Decidable Fragments of FirstOrder and FixedPoint Logic


 Marshall Randall
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1 From prefixvocabulary classes to guarded logics Aachen University
2 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 firstorder 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
3 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 firstorder 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
4 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 firstorder 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,...
5 Early results: decision procedures Decision algorithms for fragments of firstorder logic the monadic predicate calculus (Löwenheim 1915, Kalmár 1929) the prefix class (Bernays Schönfinkel 1928, Ramsey 1932) the prefix class (Ackermann 1928) the 2 prefix class, without equality (Gödel, Kalmár, Schütte )
6 Early results: decision procedures Decision algorithms for fragments of firstorder logic the monadic predicate calculus (Löwenheim 1915, Kalmár 1929) the prefix class (Bernays Schönfinkel 1928, Ramsey 1932) the prefix class (Ackermann 1928) the 2 prefix class, without equality (Gödel, Kalmár, Schütte ) These are often called the classical solvable cases of the decision problem.
7 Early results: reduction classes A fragment X FO is a reduction class if there is a computable function f : FO X such that ψ satisfiable f (ψ) satisfiable Early reduction classes: relational sentences without = (the pure predicate calculus) dyadic sentences (only predicates of arity 2) sentences, or relational sentences (Skolem normal form) only one binary predicate (Kalmár 1936) relational 3 sentences (Gödel 1933)
8 The undecidability of firstorder logic Theorem (Church 1936, Turing 1937) The satisfiability problem for firstorder logic is undecidable.
9 The undecidability of firstorder logic Theorem (Church 1936, Turing 1937) The satisfiability problem for firstorder logic is undecidable. Theorem (Trakhtenbrot 1950, Craig 1950) The satisfiability problem for firstorder logic on finite structures is undecidable.
10 The undecidability of firstorder logic Theorem (Church 1936, Turing 1937) The satisfiability problem for firstorder logic is undecidable. Theorem (Trakhtenbrot 1950, Craig 1950) The satisfiability problem for firstorder logic on finite structures is undecidable. = there is no sound and complete proof system for validity on finite structures
11 The undecidability of firstorder logic Theorem (Church 1936, Turing 1937) The satisfiability problem for firstorder logic is undecidable. Theorem (Trakhtenbrot 1950, Craig 1950) The satisfiability problem for firstorder 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.
12 The decision problem as a classification problem For which classes X FO is Sat(X) decidable, which are reduction classes? Similarly for satisfiablity on finite structures
13 The decision problem as a classification problem For which classes X FO is Sat(X) decidable, which are reduction classes? Similarly for satisfiablity on finite structures Which classes have the finite model property, which contain infinity axioms?
14 The decision problem as a classification problem For which classes X FO is Sat(X) decidable, which are reduction classes? Similarly for satisfiablity on finite structures Which classes have the finite model property, which contain infinity axioms? Complexity of decidable cases?
15 The decision problem as a classification problem For which classes X FO is Sat(X) decidable, which are reduction classes? Similarly for satisfiablity on finite structures Which classes have the finite model property, which contain infinity axioms? 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.
16 The decision problem as a classification problem For which classes X FO is Sat(X) decidable, which are reduction classes? Similarly for satisfiablity on finite structures Which classes have the finite model property, which contain infinity axioms? 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.
17 Prefixvocabulary classes [Π, (p 1, p 2,...), (f 1, f 2,...)] (=)
18 Prefixvocabulary classes [Π, (p 1, p 2,...), (f 1, f 2,...)] (=) Π is a word over {,,, }, describing set of quantifier prefixes
19 Prefixvocabulary classes [Π, (p 1, p 2,...), (f 1, f 2,...)] (=) Π is a word over {,,, }, describing set of quantifier prefixes p m, f m ω indicate how many relation and function symbols of arity m may occur
20 Prefixvocabulary classes [Π, (p 1, p 2,...), (f 1, f 2,...)] (=) Π is a word over {,,, }, describing set of quantifier prefixes p m, f m ω indicate how many relation and function symbols of arity m may occur presence or absence of = indicates whether the formulae may contain equality
21 Prefixvocabulary classes [Π, (p 1, p 2,...), (f 1, f 2,...)] (=) Π is a word over {,,, }, describing set of quantifier prefixes p m, f m ω indicate how many relation and function symbols of arity m may occur presence or absence of = indicates whether the formulae may contain equality Example: [, (ω, 1), all] =
22 Prefixvocabulary classes [Π, (p 1, p 2,...), (f 1, f 2,...)] (=) Π is a word over {,,, }, describing set of quantifier prefixes p m, f m ω indicate how many relation and function symbols of arity m may occur presence or absence of = indicates whether the formulae may contain equality Example: [, (ω, 1), all] = sentences x 1... x m y z 1... z n φ where φ is quantifierfree 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.
23 The complete classification: undecidable cases A: Pure predicate logic (without functions, without =) (1) [, (ω, 1), (0)] (Kahr 1962) (2) [ 3, (ω, 1), (0)] (Surányi 1959) (3) [, (0, 1), (0)] (KalmárSurányi 1950) (4) [, (0, 1), (0)] (Denton 1963) (5) [, (0, 1), (0)] (Gurevich 1966) (6) [ 3, (0, 1), (0)] (KalmárSurányi 1947) (7) [, (0, 1), (0)] (KostyrkoGenenz 1964) (8) [, (0, 1), (0)] (Surányi 1959) (9) [ 3, (0, 1), (0)] (Surányi 1959)
24 The complete classification: undecidable cases B: Classes with functions or equality (10) [, (0), (2)] = (Gurevich 1976) (11) [, (0), (0, 1)] = (Gurevich 1976) (12) [ 2, (0, 1), (1)] (Gurevich 1969) (13) [ 2, (1), (0, 1)] (Gurevich 1969) (14) [ 2, (ω, 1), (0)] = (Goldfarb 1984) (15) [ 2, (0, 1), (0)] = (Goldfarb 1984) (16) [ 2, (0, 1), (0)] = (Goldfarb 1984)
25 The complete classification: decidable cases (Exclude the trivial classes: finite prefix and finite relational vocabulary) A: Classes with the finite model property (1) [, all, (0)] = (Bernays, Schönfinkel 1928) (2) [ 2, all, (0)] (Gödel 1932, Kalmár 1933, Schütte 1934) (3) [all, (ω), (ω)] (Löb 1967, Gurevich 1969) (4) [, all, all] (Gurevich 1973) (5) [, all, all] = (Gurevich 1976) B: Classes with infinity axioms (6) [all, (ω), (1)] = (Rabin 1969) (7) [, all, (1)] = (Shelah 1977)
26 Why prefixvocabulary classes? historical reasons
27 Why prefixvocabulary classes? historical reasons simple, syntactically defined classes
28 Why prefixvocabulary classes? historical reasons simple, syntactically defined classes a finite classification can a priori be proved to exist.
29 Why prefixvocabulary classes? historical reasons simple, syntactically defined classes a finite classification can a priori be proved to exist. Classifiability Theorem (Gurevich) Let P be a property of prefix vocabulary classes that is closed under subsets and under union: X P, Y X = Y P X, Y P = X Y P Then there is a finite list MIN of prefix vocabulary classes such that X P Y MIN : Y X
30 Why prefixvocabulary classes? historical reasons simple, syntactically defined classes a finite classification can a priori be proved to exist. Classifiability Theorem (Gurevich) Let P be a property of prefix vocabulary classes that is closed under subsets and under union: X P, Y X = Y P X, Y P = X Y P Then there is a finite list MIN of prefix vocabulary classes such that Take P = {X : Sat(X) decidable}. X P Y MIN : Y X = there is a finite list of minimal undecidable prefixvocabulary classes.
31 Why prefixvocabulary classes? historical reasons simple, syntactically defined classes a finite classification can a priori be proved to exist.
32 Why prefixvocabulary classes? historical reasons simple, syntactically defined classes a finite classification can a priori be proved to exist. Good question!
33 Why prefixvocabulary classes? historical reasons simple, syntactically defined classes a finite classification can a priori be proved to exist. Good question! Modern applications of logic in computer science often lead to quite different classes. For instance: modal logics, twovariable logics, guarded logics. On the other side the restriction to firstorder logic is often too restrictive. Consider instead (fragments of) fixed point logics
34 ML: propositional modal logic Transition systems = Kripke structures = labeled graphs K = ( V, (E a ) a A, (P i ) i I ) states elements actions binary relations atomic propositions unary relations Syntax of ML: ψ ::= P i P i ψ ψ ψ ψ a ψ [a]ψ Example: P 1 a (P 2 [b]p 1 ) Semantics: [[ψ]] K = {v : K, v = ψ} = {v : ψ holds at state v in K}. K, v = a ψ [a]ψ : K, w = ψ for some all w with (v, w) E a
35 ML as a fragment of firstorder logic There is standard translation, taking every ψ ML to a formula ψ (x) FO such that K, v = ψ K = ψ (v) P i P i x a ψ y(e a xy ψ (y)) [a]ψ y(e a xy ψ (y))
36 Properties of modal logic Good algorithmic properties:  Sat(ML) is decidable and PSPACEcomplete  efficient model checking: time O( ψ K )  automatabased algorithms
37 Properties of modal logic Good algorithmic properties:  Sat(ML) is decidable and PSPACEcomplete  efficient model checking: time O( ψ K )  automatabased algorithms Interesting modeltheoretic properties:  invariance under bisimulation: K, v = ψ, K, v K, v = K, v = ψ in fact ML is the bisimulationinvariant fragment of firstorder logic  ML has the tree model property  ML has the finite model property ...
38 Limitations of propositional modal logic ML has very limited expressive power  no equality  no unbounded quantification  no possibility to define new transition relations, not even on the quantifierfree level  no counting mechanism  no recursion mechanism: reachability problems, game problems, etc. are not expressible
39 Limitations of propositional modal logic ML has very limited expressive power  no equality  no unbounded quantification  no possibility to define new transition relations, not even on the quantifierfree level  no counting mechanism  no recursion mechanism: reachability problems, game problems, etc. are not expressible Note: The absence of a recursion mechanism is also a limitation of FO For numerous applications, in particular for reasoning about computation, reachability problems and game problems are essential. = numerous logics that extend ML by recursion mechanisms
40 Computation tree logic CTL ML + path quantification + temporal operators on paths CTL is a very important logic for hardware verification. Good balance between expressive power and algorithmic manageability A P U Q ( paths) (P until Q)
41 Computation tree logic CTL ML + path quantification + temporal operators on paths CTL is a very important logic for hardware verification. Good balance between expressive power and algorithmic manageability A P U Q ( paths) (P until Q) E G P ( path) always P A F P ( paths) eventually P
42 Computation tree logic CTL ML + path quantification + temporal operators on paths CTL is a very important logic for hardware verification. Good balance between expressive power and algorithmic manageability A P U Q ( paths) (P until Q) E G P ( path) always P A F P ( paths) eventually P A G A F P ( paths) always ( paths) eventually P ( paths) infinitely often P
43 L µ : modal µcalculus ML + least and greatest fixed points For any definable monotone relational operator F φ : X {w : φ(x) true at w} make also the least and the greatest fixed point of F φ definable: K, w = µx.φ w {X : F φ (X) = X} (least fixed point) K, w = νx.φ w {X : F φ (X) = X} (greatest fixed point)
44 L µ : Examples K, w = νx. a X there is an infinite apath from w in K K, w = µx. P [a]x every infinite apath from w eventually hits P
45 L µ : Examples K, w = νx. a X there is an infinite apath from w in K K, w = µx. P [a]x every infinite apath from w eventually hits P Two player game on graph G = (V, E). Players alternate, moves along edges, who cannot move, loses. Player 0 has winning strategy for game G from position w G, w = µx. X
46 L µ : Examples K, w = νx. a X there is an infinite apath from w in K K, w = µx. P [a]x every infinite apath from w eventually hits P Two player game on graph G = (V, E). Players alternate, moves along edges, who cannot move, loses. Player 0 has winning strategy for game G from position w G, w = µx. X K, w = νx µy. ((P X) Y) on some path from w, P occurs infinitely often
47 Importance of CTL and modal µcalculus ML CTL L µ µcalculus encompasses most of the popular logics used in hardware verification: LTL, CTL, CTL, PDL,..., and also many logics from other fields: game logic, description logics, etc. reasonably good algorithmic properties:  satisfiability problem decidable (EXPTIMEcomplete)  efficient model checking for CTL and other important fragments of µcalculus  automatabased algorithms nice modeltheoretic properties:  finite model property  tree model property L µ is the bisimulationinvariant fragment of MSO
48 Summery: properties of modal logics The basic modal logic ML has good algorithmic and modeltheoretic properties, but for most applications in computer science, it is not expressive enough
49 Summery: properties of modal logics The basic modal logic ML has good algorithmic and modeltheoretic properties, but for most applications in computer science, it is not expressive enough There are numerous modal logics, that extend ML by recursion mechanisms and other features. These extensions are practically important because the provide the necessary expressive power for applications, while preserving the good algorithmic properties. Of particular importance for theoretical studies: modal µcalculus (modal fixed point logic)
50 Summery: properties of modal logics The basic modal logic ML has good algorithmic and modeltheoretic properties, but for most applications in computer science, it is not expressive enough There are numerous modal logics, that extend ML by recursion mechanisms and other features. These extensions are practically important because the provide the necessary expressive power for applications, while preserving the good algorithmic properties. Of particular importance for theoretical studies: modal µcalculus (modal fixed point logic) Question. (Vardi 96) Why have modal logics so robust decidability properties?
51 Bounded variable logics FO k : relational firstorder logic with only k variables
52 Bounded variable logics FO k : relational firstorder logic with only k variables we may reuse variables: there is a path of length 17 FO 2 x y(exy x(eyx y(exy yexy ) )))
53 Bounded variable logics FO k : relational firstorder logic with only k variables we may reuse variables: there is a path of length 17 FO 2 x y(exy x(eyx y(exy yexy ) ))) Note: instead of the number of variables, we can restrict the width width(ψ): maximal number of free variables in subformulae of ψ Proposition. FO k {ψ FO : width(ψ) k} Proposition. Sat(FO k ) is undecidable for all k 3 indeed, even the prefix class is undecidable Question: What about FO 2?
54 Decidability of FO 2 Theorem (Mortimer 75) FO 2 has the finite model property. Mortimer s proof implies: every satisfiable ψ FO 2 has a model with at most 2 2O( ψ ) elements. = Sat(FO 2 ) 2NEXPTIME
55 Decidability of FO 2 Theorem (Mortimer 75) FO 2 has the finite model property. Mortimer s proof implies: every satisfiable ψ FO 2 has a model with at most 2 2O( ψ ) elements. = Sat(FO 2 ) 2NEXPTIME Theorem (Lewis 80, Fürer 84) Sat(FO 2 ) is NEXPTIMEhard. holds even for sentences of form x y α x y β (α and β quantifierfree), with only monadic predicates
56 Decidability of FO 2 Theorem (Mortimer 75) FO 2 has the finite model property. Mortimer s proof implies: every satisfiable ψ FO 2 has a model with at most 2 2O( ψ ) elements. = Sat(FO 2 ) 2NEXPTIME Theorem (Lewis 80, Fürer 84) Sat(FO 2 ) is NEXPTIMEhard. holds even for sentences of form x y α x y β (α and β quantifierfree), with only monadic predicates Theorem (Grädel, Kolaitis, Vardi 97) Every satisfiable ψ FO 2 has a model with at most 2 O( ψ ) elements. Corollary Sat(FO 2 ) is NEXPTIMEcomplete
57 Modal logic and twovariable logic ML is a fragment of FO 2. The standard translation, taking ψ ML to ψ (x) FO does not need more than two variables: P i P i x a ψ y(e a xy ψ (y)) [a]ψ y(e a xy ψ (y))
58 Modal logic and twovariable logic ML is a fragment of FO 2. The standard translation, taking ψ ML to ψ (x) FO does not need more than two variables: P i P i x a ψ y(e a xy ψ (y)) [a]ψ y(e a xy ψ (y)) Traditionally the embedding into twovariable logic has been used as an explanation for the good algorithmic properties of modal logics. Is this explanation convincing?
59 Modal logic and twovariable logic ML is a fragment of FO 2. The standard translation, taking ψ ML to ψ (x) FO does not need more than two variables: P i P i x a ψ y(e a xy ψ (y)) [a]ψ y(e a xy ψ (y)) Traditionally the embedding into twovariable logic has been used as an explanation for the good algorithmic properties of modal logics. Is this explanation convincing? Answer: No!!!
60 Stronger twovariable logics Challenge: Extend FO 2 in a similar way, as CTL and µcalculus extend propositional modal logic. Are the resulting logics still decidable? FO 2?? ML CTL L µ
61 Stronger twovariable logics TC 2 = FO 2 + transitive closures (TC φ)(x, y) CL 2 = least common natural extension of CTL and FO 2 FO 2 + β(x, y) φ + [β(x, y)] φ LFP 2 = FO 2 monadic least and greatest fixed points [lfp Tx. φ(t, x)](x) Theorem. (Grädel, Otto, Rosen) Sat(TC 2 ), Sat(CL 2 ), and Sat(LFP 2 ) are highly undecidable (Σ 1 1hard).
62 Twovariable logics FO 2 CL 2 LFP 2 TC 2. ML CTL L µ Twovariable logics do not have the robust decidability properties of modal logics. Is there another explanation?
63 The guarded fragment of firstorder logic (GF) Fragment of firstorder logic with only guarded quantification y(α(x, y) φ(x, y)) y(α(x, y) φ(x, y)) with guards α : atomic formulae containing all free variables of φ
64 The guarded fragment of firstorder logic (GF) Fragment of firstorder logic with only guarded quantification y(α(x, y) φ(x, y)) y(α(x, y) φ(x, y)) with guards α : atomic formulae containing all free variables of φ Generalizes modal quantification: ML GF FO a φ y(e a xy φ(y)) [a]φ y(e a xy φ(y))
65 Modeltheoretic and algorithmic properties of GF Satisfiability for GF is decidable (Andréka, van Benthem, Németi) GF has finite model property (Andréka, Hodkinson, Németi; Grädel) GF has (generalized) tree model property: every satisfiable formula has model of small tree width Sat(GF) is 2EXPTIMEcomplete and EXPTIMEcomplete for formulae of bounded width (Grädel) (Grädel) Guarded logics have small model checking games: G(A, ψ) = O( ψ A ) = efficient gamebased model checking algorithms GF is invariant under guarded bisimulation
66 A thesis and a challenge Thesis: (Andréka, van Benthem, Németi) The guarded nature of quantification in modal logics is the real reason for their good algorithmic and modeltheoretic properties.
67 A thesis and a challenge Thesis: (Andréka, van Benthem, Németi) The guarded nature of quantification in modal logics is the real reason for their good algorithmic and modeltheoretic properties. Results on GF provide some evidence for this thesis. Real test: algorithmic properties of stronger guarded logics than GF. Challenge: Extend GF in a similar way, as µcalculus extends ML. Is the resulting guarded fixedpoint logic still decidable? L µ? LFP ML GF FO
68 A thesis and a challenge Thesis: (Andréka, van Benthem, Németi) The guarded nature of quantification in modal logics is the real reason for their good algorithmic and modeltheoretic properties. Results on GF provide some evidence for this thesis. Real test: algorithmic properties of stronger guarded logics than GF. Challenge: Extend GF in a similar way, as µcalculus extends ML. Is the resulting guarded fixedpoint logic still decidable? L µ µgf LFP ML GF FO
69 Guarded fixedpoint logic µgf = GF + least and greatest fixed points
70 Guarded fixedpoint logic µgf = GF + least and greatest fixed points For every formula ψ(t, x 1... x k ), where T is a kary relation variable, occuring only positive in ψ, build formulae [lfp Tx. ψ](x) and [gfp Tx. ψ](x)
71 Guarded fixedpoint logic µgf = GF + least and greatest fixed points For every formula ψ(t, x 1... x k ), where T is a kary relation variable, occuring only positive in ψ, build formulae [lfp Tx. ψ](x) and [gfp Tx. ψ](x) On every structure A, ψ(t, x) defines monotone operator ψ A : P(A k ) P(A k ) T {a : (A, T) = ψ(t, a)} ψ A has a least fixed point lfp(ψ A ) and a greatest fixed point gfp(ψ A ) A = [lfp Tx. ψ(t, x)](a) : a lfp(ψ A ) A = [gfp Tx. ψ(t, x)](a) : a gfp(ψ A )
72 Guarded fixed point logic Example. ψ := xyfxy ( xy. Fxy) zfyz ( xy. Fxy) [lfp Wy. ( x. Fxy)Wx] } {{ } (y) {y : there is no infinite chain y y 1 y 2 } ψ is satisfiable: (ω, <) = ψ ψ has no finite models Proposition. µgf does not have the finite model property.
73 Decidability of guarded fixed point logic Theorem. (Grädel, Walukiewicz) Sat(µGF) 2EXPTIMEcomplete, and EXPTIMEcomplete for formulae of bounded width. same complexity bounds as for GF: no penalty for fixed points! Proof. Relies on generalized tree model property: Every satisfiable ψ µgf has a model of small tree width Reduction of Sat(µGF) to emptiness problem for certain automata (alternating twoway tree automata with parity acceptance condition).
74 Final remarks The world of guarded logics is much richer than shown in this talk:  more liberal guardedness conditions: loosely guarded, clique guarded, action guarded, packed logics  guarded fragments of other logics, e.g. GSO, Datalog LITE,...
75 Final remarks The world of guarded logics is much richer than shown in this talk:  more liberal guardedness conditions: loosely guarded, clique guarded, action guarded, packed logics  guarded fragments of other logics, e.g. GSO, Datalog LITE,... While modal logics talk only about graphs, guarded logics are applicable to arbitrary relational structures. Many of the mathematical tools used for analysing modal logics have their counterpart in the world of guarded logics: bisimulation, automata,...
76 Final remarks The world of guarded logics is much richer than shown in this talk:  more liberal guardedness conditions: loosely guarded, clique guarded, action guarded, packed logics  guarded fragments of other logics, e.g. GSO, Datalog LITE,... While modal logics talk only about graphs, guarded logics are applicable to arbitrary relational structures. Many of the mathematical tools used for analysing modal logics have their counterpart in the world of guarded logics: bisimulation, automata,... Conclusion. Guarded logics provide a general and versatile family of logical systems that generalise, and to a large extent explain the good algorithmic and modeltheoretic properties of modal logics.
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