Decidable Fragments of FirstOrder and FixedPoint Logic


 Marshall Randall
 1 years ago
 Views:
Transcription
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.
FixedPoint Logics and Computation
1 FixedPoint Logics and Computation Symposium on the Unusual Effectiveness of Logic in Computer Science University of Cambridge 2 Mathematical Logic Mathematical logic seeks to formalise the process of
More informationAlgorithmic Software Verification
Algorithmic Software Verification (LTL Model Checking) Azadeh Farzan What is Verification Anyway? Proving (in a formal way) that program satisfies a specification written in a logical language. Formal
More informationFirstOrder Logics and Truth Degrees
FirstOrder Logics and Truth Degrees George Metcalfe Mathematics Institute University of Bern LATD 2014, Vienna Summer of Logic, 1519 July 2014 George Metcalfe (University of Bern) FirstOrder Logics
More informationPath Querying on Graph Databases
Path Querying on Graph Databases Jelle Hellings Hasselt University and transnational University of Limburg 1/38 Overview Graph Databases Motivation Walk Logic Relations with FO and MSO Relations with CTL
More informationON FUNCTIONAL SYMBOLFREE LOGIC PROGRAMS
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical and Mathematical Sciences 2012 1 p. 43 48 ON FUNCTIONAL SYMBOLFREE LOGIC PROGRAMS I nf or m at i cs L. A. HAYKAZYAN * Chair of Programming and Information
More informationCHAPTER 7 GENERAL PROOF SYSTEMS
CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes
More informationUndecidable FirstOrder Theories of Affine Geometries
Undecidable FirstOrder Theories of Affine Geometries Antti Kuusisto University of Tampere Joint work with Jeremy Meyers (Stanford University) and Jonni Virtema (University of Tampere) Tarski s Geometry
More informationSummary Last Lecture. Automated Reasoning. Outline of the Lecture. Definition sequent calculus. Theorem (Normalisation and Strong Normalisation)
Summary Summary Last Lecture sequent calculus Automated Reasoning Georg Moser Institute of Computer Science @ UIBK Winter 013 (Normalisation and Strong Normalisation) let Π be a proof in minimal logic
More informationominimality and Uniformity in n 1 Graphs
ominimality and Uniformity in n 1 Graphs Reid Dale July 10, 2013 Contents 1 Introduction 2 2 Languages and Structures 2 3 Definability and Tame Geometry 4 4 Applications to n 1 Graphs 6 5 Further Directions
More informationMATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. III  Logic and Computer Science  Phokion G. Kolaitis
LOGIC AND COMPUTER SCIENCE Phokion G. Kolaitis Computer Science Department, University of California, Santa Cruz, CA 95064, USA Keywords: algorithm, Armstrong s axioms, complete problem, complexity class,
More informationSoftware Modeling and Verification
Software Modeling and Verification Alessandro Aldini DiSBeF  Sezione STI University of Urbino Carlo Bo Italy 34 February 2015 Algorithmic verification Correctness problem Is the software/hardware system
More informationCODING TRUE ARITHMETIC IN THE MEDVEDEV AND MUCHNIK DEGREES
CODING TRUE ARITHMETIC IN THE MEDVEDEV AND MUCHNIK DEGREES PAUL SHAFER Abstract. We prove that the firstorder theory of the Medvedev degrees, the firstorder theory of the Muchnik degrees, and the thirdorder
More informationAlgebraic Recognizability of Languages
of Languages LaBRI, Université Bordeaux1 and CNRS MFCS Conference, Prague, August 2004 The general problem Problem: to specify and analyse infinite sets by finite means The general problem Problem: to
More informationA Propositional Dynamic Logic for CCS Programs
A Propositional Dynamic Logic for CCS Programs Mario R. F. Benevides and L. Menasché Schechter {mario,luis}@cos.ufrj.br Abstract This work presents a Propositional Dynamic Logic in which the programs are
More informationDescription Logics for Conceptual Data Modeling in UML
Description Logics for Conceptual Data Modeling in UML Diego Calvanese, Giuseppe De Giacomo Dipartimento di Informatica e Sistemistica Università di Roma La Sapienza ESSLLI 2003 Vienna, August 18 22, 2003
More informationFormal Verification and Lineartime Model Checking
Formal Verification and Lineartime Model Checking Paul Jackson University of Edinburgh Automated Reasoning 21st and 24th October 2013 Why Automated Reasoning? Intellectually stimulating and challenging
More informationModel Checking II Temporal Logic Model Checking
1/32 Model Checking II Temporal Logic Model Checking Edmund M Clarke, Jr School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 2/32 Temporal Logic Model Checking Specification Language:
More information(LMCS, p. 317) V.1. First Order Logic. This is the most powerful, most expressive logic that we will examine.
(LMCS, p. 317) V.1 First Order Logic This is the most powerful, most expressive logic that we will examine. Our version of firstorder logic will use the following symbols: variables connectives (,,,,
More informationTemporal Logics. Computation Tree Logic
Temporal Logics CTL: definition, relationship between operators, adequate sets, specifying properties, safety/liveness/fairness Modeling: sequential, concurrent systems; maximum parallelism/interleaving
More informationTheory of Automated Reasoning An Introduction. AnttiJuhani Kaijanaho
Theory of Automated Reasoning An Introduction AnttiJuhani Kaijanaho Intended as compulsory reading for the Spring 2004 course on Automated Reasononing at Department of Mathematical Information Technology,
More informationTesting LTL Formula Translation into Büchi Automata
Testing LTL Formula Translation into Büchi Automata Heikki Tauriainen and Keijo Heljanko Helsinki University of Technology, Laboratory for Theoretical Computer Science, P. O. Box 5400, FIN02015 HUT, Finland
More informationComputational Logic and Cognitive Science: An Overview
Computational Logic and Cognitive Science: An Overview Session 1: Logical Foundations Technical University of Dresden 25th of August, 2008 University of Osnabrück Who we are Helmar Gust Interests: Analogical
More informationScalable Automated Symbolic Analysis of Administrative RoleBased Access Control Policies by SMT solving
Scalable Automated Symbolic Analysis of Administrative RoleBased Access Control Policies by SMT solving Alessandro Armando 1,2 and Silvio Ranise 2, 1 DIST, Università degli Studi di Genova, Italia 2 Security
More informationA first step towards modeling semistructured data in hybrid multimodal logic
A first step towards modeling semistructured data in hybrid multimodal logic Nicole Bidoit * Serenella Cerrito ** Virginie Thion * * LRI UMR CNRS 8623, Université Paris 11, Centre d Orsay. ** LaMI UMR
More informationDEFINABLE TYPES IN PRESBURGER ARITHMETIC
DEFINABLE TYPES IN PRESBURGER ARITHMETIC GABRIEL CONANT Abstract. We consider the first order theory of (Z, +,
More informationSoftware Verification and Testing. Lecture Notes: Temporal Logics
Software Verification and Testing Lecture Notes: Temporal Logics Motivation traditional programs (whether terminating or nonterminating) can be modelled as relations are analysed wrt their input/output
More informationComposing Schema Mappings: SecondOrder Dependencies to the Rescue
Composing Schema Mappings: SecondOrder Dependencies to the Rescue Ronald Fagin IBM Almaden Research Center fagin@almaden.ibm.com Phokion G. Kolaitis UC Santa Cruz kolaitis@cs.ucsc.edu WangChiew Tan UC
More informationComposing Schema Mappings: An Overview
Composing Schema Mappings: An Overview Phokion G. Kolaitis UC Santa Scruz & IBM Almaden Joint work with Ronald Fagin, Lucian Popa, and WangChiew Tan The Data Interoperability Challenge Data may reside
More informationlogic language, static/dynamic models SAT solvers Verified Software Systems 1 How can we model check of a program or system?
5. LTL, CTL Last part: Alloy logic language, static/dynamic models SAT solvers Today: Temporal Logic (LTL, CTL) Verified Software Systems 1 Overview How can we model check of a program or system? Modeling
More informationhttp://aejm.ca Journal of Mathematics http://rema.ca Volume 1, Number 1, Summer 2006 pp. 69 86
Atlantic Electronic http://aejm.ca Journal of Mathematics http://rema.ca Volume 1, Number 1, Summer 2006 pp. 69 86 AUTOMATED RECOGNITION OF STUTTER INVARIANCE OF LTL FORMULAS Jeffrey Dallien 1 and Wendy
More informationGoal of the Talk Theorem The class of languages recognisable by T coalgebra automata is closed under taking complements.
Complementation of Coalgebra Automata Christian Kissig (University of Leicester) joint work with Yde Venema (Universiteit van Amsterdam) 07 Sept 2009 / Universitá degli Studi di Udine / CALCO 2009 Goal
More informationUniversität Stuttgart Fakultät Informatik, Elektrotechnik und Informationstechnik
Universität Stuttgart Fakultät Informatik, Elektrotechnik und Informationstechnik Infinite State ModelChecking of Propositional Dynamic Logics Stefan Göller and Markus Lohrey Report Nr. 2006/04 Institut
More informationCS Master Level Courses and Areas COURSE DESCRIPTIONS. CSCI 521 RealTime Systems. CSCI 522 High Performance Computing
CS Master Level Courses and Areas The graduate courses offered may change over time, in response to new developments in computer science and the interests of faculty and students; the list of graduate
More informationIntroduction to Software Verification
Introduction to Software Verification Orna Grumberg Lectures Material winter 201314 Lecture 4 5.11.13 Model Checking Automated formal verification: A different approach to formal verification Model Checking
More informationDevelopment of dynamically evolving and selfadaptive software. 1. Background
Development of dynamically evolving and selfadaptive software 1. Background LASER 2013 Isola d Elba, September 2013 Carlo Ghezzi Politecnico di Milano DeepSE Group @ DEIB 1 Requirements Functional requirements
More informationSchedule. Logic (master program) Literature & Online Material. gic. Time and Place. Literature. Exercises & Exam. Online Material
OLC mputational gic Schedule Time and Place Thursday, 8:15 9:45, HS E Logic (master program) Georg Moser Institute of Computer Science @ UIBK week 1 October 2 week 8 November 20 week 2 October 9 week 9
More informationSJÄLVSTÄNDIGA ARBETEN I MATEMATIK
SJÄLVSTÄNDIGA ARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET Automated Theorem Proving av Tom Everitt 2010  No 8 MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET, 106 91 STOCKHOLM
More informationDegrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets.
Degrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets. Logic is the study of reasoning. A language of propositions is fundamental to this study as well as true
More informationMonitoring Metric Firstorder Temporal Properties
Monitoring Metric Firstorder Temporal Properties DAVID BASIN, FELIX KLAEDTKE, SAMUEL MÜLLER, and EUGEN ZĂLINESCU, ETH Zurich Runtime monitoring is a general approach to verifying system properties at
More informationStatic analysis of parity games: alternating reachability under parity
8 January 2016, DTU Denmark Static analysis of parity games: alternating reachability under parity Michael Huth, Imperial College London Nir Piterman, University of Leicester Jim HuanPu Kuo, Imperial
More informationA CTLBased Logic for Program Abstractions
A CTLBased Logic for Program Abstractions Martin Lange 1 and Markus Latte 2 1 Dept. of Computer Science, University of Kassel, Germany 2 Dept. of Computer Science, LudwigMaximiliansUniversity Munich,
More informationModel Checking: An Introduction
Announcements Model Checking: An Introduction Meeting 2 Office hours M 1:30pm2:30pm W 5:30pm6:30pm (after class) and by appointment ECOT 621 Moodle problems? Fundamentals of Programming Languages CSCI
More informationWhich Semantics for Neighbourhood Semantics?
Which Semantics for Neighbourhood Semantics? Carlos Areces INRIA Nancy, Grand Est, France Diego Figueira INRIA, LSV, ENS Cachan, France Abstract In this article we discuss two alternative proposals for
More informationBounded Treewidth in Knowledge Representation and Reasoning 1
Bounded Treewidth in Knowledge Representation and Reasoning 1 Reinhard Pichler Institut für Informationssysteme Arbeitsbereich DBAI Technische Universität Wien Luminy, October 2010 1 Joint work with G.
More informationThis asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements.
3. Axioms of Set theory Before presenting the axioms of set theory, we first make a few basic comments about the relevant first order logic. We will give a somewhat more detailed discussion later, but
More informationIntroducing Formal Methods. Software Engineering and Formal Methods
Introducing Formal Methods Formal Methods for Software Specification and Analysis: An Overview 1 Software Engineering and Formal Methods Every Software engineering methodology is based on a recommended
More informationAutomated Theorem Proving  summary of lecture 1
Automated Theorem Proving  summary of lecture 1 1 Introduction Automated Theorem Proving (ATP) deals with the development of computer programs that show that some statement is a logical consequence of
More informationSyntax and Semantics for Business Rules
Syntax and Semantics for Business Rules Xiaofan Liu 1 2, Natasha Alechina 1, and Brian Logan 1 1 School of Computer Science, University of Nottingham, Nottingham, NG8 1BB, UK 2 School of Computer and Communication,
More informationFoundational Proof Certificates
An application of proof theory to computer science INRIASaclay & LIX, École Polytechnique CUSO Winter School, Proof and Computation 30 January 2013 Can we standardize, communicate, and trust formal proofs?
More informationDECISION PROBLEMS OVER INFINITE GRAPHS HIGHERORDER PUSHDOWN SYSTEMS AND SYNCHRONIZED PRODUCTS
DECISION PROBLEMS OVER INFINITE GRAPHS HIGHERORDER PUSHDOWN SYSTEMS AND SYNCHRONIZED PRODUCTS Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der RheinischWestfälischen Technischen
More informationStatic Program Transformations for Efficient Software Model Checking
Static Program Transformations for Efficient Software Model Checking Shobha Vasudevan Jacob Abraham The University of Texas at Austin Dependable Systems Large and complex systems Software faults are major
More informationSpecification and Analysis of Contracts Lecture 1 Introduction
Specification and Analysis of Contracts Lecture 1 Introduction Gerardo Schneider gerardo@ifi.uio.no http://folk.uio.no/gerardo/ Department of Informatics, University of Oslo SEFM School, Oct. 27  Nov.
More informationRigorous Software Development CSCIGA 3033009
Rigorous Software Development CSCIGA 3033009 Instructor: Thomas Wies Spring 2013 Lecture 11 Semantics of Programming Languages Denotational Semantics Meaning of a program is defined as the mathematical
More informationCertain Answers as Objects and Knowledge
Certain Answers as Objects and Knowledge Leonid Libkin School of Informatics, University of Edinburgh Abstract The standard way of answering queries over incomplete databases is to compute certain answers,
More informationThe Model Checker SPIN
The Model Checker SPIN Author: Gerard J. Holzmann Presented By: Maulik Patel Outline Introduction Structure Foundation Algorithms Memory management Example/Demo SPINIntroduction Introduction SPIN (Simple(
More informationAN INTUITIONISTIC EPISTEMIC LOGIC FOR ASYNCHRONOUS COMMUNICATION. Yoichi Hirai. A Master Thesis
AN INTUITIONISTIC EPISTEMIC LOGIC FOR ASYNCHRONOUS COMMUNICATION by Yoichi Hirai A Master Thesis Submitted to the Graduate School of the University of Tokyo on February 10, 2010 in Partial Fulfillment
More informationNondeterministic Semantics and the Undecidability of Boolean BI
1 Nondeterministic Semantics and the Undecidability of Boolean BI DOMINIQUE LARCHEYWENDLING, LORIA CNRS, Nancy, France DIDIER GALMICHE, LORIA Université Henri Poincaré, Nancy, France We solve the open
More informationAikaterini Marazopoulou
Imperial College London Department of Computing Tableau Compiled Labelled Deductive Systems with an application to Description Logics by Aikaterini Marazopoulou Submitted in partial fulfilment of the requirements
More informationTruth as Modality. Amsterdam Workshop on Truth 13 March 2013 truth be told again. Theodora Achourioti. ILLC University of Amsterdam
Truth as Modality Amsterdam Workshop on Truth 13 March 2013 truth be told again Theodora Achourioti ILLC University of Amsterdam Philosophical preliminaries i The philosophical idea: There is more to truth
More informationReading 13 : Finite State Automata and Regular Expressions
CS/Math 24: Introduction to Discrete Mathematics Fall 25 Reading 3 : Finite State Automata and Regular Expressions Instructors: Beck Hasti, Gautam Prakriya In this reading we study a mathematical model
More informationOn the Modeling and Verification of SecurityAware and ProcessAware Information Systems
On the Modeling and Verification of SecurityAware and ProcessAware Information Systems 29 August 2011 What are workflows to us? Plans or schedules that map users or resources to tasks Such mappings may
More informationx < y iff x < y, or x and y are incomparable and x χ(x,y) < y χ(x,y).
12. Large cardinals The study, or use, of large cardinals is one of the most active areas of research in set theory currently. There are many provably different kinds of large cardinals whose descriptions
More informationFunctional Programming. Functional Programming Languages. Chapter 14. Introduction
Functional Programming Languages Chapter 14 Introduction Functional programming paradigm History Features and concepts Examples: Lisp ML 1 2 Functional Programming Functional Programming Languages The
More informationGlobal Properties of the Turing Degrees and the Turing Jump
Global Properties of the Turing Degrees and the Turing Jump Theodore A. Slaman Department of Mathematics University of California, Berkeley Berkeley, CA 947203840, USA slaman@math.berkeley.edu Abstract
More informationAutomatic Verification of DataCentric Business Processes
Automatic Verification of DataCentric Business Processes Alin Deutsch UC San Diego Richard Hull IBM Yorktown Heights Fabio Patrizi Sapienza Univ. of Rome Victor Vianu UC San Diego Abstract We formalize
More informationTreebank Search with Tree Automata MonaSearch Querying Linguistic Treebanks with Monadic Second Order Logic
Treebank Search with Tree Automata MonaSearch Querying Linguistic Treebanks with Monadic Second Order Logic Authors: H. Maryns, S. Kepser Speaker: Stephanie Ehrbächer July, 31th Treebank Search with Tree
More informationSpace and Circuit Complexity of Monadic SecondOrder Definable Problems on TreeDecomposable Structures. Michael Elberfeld
Space and Circuit Complexity of Monadic SecondOrder Definable Problems on TreeDecomposable Structures Michael Elberfeld From the Institute of Theoretical Computer Science of the University of Lübeck
More informationSMALL SKEW FIELDS CÉDRIC MILLIET
SMALL SKEW FIELDS CÉDRIC MILLIET Abstract A division ring of positive characteristic with countably many pure types is a field Wedderburn showed in 1905 that finite fields are commutative As for infinite
More informationAutomatabased Verification  I
CS3172: Advanced Algorithms Automatabased Verification  I Howard Barringer Room KB2.20: email: howard.barringer@manchester.ac.uk March 2006 Supporting and Background Material Copies of key slides (already
More informationEQUATIONAL LOGIC AND ABSTRACT ALGEBRA * ABSTRACT
EQUATIONAL LOGIC AND ABSTRACT ALGEBRA * Taje I. Ramsamujh Florida International University Mathematics Department ABSTRACT Equational logic is a formalization of the deductive methods encountered in studying
More informationStatic Analysis of XML Processing with Data Values
Static Analysis of XML Processing with Data Values Luc Segoufin INRIA and Université Paris 11 http://wwwrocq.inria.fr/~segoufin 1 Introduction Most theoretical work on static analysis for XML and its
More informationPredicate Calculus. There are certain arguments that seem to be perfectly logical, yet they cannot be expressed by using propositional calculus.
Predicate Calculus (Alternative names: predicate logic, first order logic, elementary logic, restricted predicate calculus, restricted functional calculus, relational calculus, theory of quantification,
More informationModel checking test models. Author: Kevin de Berk Supervisors: Prof. dr. Wan Fokkink, dr. ir. Machiel van der Bijl
Model checking test models Author: Kevin de Berk Supervisors: Prof. dr. Wan Fokkink, dr. ir. Machiel van der Bijl February 14, 2014 Abstract This thesis is about model checking testing models. These testing
More informationParametric Domaintheoretic models of Linear Abadi & Plotkin Logic
Parametric Domaintheoretic models of Linear Abadi & Plotkin Logic Lars Birkedal Rasmus Ejlers Møgelberg Rasmus Lerchedahl Petersen IT University Technical Report Series TR007 ISSN 600 600 February 00
More informationA Logic Approach for LTL System Modification
A Logic Approach for LTL System Modification Yulin Ding and Yan Zhang School of Computing & Information Technology University of Western Sydney Kingswood, N.S.W. 1797, Australia email: {yding,yan}@cit.uws.edu.au
More informationA Beginner s Guide to Modern Set Theory
A Beginner s Guide to Modern Set Theory Martin Dowd Product of Hyperon Software PO Box 4161 Costa Mesa, CA 92628 www.hyperonsoft.com Copyright c 2010 by Martin Dowd 1. Introduction..... 1 2. Formal logic......
More informationThe Complexity of Description Logics with Concrete Domains
The Complexity of Description Logics with Concrete Domains Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der RheinischWestfälischen Technischen Hochschule Aachen zur Erlangung des
More informationThe Course. http://www.cse.unsw.edu.au/~cs3153/
The Course http://www.cse.unsw.edu.au/~cs3153/ Lecturers Dr Peter Höfner NICTA L5 building Prof Rob van Glabbeek NICTA L5 building Dr Ralf Huuck NICTA ATP building 2 Plan/Schedule (1) Where and When Tuesday,
More informationComputability Theory
CSC 438F/2404F Notes (S. Cook and T. Pitassi) Fall, 2014 Computability Theory This section is partly inspired by the material in A Course in Mathematical Logic by Bell and Machover, Chap 6, sections 110.
More informationOn strong fairness in UNITY
On strong fairness in UNITY H.P.Gumm, D.Zhukov Fachbereich Mathematik und Informatik Philipps Universität Marburg {gumm,shukov}@mathematik.unimarburg.de Abstract. In [6] Tsay and Bagrodia present a correct
More informationTuring Degrees and Definability of the Jump. Theodore A. Slaman. University of California, Berkeley. CJuly, 2005
Turing Degrees and Definability of the Jump Theodore A. Slaman University of California, Berkeley CJuly, 2005 Outline Lecture 1 Forcing in arithmetic Coding and decoding theorems Automorphisms of countable
More informationXML with Incomplete Information
XML with Incomplete Information Pablo Barceló Leonid Libkin Antonella Poggi Cristina Sirangelo Abstract We study models of incomplete information for XML, their computational properties, and query answering.
More informationTreerepresentation of set families and applications to combinatorial decompositions
Treerepresentation of set families and applications to combinatorial decompositions BinhMinh BuiXuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no
More informationSemantics and Verification of Software
Semantics and Verification of Software Lecture 21: Nondeterminism and Parallelism IV (Equivalence of CCS Processes & WrapUp) Thomas Noll Lehrstuhl für Informatik 2 (Software Modeling and Verification)
More informationOffline 1Minesweeper is NPcomplete
Offline 1Minesweeper is NPcomplete James D. Fix Brandon McPhail May 24 Abstract We use Minesweeper to illustrate NPcompleteness proofs, arguments that establish the hardness of solving certain problems.
More informationDegrees that are not degrees of categoricity
Degrees that are not degrees of categoricity Bernard A. Anderson Department of Mathematics and Physical Sciences Gordon State College banderson@gordonstate.edu www.gordonstate.edu/faculty/banderson Barbara
More informationPredicate Logic Review
Predicate Logic Review UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane 1 Grammar A term is an individual constant or a variable. An individual constant is a lowercase letter from the beginning
More informationOntologies and the Web Ontology Language OWL
Chapter 7 Ontologies and the Web Ontology Language OWL vocabularies can be defined by RDFS not so much stronger than the ER Model or UML (even weaker: no cardinalities) not only a conceptual model, but
More informationLights and Darks of the StarFree Star
Lights and Darks of the StarFree Star Edward Ochmański & Krystyna Stawikowska Nicolaus Copernicus University, Toruń, Poland Introduction: star may destroy recognizability In (finitely generated) trace
More informationFabio Patrizi DIS Sapienza  University of Rome
Fabio Patrizi DIS Sapienza  University of Rome Overview Introduction to Services The Composition Problem Two frameworks for composition: Non dataaware services Dataaware services Conclusion & Research
More informationExpressive powver of logical languages
July 18, 2012 Expressive power distinguishability The expressive power of any language can be measured through its power of distinction or equivalently, by the situations it considers indistinguishable.
More informationAnalyzing Firstorder Role Based Access Control
Analyzing Firstorder Role Based Access Control Carlos Cotrini, Thilo Weghorn, David Basin, and Manuel Clavel Department of Computer Science ETH Zurich, Switzerland {basin, ccarlos, thiloweghorn}@infethzch
More informationGenerating models of a matched formula with a polynomial delay
Generating models of a matched formula with a polynomial delay Petr Savicky Institute of Computer Science, Academy of Sciences of Czech Republic, Pod Vodárenskou Věží 2, 182 07 Praha 8, Czech Republic
More informationTest Case Generation for Ultimately Periodic Paths Joint work with Saddek Bensalem Hongyang Qu Stavros Tripakis Lenore Zuck Accepted to HVC 2007 How to find the condition to execute a path? (weakest precondition
More informationFormal Specifications of Security Policy Models
Wolfgang Thumser TSystems GEI GmbH Overview and Motivation Structure of talk Position of FSPM within CC 3.1 and CEM 3.1 CC Requirements CEM Requirements National Scheme Realization of FSPM in terms of
More informationSchema Mappings and Data Exchange
Schema Mappings and Data Exchange Phokion G. Kolaitis University of California, Santa Cruz & IBM ResearchAlmaden EASSLC 2012 Southwest University August 2012 1 Logic and Databases Extensive interaction
More informationTableau Algorithms for Description Logics
Tableau Algorithms for Description Logics Franz Baader Theoretical Computer Science Germany Short introduction to Description Logics (terminological KR languages, concept languages, KLONElike KR languages,...).
More informationFirstOrder Theories
FirstOrder Theories Ruzica Piskac Max Planck Institute for Software Systems, Germany piskac@mpisws.org Seminar on Decision Procedures 2012 Ruzica Piskac FirstOrder Theories 1 / 39 Acknowledgments Theories
More informationLecture 8: Resolution theoremproving
Comp24412 Symbolic AI Lecture 8: Resolution theoremproving Ian PrattHartmann Room KB2.38: email: ipratt@cs.man.ac.uk 2014 15 In the previous Lecture, we met SATCHMO, a firstorder theoremprover implemented
More informationLogical Foundations of Relational Data Exchange
Logical Foundations of Relational Data Exchange Pablo Barceló Department of Computer Science, University of Chile pbarcelo@dcc.uchile.cl 1 Introduction Data exchange has been defined as the problem of
More information