# ON FUNCTIONAL SYMBOL-FREE LOGIC PROGRAMS

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical and Mathematical Sciences p ON FUNCTIONAL SYMBOL-FREE LOGIC PROGRAMS I nf or m at i cs L. A. HAYKAZYAN * Chair of Programming and Information Technologies YSU Armenia In the present article logic programs (both with and without negation) that do not use functional symbols are studied. Three algorithmic problems for functional symbol-free programs are investigated: the existence of a solvable interpreter the problem of Δ-equivalence and the problem of logical equivalence. The first two problems are known to be decidable for functional symbol-free definite programs. We show that the third one is also decidable for such programs. In contrast all three problems are shown to be undedicable for functional symbol-free general programs. Keywords: logic programming functional symbol-free programs algorithmic problems. 1. Introduction. For basic definitions in logic programming we refer the reader to [1]. We study logic programs that do not use functional symbols of arity 1 (henceforth referred as FSF programs). For general programs (i.e. programs with negation) we use Clark s completion semantics from [2]. In logic programming it is customary to consider only Herbrand interpretation. We follow this convention and consider only Herbrand interpretations. While for formulas without built-in predicates (such as definite programs) this does not make a difference it makes a difference for formulas with built-in predicates (such as completions of general programs). Some of fundamental algorithmic problems for FSF programs (and indeed for any class of logic programs) are the following ones. Is there a solvable interpreter? Is -equivalence decidable? (Two programs are called -equivalent if their semantics coincide for each permitted query see [3]). Is logical equivalence (on Herbrand interpretations) of such programs decidable? The first two problems are known to have positive answers for FSF definite programs (see [4]). In the present paper we show that the third one also has positive answer for such programs. We also show that in contrast to this all three problems have negative answers for FSF general programs. Let us introduce some notation that will be used throughout the paper. We fix a first order language L with a countable set of constant symbols and countable sets of functional and predicate symbols of each arity. Programs and queries in L (both definite and general) will be denoted by P and Q respectively (possibly with some subscripts or superscripts). This will not cause confusion since we study *

2 44 Proc. of the Yerevan State Univ. Phys. and Mathem. Sci p definite and general programs in different sections. All programs are assumed to be functional symbol-free. Occasionally we will need to refer to the language consisting of constant functional and predicate symbols used in some well-formed formula F. In this case the language in question is denoted by L F. 2. Definite Programs. In this section we study definite programs (i.e. programs without negation). Definite programs possess some nice properties. In particular each definite program P is satisfiable and moreover has a least Herbrand model M P. This model is actually the set of ground atoms that logically follow from the program. The paper [4] characterizes least Herbrand models of definite programs through so called templates. Intuitively a template K of M P is a set of atoms (with certain minimality properties) such that every atom A M P is an instance of an atom B K. It is shown in [4] that all templates of the least model of a definite program are congruent (i.e. can be obtained from one another through variable renaming). It turns out that least Herbrand models of FSF definite programs have finite templates (see [4]). The following theorem is a direct consequence of this. T h eo r e m There is an algorithm for deciding if a definite query follows from a definite FSF program. The technique of templates is also applicable to the problem of so called Δ-equivalence. Two definite programs P 1 and P 2 are called Δ-equivalent if every query is a logical consequence of one program iff it is a logical consequence of the other one. It follows that definite programs would be Δ-equivalent iff the templates of their least models are congruent. This implies the next theorem. T h eo r e m The problem of Δ-equivalence is decidable for FSF definite programs (see [4]). Here we show that logical equivalence of FSF definite programs is also decidable. T h eo r e m 2.3. There is an algorithm for deciding if two FSF definite programs are logically equivalent. Proof. Let P 1 and P 2 be FSF definite programs. The formula P 1 P 2 is logically valid iff so are P 1 P 2 and P 2 P 1. By symmetry considerations we only show how to check the validity P 1 P 2. The formula P 1 P 2 is valid iff P1 P2 is unsatisfiable. Now P 1 is a universal formula and P2 is equivalent to an existential formula. Let P 1 be x1... xnf 1( x1... xn ) and P2 be equivalent to y1... ymf2 ( y1... ym) where F 1 and F 2 are quantifier free. We can assume that x 1 x n are different to y1... y m and then P1 P2 is logically equivalent to y1... ymx 1... xn ( F2 ( y1... ym) F1 ( x1... x n )). But by Skolem s Theorem the latter is satisfiable iff so is x1... xn( F2 ( c1... cm ) F1 ( x1... xn)) where c... 1 c n are new constant symbols. Denote the last formula by G. Now G is a universal formula and hence has a Herbrand model in L G if satisfiable. But since G does not use functional symbols the Herbrand universe of L G is finite and hence there are finitely many Herbrand interpretations. So we can check all Herbrand interpretations and see if any of them satisfies G. 3. General Programs. Theorems and 2.3 of the previous section demonstrate that all three algorithmic problems are decidable for FSF definite programs. This means that definite logic programs become significantly less expressive if the usage of functional symbols is forbidden. In contrast first-order logic does not lose its expressiveness without functional symbols. So the special form of definite clauses becomes crucial in the decidability results of the previous section.

3 Proc. of the Yerevan State Univ. Phys. and Mathem. Sci p In this section we study the above mentioned algorithmic problems for FSF general programs. We show that in contrast to definite programs all three problems are undecidable for FSF general programs. As stated before we employ Clark s completion semantics for general programs (see [2]). In addition we impose the restriction to Herbrand models of completion. This means that a query Q (or its negation) is considered to be a consequence of the completion comp(p) of a general program P if Q is true in all Herbrand models of comp(p). However for FSF programs the restriction to Herbrand models can be simplified. Proposition 3.1. Let F be a first-order formula without functional symbols possibly using equality. Then F has a Herbrand model if it has a countable model where each constant symbol is interpreted by a distinct element. Proof. Let M be a countable model of F. Denote by D the domain of M. Let f be a bijection between the Herbrand universe of L and D taking the constant symbols used in F into their interpretations in M. Define a Herbrand interpretation by assigning to the atom p( t1... t n ) the truth value of p M ( f ( t1)... f ( t n)). It is routine to check that this interpretation is a model of F. It is shown in [5] that the expressiveness of general programs is not extended if we allow arbitrary formulas in bodies of clauses. To make this precise recall the way comp(p) is obtained from P. First we rewrite each clause p( t1... tn ) S in the general form p( x1... xn) y1... ym ( t1 x1... tn xn S) where x... 1 x n are new variables and y... 1 y m are the variables of the original clause. If p(x 1 x n ) E 1 p(x 1 x n ) E k are all the general forms of clauses with p in their heads then the definition of p would be the formula x1... xn( p( x1... xn) E1... Ek ). As usual the empty disjunction is understood as a logical falsehood. The completion comp(p) of P is the set of definitions of predicate symbols that occur in P. This process does not in any way depend on the bodies of programs being conjunctions of literals. It is tempting to generalize general programs even further by allowing arbitrary formulas in their bodies. However as shown in [5] given such a program P we can construct a general program P such that comp(p') is a conservative extension of comp(p). Recall the construction presented in [5]. Let F be a formula of first-order logic without equality. We can assume that F is built using and. Let A be an atom whose variables are different to bound variables of F. Transform the clause A F using the following rules: Replace A F1 F2 by two clauses A F 1 and A F 2. Replace A xf by A F. Replace A F by two clauses A p( x1... x n ) and p(x 1 x n ) F where x 1 x n are the free variables of F and p is a new predicate symbol. Let P be a program with arbitrary formulas in bodies of clauses and let P' be obtained from P by repeatedly applying the above transformation. It is shown in [5] that each model of comp(p') is a model of comp(p) and further each model of comp(p') is obtained from a model of comp(p) by uniquely defining interpretations

4 46 Proc. of the Yerevan State Univ. Phys. and Mathem. Sci p of extra predicate symbols. We will refer to this transformation as Lloyd-Topor transformation. It is widely known that completions of general programs may be inconsistent. For example the completion of the program consisting of the single clause p( x) p( x) is the formula x( p( x) p( x)) which does not have a Herbrand model (or any other model). Such programs present a little interest since any query is a consequence of their completions. The next theorem shows that there is no algorithm for deciding this property of FSF general programs. T h eo r e m There is no algorithm for deciding if comp(p) has a Herbrand model for a given FSF general program P. Proof. Let F be a closed formula of pure first-order logic (i.e. without functional and constant symbols and without equality). Let p be a new predicate symbol and a be a new constant symbol. Denote by P the general program obtained from { p( a) F p( a)} { q( x1... xn ) q( x1... xn) : q is n-ary predicate symbol used in F} by Lloyd-Topor transformation. Then comp(p) has a Herbrand model comp(p) has a countable model (since comp(p) does not use functional and constant symbols other than a ) p( a) ( F p( a)) has a countable model (by Lloyd-Topor transformation) p( a) F has a countable model (since the latter is logically equivalent to the former) F has a countable model (since F does not use p and a) F is consistent (by Löwenheim Skolem Theorem). Thus by having an algorithm for deciding if comp( P ) has a Herbrand model we can decide pure first-order logic. But this is not possible by Church s Theorem. This result in itself guaranties that it is algorithmically impossible to decide if a query is a consequence of the completion of a FSF general program. Indeed comp(p) would have a Herbrand model iff the query p( a) p( a) is not a logical consequence of it. However this kind of undecidability is not very practical since in practice one often writes programs with consistent completions. Thus one can pose the question of deciding if a query (or the negation of a query) is a conesquence of the completion of FSF general program with an additional assumption that the completion is consistent. However this problem also turns out to be undecidable. T h eo r e m There is no algorithm that takes as input FSF general program P and a general query Q with the following properties: if comp(p) has a Herbrand model and comp( P) Q holds on Herbrand interpretations then the algorithm halts with the answer Yes ; if comp(p) has a Herbrand model and comp( P) Q holds on Herbrand interpretations then the algorithm halts with the answer No. Proof. Consider Robinson arithmetic [6] which is a finitely axiomatizable and Σ 1 -complete subtheory of Peano Arithmetic in the language with equality constant symbol 0 and functional symbols s + and for the successor addition and multiplication respectively. Denote by R the conjunction of axioms of Robinson arithmetic. Let F(x) denote the Σ 1 formula representing the predicate the Turing machine with number x halts on its number. Since R is Σ 1 -complete the Turing Σ 1 -completness means that every true Σ 1 formula is provable. The formula is called Σ 1 if there is only one unbounded quantifier. Bounded quantifier are quantifiers of the form x y and x y. Robinson arithmetic is often denoted by Q but we have reserved this letter for queries.

5 Proc. of the Yerevan State Univ. Phys. and Mathem. Sci p machine with number n halts on its number iff R =F(s n (0)). Now we want to encode R and F as a functional symbol-free general program. In Lloyd-Topor transformation bodies of clauses are not allowed to contain equality. But this is easy to workaround. We substitute atoms t 1 =t 2 with eq(t 1 t 2 ) where eq is a new binary predicate symbol. Later we will add to the program the clause eq(xx). Then the completion of the program will contain y z( eq( y z) x( x y x z)) which means that eq is interpreted as the equality. Next to get rid of functional symbols we use a well-known technique of substituting functional symbols with predicate symbols representing their graphs. For an n-ary functional symbol f we peak a new n 1-ary predicate symbol p f which satisfies x... x y( p ( x... x y)) 1 n f 1 n x... x y z( p ( x... x y) p ( x... x z) eq( y z)). 1 n f 1 n f 1 n Denote the conjunction of these two formulas as Def ( f ). Then we can replace atoms A( f ( t1... t n)) with x( p f ( t1... tn x) A( x)) until no functional symbol is left. Denote by R' and F' the formulas obtained from R and F respectively by replacing equality with eq and functional symbols s + and with p s p + and p. Let Def(R) denote the conjunction of Def(p s ) Def(p + ) and Def ( p ). Then the Turing machine with number n halts on its number iff R Def ( R) x y( eq( x y) x y) x1... xn ( ps (0 x)... ps ( xn 1 xn) F( xn)). Let p and q be new unary predicate symbols and P be obtained from p(0) ( R Def ( R)) p(0) q( x) F( x) eq( x x) ps ( x y) ps ( x y) p ( x y z) p ( x y z) p ( x y z) p ( x y z) by Lloyd-Topor transformation. Then comp(p) is a conservative extension of p(0) R Def ( R) x( q( x) F( x)) x y( eq( x y) x y). It is clear that comp(p) has a model (natural numbers with appropriate interpretations of predicate symbols) and even a Herbrand model (since this model is countable and there is only one constant symbol used). Let Q denote the query x1... xn ( ps (0 x1 )... ps( xn 1 xn) q( xn )). Now if the Turing machine with number n halts on its number then comp( P) Q on all models and in particular on Herbrand models. On the other hand if the Turing machine with number n does not halt on its number then the set of natural numbers (with appropriate interpretations of predicate symbols) is a model of comp( P) Q. But then comp( P) Q has a Herbrand model since there is only one constant symbol used. It follows that having an algorithm with desired properties we can solve the halting problem which is impossible. Corollary 1 from Theorem 3.2. There is no algorithm that takes as input FSF general program P and a general query Q with the following properties: if comp(p) has a Herbrand model and comp(p) Q holds on Herbrand interpretations then the algorithm halts with the answer Yes ;

6 48 Proc. of the Yerevan State Univ. Phys. and Mathem. Sci p if comp(p) has a Herbrand model and comp( P) Q holds on Herbrand interpretations then the algorithm halts with the answer No. Proof. Let P be a FSF general program whose completion has a Herbrand model and Q be a general query. Denote by P the program obtained from P by adding the clause p( a) Q where p is a new predicate symbol and a is a new constant symbol. Denote by Q the query p( a). It is easy to see that comp(p') has a Herbrand model and comp( P) Q iff comp( P) Q (on Herbrand as well as on all interpretations). Corollary 2 from Theorem 3.2. There is no algorithm that takes as input FSF general program P and a general query Q with the following properties: if comp(p) has a Herbrand model and comp( P) Q holds on Herbrand interpretations then the algorithm halts with the answer Yes ; if comp(p) has a Herbrand model and comp( P) Q holds on Herbrand interpretations then the algorithm halts with the answer No ; if comp(p) has a Herbrand model and none of the above holds then the algorithm halts with the answer Undefined. General programs P 1 and P 2 are called Δ-equivalent if for every query Q we have comp( P1 ) Q comp( P2 ) Q and comp( P1 ) Q comp( P2 ) Q (the relation is restricted to Herbrand interpretations). Using Theorem 3.2 it is easy to prove that neither Δ-equivalence nor logical equivalence of completions (on Herbrand interpretations) is decidable for FSF general programs (even for programs whose completions have Herbrand models). Indeed given a general program P and a general query Q define programs P 1 and P 2 by adding to P clauses p( a) and p( a) Q respectively where p is a new predicate symbol and a is a new constant symbol. Clearly comp( P 1) and comp(p 2 ) are consistent if so is comp(p). It is easy to see that comp( P) Q on Herbrand interpretations iff comp(p 1 ) and comp(p 2 ) are Δ-equivalent iff comp(p 1 ) and comp(p 2 ) are logically equivalent on Herbrand interpretations. As a corollary we have the following two Theorems. T h eo r e m There is no algorithm for deciding Δ-equivalence of FSF general programs (even for programs whose completions have Herbrand models). T h eo r e m There is no algorithm for deciding logical equivalence (on Herbrand interpretations) of completions of FSF general programs (even for programs whose completions have Herbrand models). Received R E FE R E N C E S 1. Lloyd J. Foundations of Logic Programming. Springer-Verlag p. 2. Clark K.L. Negation as Failure. In Gallaire H. and Minker J. (eds). Logic and Data Bases. New York: Plenum Press 1978 p Nigiyan S.A. Khachoyan L.O. // Programming and Computer Software 1997 v. 23 p Nigiyan S.A. Khachoyan L.O. // Dokladi NAN Armenii 1999 v p (in Russian). 5. Lloyd J. Topor R. // Journal of Logic Programming 1984 v. 1 p Robinson R.M. Proceedings of International Congress of Mathematics 1950 p

7 Proc. of the Yerevan State Univ. Phys. and Mathem. Sci p Լ. Ա. Հայկազյան. Ֆունկցիոնալ սիմվոլներ չպարունակող տրամաբանական ծրագրերի մասին էջ Հոդվածում ուսումնասիրվում են տրամաբանական ծրագրեր (ժխտում պարունակող և չպարունակող) որոնք չեն օգտագործում ֆունկցիոնալ սիմվոլներ: Քննարկվում են հետևյալ երեք ալգորիթմական խնդիրները. լուծելի ինտերպրետատորի առկայությունը Δ-համարժեքության խնդիրը և տրամաբական ծրագրերի սովորական համարժեքության խնդիրը: Հայտնի է առաջին երկու խնդիրների լուծելիությունը` ժխտում չպարունակող ծրագրերի համար: Սույն աշխա-տանքում ցույց է տրվում երրորդ խնդիրի լուծելիությունը ժխտում չպարու-նակող ծրագրերի համար ինչպես նաև բոլոր երեք խնդիրների անլուծելիությունը ժխտում պարունակող ծրագրերի համար:

### CHAPTER 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

### (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 first-order logic will use the following symbols: variables connectives (,,,,

### SJÄLVSTÄNDIGA 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

### INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

### The Recovery of a Schema Mapping: Bringing Exchanged Data Back

The Recovery of a Schema Mapping: Bringing Exchanged Data Back MARCELO ARENAS and JORGE PÉREZ Pontificia Universidad Católica de Chile and CRISTIAN RIVEROS R&M Tech Ingeniería y Servicios Limitada A schema

### Theory of Computation Lecture Notes

Theory of Computation Lecture Notes Abhijat Vichare August 2005 Contents 1 Introduction 2 What is Computation? 3 The λ Calculus 3.1 Conversions: 3.2 The calculus in use 3.3 Few Important Theorems 3.4 Worked

### EQUATIONAL 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

### 4 Domain Relational Calculus

4 Domain Relational Calculus We now present two relational calculi that we will compare to RA. First, what is the difference between an algebra and a calculus? The usual story is that the algebra RA is

### First-Order Logics and Truth Degrees

First-Order Logics and Truth Degrees George Metcalfe Mathematics Institute University of Bern LATD 2014, Vienna Summer of Logic, 15-19 July 2014 George Metcalfe (University of Bern) First-Order Logics

### 2. The Language of First-order Logic

2. The Language of First-order Logic KR & R Brachman & Levesque 2005 17 Declarative language Before building system before there can be learning, reasoning, planning, explanation... need to be able to

### CS 3719 (Theory of Computation and Algorithms) Lecture 4

CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 Church-Turing thesis Let s recap how it all started. In 1990, Hilbert stated a

### Validity Checking. Propositional and First-Order Logic (part I: semantic methods)

Validity Checking Propositional and First-Order Logic (part I: semantic methods) Slides based on the book: Rigorous Software Development: an introduction to program verification, by José Bacelar Almeida,

### Handout #1: Mathematical Reasoning

Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or

### Lecture 8: Resolution theorem-proving

Comp24412 Symbolic AI Lecture 8: Resolution theorem-proving Ian Pratt-Hartmann Room KB2.38: email: ipratt@cs.man.ac.uk 2014 15 In the previous Lecture, we met SATCHMO, a first-order theorem-prover implemented

### Computational Methods for Database Repair by Signed Formulae

Computational Methods for Database Repair by Signed Formulae Ofer Arieli (oarieli@mta.ac.il) Department of Computer Science, The Academic College of Tel-Aviv, 4 Antokolski street, Tel-Aviv 61161, Israel.

### There is no degree invariant half-jump

There is no degree invariant half-jump Rod Downey Mathematics Department Victoria University of Wellington P O Box 600 Wellington New Zealand Richard A. Shore Mathematics Department Cornell University

### o-minimality and Uniformity in n 1 Graphs

o-minimality 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

### Introducing Functions

Functions 1 Introducing Functions A function f from a set A to a set B, written f : A B, is a relation f A B such that every element of A is related to one element of B; in logical notation 1. (a, b 1

### Composing Schema Mappings: Second-Order Dependencies to the Rescue

Composing Schema Mappings: Second-Order Dependencies to the Rescue Ronald Fagin IBM Almaden Research Center fagin@almaden.ibm.com Phokion G. Kolaitis UC Santa Cruz kolaitis@cs.ucsc.edu Wang-Chiew Tan UC

### Computability 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 1-10.

### Consistency, completeness of undecidable preposition of Principia Mathematica. Tanmay Jaipurkar

Consistency, completeness of undecidable preposition of Principia Mathematica Tanmay Jaipurkar October 21, 2013 Abstract The fallowing paper discusses the inconsistency and undecidable preposition of Principia

### THE TURING DEGREES AND THEIR LACK OF LINEAR ORDER

THE TURING DEGREES AND THEIR LACK OF LINEAR ORDER JASPER DEANTONIO Abstract. This paper is a study of the Turing Degrees, which are levels of incomputability naturally arising from sets of natural numbers.

### One More Decidable Class of Finitely Ground Programs

One More Decidable Class of Finitely Ground Programs Yuliya Lierler and Vladimir Lifschitz Department of Computer Sciences, University of Texas at Austin {yuliya,vl}@cs.utexas.edu Abstract. When a logic

### The Foundations: Logic and Proofs. Chapter 1, Part III: Proofs

The Foundations: Logic and Proofs Chapter 1, Part III: Proofs Rules of Inference Section 1.6 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments

### A 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......

### Summary 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

### NP-Completeness and Cook s Theorem

NP-Completeness and Cook s Theorem Lecture notes for COM3412 Logic and Computation 15th January 2002 1 NP decision problems The decision problem D L for a formal language L Σ is the computational task:

### Scalable Automated Symbolic Analysis of Administrative Role-Based Access Control Policies by SMT solving

Scalable Automated Symbolic Analysis of Administrative Role-Based Access Control Policies by SMT solving Alessandro Armando 1,2 and Silvio Ranise 2, 1 DIST, Università degli Studi di Genova, Italia 2 Security

### DEGREES OF CATEGORICITY AND THE HYPERARITHMETIC HIERARCHY

DEGREES OF CATEGORICITY AND THE HYPERARITHMETIC HIERARCHY BARBARA F. CSIMA, JOHANNA N. Y. FRANKLIN, AND RICHARD A. SHORE Abstract. We study arithmetic and hyperarithmetic degrees of categoricity. We extend

### 1.4 Predicates and Quantifiers

Can we express the following statement in propositional logic? Every computer has a CPU No 1 Can we express the following statement in propositional logic? No Every computer has a CPU Let's see the new

### Biinterpretability up to double jump in the degrees

Biinterpretability up to double jump in the degrees below 0 0 Richard A. Shore Department of Mathematics Cornell University Ithaca NY 14853 July 29, 2013 Abstract We prove that, for every z 0 0 with z

### Mathematical Induction

Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. E.g., we would like to show: N 1 : is a number 1 : has the feature Φ ( x)(n 1 x! 1 x) How

### 2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.

2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then

### Introduction to Logic in Computer Science: Autumn 2006

Introduction to Logic in Computer Science: Autumn 2006 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss 1 Plan for Today Now that we have a basic understanding

### Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

### Lecture 2: Universality

CS 710: Complexity Theory 1/21/2010 Lecture 2: Universality Instructor: Dieter van Melkebeek Scribe: Tyson Williams In this lecture, we introduce the notion of a universal machine, develop efficient universal

### Computational Models Lecture 8, Spring 2009

Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 1 Computational Models Lecture 8, Spring 2009 Encoding of TMs Universal Turing Machines The Halting/Acceptance

### Problems on Discrete Mathematics 1

Problems on Discrete Mathematics 1 Chung-Chih Li 2 Kishan Mehrotra 3 Syracuse University, New York L A TEX at January 11, 2007 (Part I) 1 No part of this book can be reproduced without permission from

### Algorithmic 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

### Extending Semantic Resolution via Automated Model Building: applications

Extending Semantic Resolution via Automated Model Building: applications Ricardo Caferra Nicolas Peltier LIFIA-IMAG L1F1A-IMAG 46, Avenue Felix Viallet 46, Avenue Felix Viallei 38031 Grenoble Cedex FRANCE

### Correspondence analysis for strong three-valued logic

Correspondence analysis for strong three-valued logic A. Tamminga abstract. I apply Kooi and Tamminga s (2012) idea of correspondence analysis for many-valued logics to strong three-valued logic (K 3 ).

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### This 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

### Foundations of mathematics

Foundations of mathematics 1. First foundations of mathematics 1.1. Introduction to the foundations of mathematics Mathematics, theories and foundations Sylvain Poirier http://settheory.net/ Mathematics

Computing and Informatics, Vol. 20, 2001,????, V 2006-Nov-6 UPDATES OF LOGIC PROGRAMS Ján Šefránek Department of Applied Informatics, Faculty of Mathematics, Physics and Informatics, Comenius University,

### Discrete Mathematics, Chapter : Predicate Logic

Discrete Mathematics, Chapter 1.4-1.5: Predicate Logic Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 1 / 23 Outline 1 Predicates

### CS510 Software Engineering

CS510 Software Engineering Propositional Logic Asst. Prof. Mathias Payer Department of Computer Science Purdue University TA: Scott A. Carr Slides inspired by Xiangyu Zhang http://nebelwelt.net/teaching/15-cs510-se

### Applications of Methods of Proof

CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are

### Parametric Domain-theoretic models of Linear Abadi & Plotkin Logic

Parametric Domain-theoretic models of Linear Abadi & Plotkin Logic Lars Birkedal Rasmus Ejlers Møgelberg Rasmus Lerchedahl Petersen IT University Technical Report Series TR-00-7 ISSN 600 600 February 00

### University of Ostrava. Reasoning in Description Logic with Semantic Tableau Binary Trees

University of Ostrava Institute for Research and Applications of Fuzzy Modeling Reasoning in Description Logic with Semantic Tableau Binary Trees Alena Lukasová Research report No. 63 2005 Submitted/to

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### Notes on Complexity Theory Last updated: August, 2011. Lecture 1

Notes on Complexity Theory Last updated: August, 2011 Jonathan Katz Lecture 1 1 Turing Machines I assume that most students have encountered Turing machines before. (Students who have not may want to look

### Certain 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,

### Turing 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

### Schedule. 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

### The Classes P and NP

The Classes P and NP We now shift gears slightly and restrict our attention to the examination of two families of problems which are very important to computer scientists. These families constitute the

### SETS, RELATIONS, AND FUNCTIONS

September 27, 2009 and notations Common Universal Subset and Power Set Cardinality Operations A set is a collection or group of objects or elements or members (Cantor 1895). the collection of the four

### CSE 135: Introduction to Theory of Computation Decidability and Recognizability

CSE 135: Introduction to Theory of Computation Decidability and Recognizability Sungjin Im University of California, Merced 04-28, 30-2014 High-Level Descriptions of Computation Instead of giving a Turing

### Mathematics for Computer Science

Mathematics for Computer Science Lecture 2: Functions and equinumerous sets Areces, Blackburn and Figueira TALARIS team INRIA Nancy Grand Est Contact: patrick.blackburn@loria.fr Course website: http://www.loria.fr/~blackbur/courses/math

### First-Order Stable Model Semantics and First-Order Loop Formulas

Journal of Artificial Intelligence Research 42 (2011) 125-180 Submitted 03/11; published 10/11 First-Order Stable Model Semantics and First-Order Loop Formulas Joohyung Lee Yunsong Meng School of Computing,

### Semantics for the Predicate Calculus: Part I

Semantics for the Predicate Calculus: Part I (Version 0.3, revised 6:15pm, April 14, 2005. Please report typos to hhalvors@princeton.edu.) The study of formal logic is based on the fact that the validity

### The Halting Problem is Undecidable

185 Corollary G = { M, w w L(M) } is not Turing-recognizable. Proof. = ERR, where ERR is the easy to decide language: ERR = { x { 0, 1 }* x does not have a prefix that is a valid code for a Turing machine

### Degrees 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

### Aikaterini 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

### Computational 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

### Logical 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

### Properties of Stabilizing Computations

Theory and Applications of Mathematics & Computer Science 5 (1) (2015) 71 93 Properties of Stabilizing Computations Mark Burgin a a University of California, Los Angeles 405 Hilgard Ave. Los Angeles, CA

### A Formalization of the Church-Turing Thesis

A Formalization of the Church-Turing Thesis Udi Boker and Nachum Dershowitz School of Computer Science, Tel Aviv University Tel Aviv 69978, Israel {udiboker,nachumd}@tau.ac.il Abstract Our goal is to formalize

### A 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

### Invertible elements in associates and semigroups. 1

Quasigroups and Related Systems 5 (1998), 53 68 Invertible elements in associates and semigroups. 1 Fedir Sokhatsky Abstract Some invertibility criteria of an element in associates, in particular in n-ary

### CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs

CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce

### High Integrity Software Conference, Albuquerque, New Mexico, October 1997.

Meta-Amphion: Scaling up High-Assurance Deductive Program Synthesis Steve Roach Recom Technologies NASA Ames Research Center Code IC, MS 269-2 Moffett Field, CA 94035 sroach@ptolemy.arc.nasa.gov Jeff Van

### We give a basic overview of the mathematical background required for this course.

1 Background We give a basic overview of the mathematical background required for this course. 1.1 Set Theory We introduce some concepts from naive set theory (as opposed to axiomatic set theory). The

facultad de informática universidad politécnica de madrid On the Confluence of CHR Analytical Semantics Rémy Haemmerlé Universidad olitécnica de Madrid & IMDEA Software Institute, Spain TR Number CLI2/2014.0

### Logic in general. Inference rules and theorem proving

Logical Agents Knowledge-based agents Logic in general Propositional logic Inference rules and theorem proving First order logic Knowledge-based agents Inference engine Knowledge base Domain-independent

### Software Verification and Testing. Lecture Notes: Z I

Software Verification and Testing Lecture Notes: Z I Motivation so far: we have seen that properties of software systems can be specified using first-order logic, set theory and the relational calculus

### Math 223 Abstract Algebra Lecture Notes

Math 223 Abstract Algebra Lecture Notes Steven Tschantz Spring 2001 (Apr. 23 version) Preamble These notes are intended to supplement the lectures and make up for the lack of a textbook for the course

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

### arxiv:1112.0829v1 [math.pr] 5 Dec 2011

How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

### THE SEARCH FOR NATURAL DEFINABILITY IN THE TURING DEGREES

THE SEARCH FOR NATURAL DEFINABILITY IN THE TURING DEGREES ANDREW E.M. LEWIS 1. Introduction This will be a course on the Turing degrees. We shall assume very little background knowledge: familiarity with

### Sudoku as a SAT Problem

Sudoku as a SAT Problem Inês Lynce IST/INESC-ID, Technical University of Lisbon, Portugal ines@sat.inesc-id.pt Joël Ouaknine Oxford University Computing Laboratory, UK joel@comlab.ox.ac.uk Abstract Sudoku

### OHJ-2306 Introduction to Theoretical Computer Science, Fall 2012 8.11.2012

276 The P vs. NP problem is a major unsolved problem in computer science It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a \$ 1,000,000 prize for the

### Bounded 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.

### A 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

### Is Sometime Ever Better Than Alway?

Is Sometime Ever Better Than Alway? DAVID GRIES Cornell University The "intermittent assertion" method for proving programs correct is explained and compared with the conventional method. Simple conventional

### Automated 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

### WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

### An example of a computable

An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

### Applied Logic in Engineering

Applied Logic in Engineering Logic Programming Technische Universität München Institut für Informatik Software & Systems Engineering Dr. Maria Spichkova M. Spichkova WS 2012/13: Applied Logic in Engineering

### Predicate Logic. Example: All men are mortal. Socrates is a man. Socrates is mortal.

Predicate Logic Example: All men are mortal. Socrates is a man. Socrates is mortal. Note: We need logic laws that work for statements involving quantities like some and all. In English, the predicate is

### Data Integration: A Theoretical Perspective

Data Integration: A Theoretical Perspective Maurizio Lenzerini Dipartimento di Informatica e Sistemistica Università di Roma La Sapienza Via Salaria 113, I 00198 Roma, Italy lenzerini@dis.uniroma1.it ABSTRACT

### Summary of Last Lecture. Automated Reasoning. Outline of the Lecture. Definition. Example. Lemma non-redundant superposition inferences are liftable

Summary Automated Reasoning Georg Moser Institute of Computer Science @ UIBK Summary of Last Lecture C A D B (C D)σ C s = t D A[s ] (C D A[t])σ ORe OPm(L) C s = t D u[s ] v SpL (C D u[t] v)σ C s t Cσ ERR

### Theory of Automated Reasoning An Introduction. Antti-Juhani Kaijanaho

Theory of Automated Reasoning An Introduction Antti-Juhani Kaijanaho Intended as compulsory reading for the Spring 2004 course on Automated Reasononing at Department of Mathematical Information Technology,

### Our goal first will be to define a product measure on A 1 A 2.

1. Tensor product of measures and Fubini theorem. Let (A j, Ω j, µ j ), j = 1, 2, be two measure spaces. Recall that the new σ -algebra A 1 A 2 with the unit element is the σ -algebra generated by the

### CODING TRUE ARITHMETIC IN THE MEDVEDEV AND MUCHNIK DEGREES

CODING TRUE ARITHMETIC IN THE MEDVEDEV AND MUCHNIK DEGREES PAUL SHAFER Abstract. We prove that the first-order theory of the Medvedev degrees, the first-order theory of the Muchnik degrees, and the third-order

### Generating 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

### Cartesian Products and Relations

Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special

### Secrecy for Bounded Security Protocols With Freshness Check is NEXPTIME-complete

Journal of Computer Security vol 16, no 6, 689-712, 2008. Secrecy for Bounded Security Protocols With Freshness Check is NEXPTIME-complete Ferucio L. Țiplea a), b), Cătălin V. Bîrjoveanu a), Constantin