Some Rewriting Systems as a Background of Proving Methods

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1 Some Rewriting Systems as a Background of Proving Methods Katalin Pásztor Varga Department of General Computer Science Eötvös Loránd University pkata@ludens.elte.hu Magda Várterész Institute of Informatics Debrecen University varteres@inf.unideb.hu October 27, 2003 Abstract In [8] two formula rewriting systems for theorem proving were investigated. Both of them were based on the duality of conjunction and disjunction. By the studied methods the formulas can be rewritten into disjunctive or conjunctive normal form to make them usable for the theorem proving methods. In [7] we gave an extension of these rewriting systems for first-order formulas. After obtaining the suitable form of the formula the next problem is to find tools to extract some information for the existentially quantified variables of the,,theorem formula from the theorem proving systems. In this paper we investigate the tableau method and the resolution calculus from this point of view. Categories and Subject Descriptors: F.4.1 [Mathematical Logic]: Mechanical theorem proving; I.2.3 [Deduction and Theorem Proving]: Resolution Key Words and Phrases: Theorem proving, resolution, semantic tableaux, formula rewriting 1 Formula Rewriting into Normal Form In this section two rewriting systems that apply the duality in the rewriting are recalled by [7] and [8]. So-called point-plus rewriting system [1] introduces two new symbols for dual logic operations, owing to which normal forms of formulas can be obtained easily and efficiently. The other one is the recursive rewriting The work of the first author was partially supported by the grant OTKA T

2 system [3] which results a generalized conjunctive or disjunctive normal form in every step. In [8] the equivalent power of these methods was shown also. As both rewriting methods gave some kind of generalized disjunctive or conjunctive normal forms during the execution, the rewriting and the theorem proving algorithms can work parallel. 1.1 PP Rewriting System Definition 1 Let A be an arbitrary first-order formula. The strings ta and f A are called marked formulas. Let us define a point-plus-expression (in short ppexpression) in an inductive way: Every marked formula, signs and are pp-expressions. If E 1,E 2 are pp-expressions, then both are pp-expressions. (E 1 + E 2 ) and (E 1 E 2 ) If E is a pp-expression and x is a variable, then xe and xe are ppexpressions. We think the operations + and are syntactically commutative and associative, but we do not think that they are syntactically idempotent. To omit certain parentheses in the pp-expressions we introduce a precedence rule: the operation is stronger than +. We remark that the outside parentheses also need not in a pp-expression. Accordingly if E 1,...,E n are pp-expressions we can write simply a point chain E 1... E n and a plus chain E E n without using new parentheses. A pp-expression E has two interpretations. In order to get the cosequent interpretation E cos and the sequent interpretation E s we shall translate the pp-signs in E into usual logical ones according to Table 1. (Note that means some true, E t f + E cos E s Table 1: Translation of the pp-signs means some false formula.) So a pp-expression E represents two distinct formulas, which are (logically equivalent to) the negations of each other. Moreover if we substitute the pp-signs in E according to Table 2 we get another pp-expression G. It is obvious that E cos and G s are the same formulas, and so are E s and G cos. We say that E 1 E 2 (E 1 is pp-equivalent to E 2 ) if E1 s is logically equivalent to E2 s (or Ecos 1 is logically equivalent to E2 cos. We investigate some logical equivalents in the pp-notation. First we give the so-called pp-rules allowing to move t and f through a pp-expression (Table 3). 2

3 E t f + G f t + Table 2: Pairs of pp-signs ta B ta tb (t ) f A B f A + f B ( f ) ta B ta +tb (t ) f A B f A f B ( f ) ta B f A +tb (t ) f A B ta f B ( f ) t A f A (t ) f A ta ( f ) t xa xta (t ) f xa x f A ( f ) t xa xta (t ) f xa x f A ( f ) t (t ) f ( f ) t (t ) f ( f ) Table 3: pp-rules Definition 2 A pp-expression is in a literal form if the marks t and f occur before atomic formulas only. If P is an atom, the pp-expressions tp and f P are called pp-literals. Definition 3 A literal point chain is called a pp-expression of the form E 1... E n, and a literal plus chain is called a pp-expression of the form when E 1,...,E n (n 1) are pp-literals. E E n, Definition 4 Let C 1,...,C m (m 1) be point chains. The pp-expression C C m, denoted +gc, is a generalized plus normal form. Let now D 1,...,D m (m 1) be plus chains. The pp-expression D 1... D m, denoted gd, is a generalized point normal form. If the point chains in a +gc, or the plus chains in a gd are literal chains, we get so-called plus or point normal forms. Our purpose is to produce normal forms, therefore also some other rules are needed. We give these in Table 4 and 5. In the equivalences of Table 5 variable x does not occur in the pp-expression E 3 freely. Normal Form Procedure 3

4 E 1 (E 2 + E 3 ) E 1 E 2 + E 1 E 3 E 1 + E 2 E 3 (E 1 + E 2 ) (E 1 + E 3 ) Table 4: pp-distributions xe 1 (x) xe 2 (x) x(e 1 (x) E 2 (x)) xe 1 (x) + E 3 x(e 1 (x) + E 3 ) xe 1 (x) E 3 x(e 1 (x) E 3 ) xe 1 (x) + xe 2 (x) x(e 1 (x) + E 2 (x)) xe 1 (x) + E 3 x(e 1 (x) + E 3 ) xe 1 (x) E 3 x(e 1 (x) E 3 ) Table 5: Quantifier pp-rules To convert an arbitrary propositional formula A into a conjunctive or disjunctive normal form, let us start with either ta or f A. We will apply the pp-rules until the pp-expression is in literal form. The result is still a generalized normal form. Now we have to use the distribution rules to get equivalent plus or point normal form. When we start with ta the cosequent interpretation of the result plus normal form is a DNF of A and of the result point normal form is a CNF of A. And if we start with f A the sequent interpretation of the result plus normal form is a CNF of A and of the result point normal form is a DNF of A. The algorithm is non-deterministic; at each stage there are choices to be made of what to do next. Example 1 From a pp-expression f (P Q) (R (P Q)) by using of the pp-rules we get the generalized normal form ( f P +tq) tr tp f Q. Then applying the pp-distribution we obtain the plus normal form f P tr tp f Q +tq tr tp f Q, which is a CNF of the propositional formula (P Q) R (P Q) in the sequent interpretation. Now we give an extension of this rewriting system for first-order formulas. Let A be a first-order formula. Transforming A into prenex normal form means taking the following steps. First we apply the pp-rules until the ppexpression ta or f A is in literal form. After renaming bounded variables of obtained pp-expression we use the laws of Table 5 to move all quantifiers to the left of its inside. The last step is to convert the matrix of this result into plus or point normal form. This algorithm is also non-deterministic. 4

5 Example 2 If we want to transform xp(x) x(q(x) R(x)) into prenex normal form, starting with f xp(x) x(q(x) R(x)) pp-expression we get xtp(x) x( f Q(x)+ f R(x)) by using of the pp-rules. Now we rename the bound variables and by the Table 5 we move the quantifiers ahead: x y(tp(x) ( f Q(y) + f R(y))). If our purpose is a prenex conjunctive normal form, then applying the pp-distribution we get x y(tp(x) f Q(y) +tp(x) f R(y)) which is a prenex CNF in the sequent interpretation. 1.2 Recursive Rewriting System The logical operations are classified into α- and β-type connectives after the cardinality of their truth sets ([3]). α 1,α 2 or β 1,β 2 are the components or immediate subformulas of formulas α or β. Any α-type formula is the conjunction and β-type formula is the disjunction of their immediate subformulas α 1,α 2 and β 1,β 2, so that α is equivalent to α 1 α 2 and β is equivalent to β 1 β 2 (Table 6). Conjunctive Disjunctive type immediate subformulas type immediate subformulas α α 1 α 2 β β 1 β 2 A B A B (A B) A B (A B) A B A B A B (A B) A B A B A B Table 6: α- and β-formulas All quantified formulas and their negations are grouped into γ- or δ-type formulas (Table 7). Universal γ xa xa Existential δ xa xa Table 7: γ- and δ-formulas Definition 5 Let A 1,...,A n be formulas. [A 1,...,A n ] 5

6 is called a generalized disjunction, A 1,...,A n a generalized conjunction of A 1,...,A n respectively. Definition 6 Let C 1,...,C m (m 1) be generalized disjunctions, then C 1,...,C m is a generalized CNF. Let D 1,...,D m (m 1) be generalized conjunctions, then is a generalized DNF. [D 1,...,D m ] Definition 7 Let A be a formula. [A] or [ A ] denotes a generalized CNF or DNF of A respectively, and means the task transforming them recursively into a more detailed generalized CNF or DNF. To have an algorithm for converting formulas into generalized CNF and generalized DNF the normal form rewriting rules given in Table 8 are used. The addition of two quantifier rules of Table 9 turns the algorithm into a firs-order version. A A gc-rules: [α] [α 1 ],[α 2 ] [β] [β 1,β 2 ] gd-rules: [ α ] [ α 1,α 2 ] [ β ] [ β 1, β 2 ] Table 8: Recursive Rewriting Rules γ γ(x) x is a free variable δ δ(s(x 1,...,x n )) s is a new Skolem function, and x 1,...,x n all the free variables of δ Normal Form Procedure. Table 9: Expansion Rules To convert an arbitrary propositional formula A to CNF, let us start with [A], to DNF with [ A ]. We will apply the normal form rewriting rules until the argument formulas in the generalized disjunction or conjunction are literals. This algorithm is non-deterministic also. 6

7 Example 3 From the generalized CNF [(P Q) R (P Q)] we get [ (P Q), R, P,Q] with using the rule [β] [β 1,β 2 ] three times. Then the rule [α] [α 1 ],[α 2 ] is used and we obtain the generalized CNF [P, R, P,Q],[ Q, R, P,Q], in which the component are literals, so it is a CNF. Example 4 Applying expansion rules in the recursive rewriting from the generalized CNF [ ( w xr(x, w, g(x, w)) w x yr(x, w, y))] we obtain the following formulas one after the other and finally we get [ w xr(x,w,g(x,w))], [ w x yr(x,w,y)], [ xr(x,a,g(x,a))], [ x yr(x,v 1,y)], [R(v 2,a,g(v 2,a))], [ yr(s(v 1 ),v 1,y)], [R(v 2,a,g(v 2,a))], [ R(s(v 1 ),v 1,v 3 )]. 2 The Resolution Principle I. For the propositional resolution deduction a (generalized) conjunctive normal form is required. If the empty clause is deducible by resolution from the set of clauses of the formula A, then A is proved (or A has a resolution refutation). In the case of general clauses the resolvent exists on an arbitrary formula also. 2.1 Propositional pp-resolution Definition 8 A propositional pp-clause is a pp-expression of the form E 1... E n, where E 1,...,E n are pp-literals, and we adapt the empty pp-clause as. A propositional pp-clause-cluster is a pp-expression C C m, where C 1,...,C m are pp-clauses, and the empty pp-clause-cluster is. 7

8 E E E+ ta + f A ta E 1 + f A E 2 ta E 1 + f A E 2 + E 1 E 2 Table 10: R-Rules We have some pp-equivalents which help us with decision (Table 10). The question, whether A is proved, can be expressed in pp-notation by the form f A. In the propositional logic, if we separate the data preparation and the deduction phase, first we rewrite the pp-expression f A into a pp-clause-cluster, then we use the R-rules in the pp-clause-cluster. Note that we can apply the pprules, the distribution rule and the R-rules in arbitrary order. Example 5 From the pp-expression f ((P Q) R (P Q)), we get f P Q + f R +tp Q by pp-rules, then using the third R-rule we obtain f R+, which is pp-equivalent to by second R-rule. 2.2 Recursive Rewriting for the Propositional Resolution Example 6 We give a resolution proof of (P Q) (R S) (P (R S)) (Q (R S)). The following is a sequence of generalized disjunctions in which each, after the first, follows from earlier lines by the application of recursive rewriting or resolution rules: (1) [ (((P Q) (R S)) (P (R S)) (Q (R S)))] (2) [(P Q) (R S)] from (1) (3) [ ((P (R S)) (Q (R S)))] from (1) (4) [P Q, R S] from (2) (5) [P, R S] from (4) (6) [Q, R S] from (4) (7) [ (P (R S)), (Q (R S))] from (3) (8) [ P, (Q (R S))] from (7) (9) [ (R S), (Q (R S))] from (7) (10) [ P, Q] from (8) (11) [ P, (R S)] from (8) (12) [ (R S), Q] from (9) (13) [ (R S), (R S)] from (9) (14) [P, Q] resolvent on R S from (5, 13) (15) [ Q] resolvent on P from (10,14) (16) [R S] resolvent on Q from (6,15) (17) [ ] resolvent on R S from (11,16) 8

9 II. For the first-order resolution principle the formula A must be in Skolem normal form. A Skolem normal formula is logically equivalent to such a conjunction formula, where the argument formulas are first-order clauses obtained from the body of the Skolem normal form by quantifier rules. These first-order clauses have no common variables. Due to the Herbrand s results such formula is unsatisfiable if and only if it is unsatisfiable over the Herbrand universe of the formula. A consequence of the,,different variables property is that the variables appearing in the clauses can get values independently from the Herbrand universe. So it is possible to develop the clauses independently into the conjunction of ground clauses and apply the propositional resolution deduction. It is possible to classify the resolvent clauses after the pairs of first-order clauses C 1,C 2 as the set of resolvents of the ground clauses C 1 µ i,c 2 µ i. µ i is a ground unification substitution for the literal, the parent clauses are resolved upon. The most general unifier (mgu) is such a substitution λ that µ i = λ ν i (for every i), where ν i is element of the Herbrand universe. In the definition of the first-order resolvent the most general unifier is the key. An other consequence of the,,different variables property is that any element of the set of clauses is usable in the resolution deduction not only once. The resolution calculus deals with first-order clauses. The universally quantified variables of the negated,,theorem formula were originally existentially quantified variables. In case of a successful resolution deduction the unification substitutions for these universally quantified variables give an answer for the restrictions of the variables of the theorem formula. This is called answer substitution [5]. The resolution strategies assure that the different restrictions coming from the theorem appear answer substitutions. We illustrate this fact with an example: Example 7 Suppose we have the following first-order clauses: x y(t (x,y) Q(x) P(x,y)), T (c,q), P(a,r), P(b,s), P(e,s),Q(a), Q(b), Q(c) symbolized by S and we have the formula x yt (x,y) as a theorem. Then the negated theorem is x y T (x, y). The possible three resolution deductions answer the question: a b (1a) T (x, y) (2a) T (c, q) (x c, y q) (3a) (1b) T (x, y) (2b) T (x,y) Q(x) P(x,y) (3b) Q(x) P(x, y) (4b) Q(a) (5b) P(a, y) (x a) (6b) P(a, r) (7b) (y r) 9

10 c (1c) T (x, y) (2c) T (x,y) Q(x) P(x,y) (3c) Q(x) P(x, y) (4c) Q(b) (5c) P(b, y) (x b) (6c) P(b, s) (7c) (y s) The formula x yt (x,y) is the consequence of the formulas S in case of the substitutions: (x c, y q),(x a, y r),(x b, y s). 2.3 The First-order pp-resolution If E is a pp-expression and x 1,...,x n are its free variables, then we denote the pp-expression x 1... x n E by.e. Definition 9 If E is quantifier-free,.e is in so-called Skolem form..c is a -clause if C is a pp-clause. A -clause-cluster is a sum of -clauses. We have some pp-equivalents, which help us with decision. For all quantifierfree pp-expression G 1,G 2, first-order formula A and quantifier-free formula A 1,A 2 we get Table 11, if there is a substitution θ so that A 1 θ = A 2 θ..ta 1 +. f A 2.(tA 1 G 1 ) +.( f A 2 G 2 ).(ta 1 G 1 ) +.( f A 2 G 2 ) +.(G 1 θ G 2 θ) Table 11: Quantifier R-Rules First we rewrite the pp-expression f A into literal form, then we get a Skolem form using the skolemization, then the prenexization procedure. Next we rewrite the pp-body into a pp-clause-cluster and move quantifiers before the pp-clauses. In the deduction phase we use the R-rules in the pp-clause-cluster. As in the propositional logic we can apply the pp-rules, the distribution rule and the R-rules in arbitrary order. Example 8 The pp-expression f x y u v(p(x, y) P(v, u)) can be transformed by pp-rules, the skolemization procedure, the distribution rule to a -clause-cluster form y f P(a,y) + y vtp(v,g(y)). After renaming the bound variables in this pp-expression we get y f P(a,y) + w vtp(v,g(w)). Resolving these two -clauses with (v a,y g(w)) substitution we obtain. 10

11 3 The Method of Tableaux I. The necessary form for the method of propositional tableaux of the formula is indifferent. In principle the method uses the construction graph of the formula which gives the construction of the direct tableau of the formula. After that the direct tableau of the subformulas of the formula are recursively constructed until obtaining the closure of every branches of the tableau. If the recursive rewriting is used, then the tableau means the generalized disjunctive normal form of the formula. On a branch a generalized conjunction appears. The closure of the tableau means that this generalized DNF is false. In case of general conjunctions the closure of the tableau is possible using an arbitrary formula. Example 9 Now we give a tableau proof of the formula (P Q) (R S) (P (R S)) (Q (R S)). The following is a sequence of generalized conjunctions in the negated formula: (1) ((P Q) (R S) (P (R S)) (Q (R S))) (2) (P Q) (R S), ((P (R S)) (Q (R S))) (3) P Q, ((P (R S)) (Q (R S))), R S, ((P (R S)) (Q (R S))) (4) P, Q, (P (R S)), P, Q, (Q (R S)), R S, (P (R S)), R S, (Q (R S)) (5) P, Q, P, (R S), P, Q, Q, (R S), R S, P, (R S), R S, Q, (R S) Note that in (5) every generalized conjunction is false. The above generalized conjunctions of (5) appear on the four branches in the tableau proof below. (1) ((P Q) (R S) (P (R S)) (Q (R S))) (2) (P Q) (R S) (3) ((P (R S)) (Q (R S))) / \ (4) (P (R S)) (5) (Q (R S)) (6) P (8) Q (7) (R S) (9) (R S) / \ / \ (10) P Q (11) R S (12) P Q (13) R S (14) P (15) P (16) Q II. For the first order tableau we use Skolem formula due to the expansion rules of Table 9, where in the direct tableau of a quantified formula special substitution is used for the quantified variable. So the Herbrand universe of the tableau 11

12 consists of the ground terms obtained from the function- and constant symbols appearing in the tableau. A branch is closed, if it contains two unifiable formulas where the unification results complement pair. These substitutions are answer substitutions if they are related to the variables of the negated theorem formula. If for a formula the tableau rule is a branching rule and the subformulas contain common variables, then remain the same variables. Consequently if such a variable is substituted on a branch, then we must execute this substitution for every appearance of this variable. In a systematic tableau universal formula is sometimes elaborated more than once on a branch or on different branches. Naturally in a new expansion new variables are introduced. To find all of the answer substitutions for tableau is not easy. In this paper we propose an algorithm to obtain answer substitutions. In case of the resolution principle all of the possible resolution deduction were generated. As a formula has different tableaux our aim is to generate such tableau in which every subformula is elaborated at least once. For that we develop the formula in generalized conjunctive normal form and the tableau will be generated using this. Example 10 Let us consider the formula of the example 7. The development: (1) x y(t (x,y) Q(x) P(x,y)), x y T (x,y), T (c,q), P(a, r), P(b, s), P(e, s), Q(a), Q(b), Q(c) (2) T (x,y) Q(x) P(x,y), T (x 1,y 1 ), T (c,q), P(a, r), P(b, s), P(e, s), Q(a), Q(b), Q(c) (3) T (x,y), T (x 1,y 1 ), T (c,q), P(a, r), P(b, s), P(e, s), Q(a), Q(b), Q(c) Q(x), T (x 1,y 1 ), T (c,q), P(a, r), P(b, s), P(e, s), Q(a), Q(b), Q(c) P(x,y), T (x 1,y 1 ), T (c,q), P(a, r), P(b, s), P(e, s), Q(a), Q(b), Q(c) 1. In the systematic tableau first the rules for α,β,δ formulas are executed. The following step is the expansion of γ formulas. After that we continue the process until find the closure of the tableau. 2. After every expansion step we regard the branches whether it is possible to close some of them using the new formula. But first we unify the new formula (if it is possible) with a formula on the branch, having the same form. If there are more than one, then we choose one of them and note the others. Then we check whether a formula exists which can be the complement pair of the chosen after unification. 3. If there are still not used unification substitutions we construct a new tableau, in which the marked substitution is changed. Example 11 Now we sketch a first-order tableau proof of the example 7. The root of the tableau coincides with S from the mentioned example. The first-order tableau: 12

13 S (1) T (x,y) (2) Q(x) (3) P(x,y) (1) T (c,q) (2) Q(c) (3) P(c,q) (x c, y q) (4) T (x 1,y 1 ) (5) T (x 3,y 3 ) (6) T (x,y) (7) Q(x) (8) P(x,y) (6) T (a,r) (7) Q(a) (8) P(a,r) (x a, y r) (9) T (x 4,y 4 ) (10) Q(a) (11) P(a,r) In the second level of this tableau there is only one substitution and in the fifth level we have two, so if we generate the second tableau, we obtain a third answer substitution. The substitutions are: (x c, y q), (x a, y r), (x b, y s). 3.1 pp-tableaux If E is a pp-expression and x 1,...,x n are its free variables, then we denote the pp-expression x 1... x n E by.e. Definition 10 If E is quantifier-free,.e is in so-called Skolem form. We have some pp-equivalents, which we can use for decision. For all ppexpression E,E 1,E 2, quantifier-free pp-expression G 1,G 2, first-order formula A and quantifier-free formula A 1,A 2 we get Table 12. E E+ E ta f A (ta + E 1 ) ( f A + E 2 ) (ta + E 1 ) ( f A + E 2 ) (E 1 + E 2 ) Table 12: T-Rules If there is a substitution θ so that A 1 θ = A 2 θ, then.ta 1. f A 2.(tA 1 + G 1 ).( f A 2 + G 2 ).(ta 1 + G 1 ).( f A 2 + G 2 ).(G 1 θ + G 2 θ) Table 13: Quantifier T-Rules The question, whether A is unsatisfiable, can be expressed in pp-notation by the form ta. In the propositional logic, if we separate the data preparation and the deduction phase, first we rewrite the pp-expression ta into a pp-clause-cluster, then we use the T-rules in the pp-clause-cluster. Otherwise we can apply the pp-rules, the distribution rule and the T-rules in any order. Example 12 From the pp-expression t ((P Q) R (P Q)) by pp-rules we get t(p Q) tr f (P Q), then using T-rules 3 and 1 we obtain, that our ppexpression is equivalent to. 13

14 In the predicate logic the situation is similar, but a bit more complicated. First we rewrite the pp-expression ta into literal form, then we get a Skolem form using the skolemization, then the prenexization procedure. Next we rewrite the pp-body into a pp-dual-clause-cluster and move quantifiers before the pp-dualclauses. In the deduction phase we use the T-rules in the pp-dual-clause-cluster. We can apply the pp-rules, the distribution rule and the T-rules in arbitrary order. Example 13 The pp-expression t x y u v(p(x, y) P(v, u)) can be transformed by pp-rules and the skolemization and prenexization procedure to the form ytp(a,y) y v f P(v,g(y)). After renaming the bound variables we obtain ytp(a,y) w v f P(v,g(w)). With (v a,y g(w)) substitution we get by T-rule 5. References [1] Dragalin, A.G., On a self-dual notation in automated reasoning, Technical Report, Debrecen, [2] Egly, U., Quantifiers and the System KE: Some Surprising Results, CSL 98, pp [3] Fitting, M., First-order logic and automated theorem proving, Spinger-Verlag, [4] Gavilanes, A., Leach, J., Martin, P.J., Free variable tableaux for a logic with term declarations, TABLEAUX 98, pp [5] Lloyd, J.,W., Foundation of Logic programming, Spinger,1987. [6] Pásztor Varga, K., Theorem Proving Method and the Computer Science, Annales Univ. Sci. Budapest, Sect. Comp , pp [7] Pásztor Varga, K., Várterész, M., A Generalized Approach to the Theorem Proving Methods, Proc. 5th International Conference on Applied Informatics, Eger, 2001, pp [8] Pásztor Varga, K., Várterész, M., Comparison and Usability of two Rewriting Systems for Theorem Proving, Pure Mathematics and Applications, Vol.13, No.1-2, 2002, pp [9] Smullyan, R.M., First-order logic, Spinger-Verlag, Postal addresses 14

15 Katalin Pásztor Varga Department of General Computer Science Eötvös Loránd University 1117 Budapest, Pázmány Péter sétány 1/C. Hungary Magda Várterész Institute of Informatics Debrecen University 4010 Debrecen, Egyetem tér 1. Hungary 15