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Decidable Fragments of First-Order and Fixed-Point Logic from Prefix-Vocabulary Classes to Guarded Logics

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Decidable Fragments of First-Order and Fixed-Point Logic from Prefix-Vocabulary Classes to Guarded Logics Decidable Fragments of First-Order and Fixed-Point Logic From prefix-vocabulary classes to guarded logics Erich Grädel [email protected]. Aachen University The classical decision problem part of Hilbert's formalist programme for the foundations of mathematics in modern terminology: find an algorithm that solves the satisfiability problem for first-order logic D. Hilbert and W. Ackermann: “Das Entscheidungsproblem ist gelöst, wenn man ein Verfahren kennt, das bei einem vorgelegten logischen Ausdruck durch endlich viele Operationen die Entscheidung über die Allgemeingültigkeit bzw. Erfüllbarkeit erlaubt. (: : :) Das Entscheidungsproblem muss als das Hauptproblem der mathematischen Logik bezeichnet werden.” Grundzüge der theoretischen Logik (1928), pp 73ff Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic The classical decision problem part of Hilbert's formalist programme for the foundations of mathematics in modern terminology: find an algorithm that solves the satisfiability problem for first-order logic D. Hilbert and W. Ackermann: “The decision problem is solved when we know a procedure that allows, for any given logical expression, to decide by finitely many operations its validity or satisfiability. (: : :) The decision problem must be considered the main problem of mathematical logic.” Grundzüge der theoretischen Logik (1928), pp 73ff Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic The classical decision problem part of Hilbert's formalist programme for the foundations of mathematics in modern terminology: find an algorithm that solves the satisfiability problem for first-order logic D. Hilbert and W. Ackermann: “The decision problem is solved when we know a procedure that allows, for any given logical expression, to decide by finitely many operations its validity or satisfiability. (: : :) The decision problem must be considered the main problem of mathematical logic.” Grundzüge der theoretischen Logik (1928), pp 73ff Similar statements by many other logicians, including Bernays, Schönfinkel, Herbrand, von Neumann, . Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic Early results: decision procedures Decision algorithms for fragments of first-order logic ² the monadic predicate calculus (Löwenheim 1915, Kalmár 1929) ² the prefix class 9¤8¤ (Bernays–Schönfinkel 1928, Ramsey 1932) ² the 9¤89¤-prefix class (Ackermann 1928) ² the 9¤829¤-prefix class, without equality (Gödel, Kalmár, Schütte 1932-1934) Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic Early results: decision procedures Decision algorithms for fragments of first-order logic ² the monadic predicate calculus (Löwenheim 1915, Kalmár 1929) ² the prefix class 9¤8¤ (Bernays–Schönfinkel 1928, Ramsey 1932) ² the 9¤89¤-prefix class (Ackermann 1928) ² the 9¤829¤-prefix class, without equality (Gödel, Kalmár, Schütte 1932-1934) These are often called the classical solvable cases of the decision problem. Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic Early results: reduction classes A fragment X ⊆ FO is a reduction class if there is a computable function f : FO ! X such that satisfiable () f ( ) satisfiable Early reduction classes: ² relational sentences without = (the pure predicate calculus) ² dyadic sentences (only predicates of arity 2) ² 8¤-sentences, or relational 8¤9¤-sentences (Skolem normal form) ² only one binary predicate (Kalmár 1936) ² relational 839¤-sentences (Gödel 1933) Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic The undecidability of first-order logic Theorem (Church 1936, Turing 1937) The satisfiability problem for first-order logic is undecidable. Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic The undecidability of first-order logic Theorem (Church 1936, Turing 1937) The satisfiability problem for first-order logic is undecidable. Theorem (Trakhtenbrot 1950, Craig 1950) The satisfiability problem for first-order logic on finite structures is undecidable. Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic The undecidability of first-order logic Theorem (Church 1936, Turing 1937) The satisfiability problem for first-order logic is undecidable. Theorem (Trakhtenbrot 1950, Craig 1950) The satisfiability problem for first-order logic on finite structures is undecidable. =) there is no sound and complete proof system for validity on finite structures Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic The undecidability of first-order logic Theorem (Church 1936, Turing 1937) The satisfiability problem for first-order logic is undecidable. Theorem (Trakhtenbrot 1950, Craig 1950) The satisfiability problem for first-order logic on finite structures is undecidable. =) there is no sound and complete proof system for validity on finite structures After the negative solution, the decision problem did not disappear, but became a classification problem. Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic The decision problem as a classification problem ² For which classes X ⊆ FO is Sat(X) decidable, which are reduction classes ? ² Similarly for satisfiablity on finite structures Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic The decision problem as a classification problem ² For which classes X ⊆ FO is Sat(X) decidable, which are reduction classes ? ² Similarly for satisfiablity on finite structures ² Which classes have the finite model property, which contain infinity axioms ? Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic The decision problem as a classification problem ² For which classes X ⊆ FO is Sat(X) decidable, which are reduction classes ? ² Similarly for satisfiablity on finite structures ² Which classes have the finite model property, which contain infinity axioms ? ² Complexity of decidable cases ? Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic The decision problem as a classification problem ² For which classes X ⊆ FO is Sat(X) decidable, which are reduction classes ? ² Similarly for satisfiablity on finite structures ² Which classes have the finite model property, which contain infinity axioms ? ² Complexity of decidable cases ? What kind of classes X ⊆ FO should be considered? There are continuum many such classes, one cannot study all of them. A direction is needed. Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic The decision problem as a classification problem ² For which classes X ⊆ FO is Sat(X) decidable, which are reduction classes ? ² Similarly for satisfiablity on finite structures ² Which classes have the finite model property, which contain infinity axioms ? ² Complexity of decidable cases ? What kind of classes X ⊆ FO should be considered? There are continuum many such classes, one cannot study all of them. A direction is needed. In view of the early results, most attention (in logic) was given to classes determined by quantifier prefix and vocabulary, i.e. the number and arity of relation and function symbols. Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic Prefix-vocabulary classes [Π; (p1; p2; : : :); (f1; f2; : : :)](=) Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic Prefix-vocabulary classes [Π; (p1; p2; : : :); (f1; f2; : : :)](=) ² Π is a word over f9; 8; 9¤; 8¤g, describing set of quantifier prefixes Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic Prefix-vocabulary classes [Π; (p1; p2; : : :); (f1; f2; : : :)](=) ² Π is a word over f9; 8; 9¤; 8¤g, describing set of quantifier prefixes ² pm; fm · ! indicate how many relation and function symbols of arity m may occur Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic Prefix-vocabulary classes [Π; (p1; p2; : : :); (f1; f2; : : :)](=) ² Π is a word over f9; 8; 9¤; 8¤g, describing set of quantifier prefixes ² pm; fm · ! indicate how many relation and function symbols of arity m may occur ² presence or absence of = indicates whether the formulae may contain equality Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic Prefix-vocabulary classes [Π; (p1; p2; : : :); (f1; f2; : : :)](=) ² Π is a word over f9; 8; 9¤; 8¤g, describing set of quantifier prefixes ² pm; fm · ! indicate how many relation and function symbols of arity m may occur ² presence or absence of = indicates whether the formulae may contain equality ¤ ¤ Example: [9 89 ; (!; 1); all]= Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic Prefix-vocabulary classes [Π; (p1; p2; : : :); (f1; f2; : : :)](=) ² Π is a word over f9; 8; 9¤; 8¤g, describing set of quantifier prefixes ² pm; fm · ! indicate how many relation and function symbols of arity m may occur ² presence or absence of = indicates whether the formulae may contain equality ¤ ¤ Example: [9 89 ; (!; 1); all]= sentences 9x1 : : : 9xm8y9z1 : : : 9zn' where ' is quantifier-free and – contains at most one binary predicate, and no predicates of arity ¸ 3, – may contain any number of monadic predicates, – may contain any number of function symbols of any arity, – may contain equality. Erich Grädel Decidable Fragments of First-Order and Fixed-Point Logic The complete classification: undecidable cases A: Pure predicate logic (without functions, without =) (1) [898; (!; 1); (0)] (Kahr 1962) (2) [839; (!; 1); (0)] (Surányi 1959) (3) [8¤9; (0; 1); (0)] (Kalmár-Surányi 1950) (4) [898¤; (0; 1); (0)] (Denton 1963) (5) [8989¤; (0; 1); (0)] (Gurevich 1966) (6) [839¤; (0; 1); (0)] (Kalmár-Surányi 1947) (7) [89¤8; (0; 1); (0)] (Kostyrko-Genenz 1964) (8) [9¤898;
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