An Automatic Theorem Prover for Relation Algebras
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1 ARA An Automatic Theorem Prover for Relation Algebras Carsten Sinz Symbolic Computation Group, WSI for Computer Science, Universität Tübingen, Germany www-sr.informatik.uni-tuebingen.de/ sinz
2 Peirce (The Logic of Relatives, 1883): The logic of relatives is highly multiform; it is characterized by innumerable inferences, and by various distinct conclusions from the same sets of premises. (...) The effect of these pecularities is that this algebra cannot be subjected to hard and fast rules like those of the Boolian calculus; and all that can be done in this place is to give a general idea of the way of working with it.
3 RA, RRA, SA: Definitions Relation Algebra (RA): Structure A = (A,+,,,,1 ) satisfying the equations (BI)-(BX) for all R, S, T A. Proper Relation Algebra: All elements of A are actually binary relations, operations are the usual set-theoretic ones. Representable Relation Algebra (RRA): A representable iff. A isomorphic to a proper relation algebra. Semi-associative Relation Algebra (SA): Associativity of (equation (BIV)) replaced by equation R 1 = (R 1) 1.
4 Finite Variable Logic n-variable logic: First order predicate logic with restricted language: only n distinct variables x 0,..., x n 1. n-variable calculus: All axioms resp. rules restricted to n variables. Cut-rule: In n-variable logic calculi with cutrule are usually stronger than their counterparts without cut-rule (e.g. sequence calculus). n-variable resolution is weaker than the n- variable sequence calculus with cut-rule. Gordeev s Reduction Predicate Calculi: n-variable calculi without cut-rule (RPC n ), equivalent in proof power to n-variable sequence calculus with cut-rule (SCC n ).
5 The ARA Prover Prover for the RPC n calculi. Front-end to convert RA equations to 3- variable sentences of first-order logic. Different reduction strategies, e.g.: ORP: oldest reduction possibility LP: complementary literal pair strategy Various additional simplification rules, e.g.: Priority for shortening rules. Subgoal generation. Pure literal deletion.
6 Calculus of Relations Basic objects: Binary relations R, S, T,... Basic operations: complementation: R xy(xr y xry) conversion: R xy(xr y yrx) rel. multiplication: R S xy(xr Sy z(xrz zsy)) abs. addition: R + S xy(xr + Sy xry xsy) relative unit: 1 xy(x1 y x = y) relational equiv.: R = S xy(xry xsy) relational incl.: R S xy(xry xsy) Derived operations: relative addition: R S R S = (R S ) relative zero: 0 0 = 1
7 Calculus of Relations (cont.) Examples: R R R transitivity xy( z(xrz zry) xry) R R symmetry xy(xry yrx) R R 1 functionality xyz(zrx zry x = y) Basic identities (Peirce 1883): 1 R R R R 0 R (S T) (R S) T R (S T) (R S) T R S T R T S T S R
8 Tarski s Axiomatization (BI) R + S = S + R (BII) R + (S + T) = (R + S) + T (BIII) (R + S) + (R + S ) = R (BIV) R (S T) = (R S) T (BV) (R + S) T = R T + S T (BVI) R 1 = R (BVII) R = R (BVIII) (R + S) = R + S (BIX) (R S) = S R (BX) R (R S) + S = S
9 RPC n Rewrite Systems (R1) A (R2) L L (R3) A A (R4) A (B C) (A B) (A C) (R5) xa xa A[x/t] (R6) xa B xa B y( yb A[x y][x/y]) (R6 ) xa xa A[ x] A[ x] replaces all literals containing a free x by, A[y x] only if x y. F is provable in RPC n iff. it can be reduced to in finitely many reduction steps. and are assumed to be associative and commutative.
10 Finite Variable Logic and Relation Algebras: The Link Tarski (1941): Every sentence of relation algebra can be translated to a 3-variable logic sentence and vice versa. Maddux (1978): 1. A sentence is valid in SA iff. its translation is provable in SCC A sentence is valid in RA iff. its translation is provable in SCC A sentence is valid in RRA iff. its translation is provable in SCC ω.
11 Experimental Results problem source class strat. proofs steps time 3.2(v) [TG87] SSA LI (vi) [TG87] SSA LI (xvii) [TG87] SSA LI (xviii) [TG87] RA AI (iii)(ǫ) [TG87] RA AI (xix) [TG87] RA LA Thm 2.7 [CT51] RA AI Thm 2.11 [CT51] RA AI Cor 2.19 [CT51] RA AI Dedekind [DG98] RA AI Cor 2.19 [CT51] RRA AI Run-times in seconds on a Sun E450 running at 400 MHz.
System Description: ara An Automatic Theorem Prover for Relation Algebras
System Description: ara An Automatic Theorem Prover for Relation Algebras Carsten Sinz Symbolic Computation Group, WSI for Computer Science, Universität Tübingen, D-72076 Tübingen, Germany sinz@informatik.uni-tuebingen.de
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