriction (VUR).
y is a global universal variable.It is clear w.r.t.skolemization technique that an existential variable can be substituted into an universal one only if all global universal variables over the scope of the existential one have been already substituted by a term.Skolem functors function in the same way.Now we can define the most general unification algorithm based on recursive conditions (extended unification in contrast to standard MGU).59 Resolution Principle and Fuzzy Logic www.intechopen.comDefinition 4. Most general unifier algorithm

Refutational resolution theorem prover for FPL Refutational
non-clausal resolution theorem prover for FPL (RRTP FPL ) is the inference system with the inference rule GR FPL and sound simplification rules for ⊥, ⊤ (standard equivalencies for logical constants).A refutational proof by definition 8 represents a proof of a formula G (goal) from the set of special axioms N. It is assumed that Sig(α)=1 for ∀α in F ∈ N ∪¬G formula, every formula in a proof has no free variable and has no quantifier for a variable not occurring in the formula.Provability and refutation degree forGR FPL T ⊢ a A iff a = {Val(w)| w is a refutational proof of A from LAx ∪ SAx}Proof.If T ⊢ a A then a = {Val(w)| wisaproofofAfromLAx∪ SAx} and for every such a proof of we can construct refutational proof as follows (Val(w) ≤ a): w 64 Fuzzy Logic -Algorithms, Techniques and Implementations www.intechopen.comDefinition 12.
a then we can apply DCF.DCF Trivial check performs a symbolic comparison of R old and R new we use the same provability degree condition.In other cases we have to add R new into the set of resolvents and we can apply DCF Kill procedure.DCF Kill searches for every R old being a logical consequence of R new and if U

Table 4 .
The simplest is trivial DCF method, which detects redundant resolvent only by its exact symbolic comparison, i.e. formulas are equivalent only if the are syntactically the same.Even it is a very rough method, Inference strategies it is computationally very simple and forms necessary essential restriction for possibly infinite inference process.The next method of DCF technique enables do detect the equivalency of a formula (potential new resolvent) by the means described above.DCF Kill technique additionally tries to remove every redundant resolvent from the set of resolvents.The important aspect of the theorem DCF lies in its simple implementation into an automated theorem prover based on general resolution.The prover handles formulas in the form of syntactical tree.It is programmed a procedure performing general resolution with two