LOGICS OF TIME AND COMPUTATION


 Candace Atkinson
 1 years ago
 Views:
Transcription
1 LOGICS OF TIME AND COMPUTATION
2
3 CSLI Lecture Notes No. 7 LOGICS OF TIME AND COMPUTATION Second Edition Revised and Expanded Robert Goldblatt CSLI CENTER FOR THE STUDY OF LANGUAGE AND INFORMATION
4 Copyright 1992 Center for the Study of Language and Information Leland Stanford Junior University Printed in the United States CIP data and other information appear at the end of the book
5 To my daughter Hannah
6
7 Preface to the First Edition These notes are based on lectures, given at Stanford in the Spring Quarter of 1986, on modal logic, emphasising temporal and dynamic logics. The main aim of the course was to study some systems that have been found relevant recently to theoretical computer science. Part One sets out the basic theory of normal modal and temporal prepositional logics, covering the canonical model construction used for completeness proofs, and the filtration method of constructing finite models and proving decidability results and completeness theorems. Part Two applies this theory to logics of discrete (integer), dense (rational), and continuous (real) time; to the temporal logic of henceforth, next, and until, as used in the study of concurrent programs; and to the prepositional dynamic logic of regular programs. Part Three is devoted to firstorder dynamic logic, and focuses on the relationship between the computational process of assignment to a variable, and the syntactic process of substitution for a variable. A completeness theorem is obtained for a proof theory with an infinitary inference rule. There is more material here than was covered in the course, partly because I have taken the opportunity to gather together a number of observations, new proofs of old theorems etc., that have occurred to me from time to time. Those familiar with the subject will observe, for instance, that in Part Two proofs of completeness for various logics of discrete and continuous time, and for the temporal logic of concurrency, as well as the discussion of Bull's theorem on normal extensions of S4.3, all differ from those that appear in the literature. In order to make the notes effective for classroom use, I have deliberately presented much of the material in the form of exercises (especially in Part One). These exercises should therefore be treated as an integral part of the text. Acknowledgements. My visit to Stanford took place during a period of sabbatical leave from the Victoria University of Wellington which was supported by both universities, and the Fulbright programme. I would like to thank Solomon Feferman and Jon Barwise for the facilities that were made available to me at that time. The CSLI provided generous access to its excellent computertypesetting system, and the Center's Editor, Dikran Karagueuzian, was particularly helpful with technical advice and assistance in the preparation of the manuscript. vu
8 Preface to the Second Edition The text for this edition has been increased by more than a third. Major additions are as follows. 7, originally concerned with incompleteness, now discusses a number of other metatheoretic topics, including firstorder definability, (in)validity in canonical frames, failure of the finite model property, and the existence of undecidable logics with decidable axiomatisation. 9 now includes a study of the " branching time" system of Computational Tree Logic, due to Clarke and Emerson, which introduces connectives that formalise reasoning about behaviour along different branches of the tree of possible future states. Completeness and decidability are shown by the method of filtration in an adaptation of ideas due to Emerson and Halpern. In 10 dynamic logic is extended by the concurrency command a fl/3, interpreted as "a and (3 executed in parallel". This is modelled by the use of "reachability relations", in which the outcome of a single execution is a set of terminal states, rather than a single state. This leads to a semantics for [ a ] and < a > which makes them independent (i.e. not interdefinable via negation). The resulting logic is shown to be finitely axiomatisable and decidable, by a new theory of canonical models and filtrations for reachability relations. A significant conceptual change involves the definition of a "logic" (p. 16), which no longer includes the rule of Uniform Substitution. Logics satisfying this rule are called Uniform, and are discussed in detail on page 23. The change causes a number of minor adaptations throughout the text. A notable technical improvement concerns the completeness proof for S4.3Dum in 8 (pp ). The original DwmLemma has been replaced by a direct proof that nonlast clusters in the filtration are simple. This has resulted in some rearrangement of the material concerning Bull's Theorem, and a simplification of the completeness theorem for the temporal logic of concurrency in 9 (pp ). Other small changes include additional material about the Diodorean modality of spacetime (p. 45), and a rewriting of the basic filtration construction for dynamic logic (p. 114) using a uniform method of proving the first filtration condition that obviates the need to establish any standardmodel conditions for the canonical model. Reformatting the text has provided the opportunity to make numerous changes in style and expression, as well as te, correct typos. I will be thankful for, if not pleased by, information about any further such errors. vuw. ac.nz Vlll
9 Contents Preface to the First Edition vii Preface to the Second Edition viii Part One: Prepositional Modal Logic 1 1. Syntax and Semantics 3 2. Proof Theory Canonical Models and Completeness Filtrations and Decidability Multimodal Languages Temporal Logic Some Topics in Metatheory 48 Part Two: Some Temporal and Computational Logics Logics with Linear Frames Temporal Logic of Concurrency Prepositional Dynamic Logic 109 Part Three: FirstOrder Dynamic Logic Assignments, Substitutions, and Quantifiers Syntax and Semantics Proof Theory Canonical Model and Completeness 162 Bibliography 169 Index 175
10
11 Part One Prepositional Modal Logic
12
13 1 Syntax and Semantics BNF The notation of BackusNaur form (BNF) will be used to define the syntax of the languages we will study. This involves specifying certain syntactic categories, and then giving recursive equations to show how the members of those categories are generated. The method can be illustrated by the syntax of standard propositional logic, which has one main category, that of the formulae. These are generated from some set of atomic formulae (or propositional variables), together with a constant _L (the falsum), by the connective > (implication). In BNF, this is expressed in one line as < formula > ::= < atomic formula > ± < formula >>< formula > The symbol ::= can be read "comprises", or "consists of", or simply "is". The vertical bar is read "or". Thus the equation says that a formula is either an atomic formula, the falsum, or an implication between two formulae. The definition becomes even more concise when we use individual letters for members of syntactic categories, in the usual way. Let $ be a denumerable set of atomic formulae, with typical member denoted p. The set of all formulae generated from $ will be denoted Fma(<?), and its members denoted A, AI, A', B,... etc. The presentation of syntax then becomes Atomic formulae: p <2> Formulae: A Fma(4>) A::=p\L\A+ A Technically, the recursive equation governs a nondeterministic rewriting procedure for generating formulae, in which any occurrence of the symbol to the left of the ::= sign can be replaced by any of the alternative expressions on the right side. Thus the two occurrences of A in the expression A» A may themselves be replaced by different expressions, and so stand for different formulae. In some BNF presentations, this is emphasised by
14 4 Prepositional Modal Logic using subscripts to distinguish different occurrences of a symbol. Then the above equation is given as Modal Formulae The language of propositional modal logic requires one additional symbol, the "box" D. The BNF definition of the set of modal formulae generated by <? is Atomic formulae: p 6 $ Formulae: A Fma(<I>) A::=p\±\A 1 >A 2 \DA Possible readings of It is necessarily true that A. It will always be true that A. It ought to be that A. It is known that A. It is believed that A. It is provable in Peano Arithmetic that A. After the program terminates, A. Other connectives These are introduced by the usual abbreviations. Negation: >A is A» _L Verum: T is il Disjunction: AI VA 2 is (>Ai) > A? Conjunction: Equivalence: A\ A AI AI <» Ai is is >(Ai > >Az) (.Ai» A%) A (A 2 "Diamond": O is Notational Convention In the case that n = 0, the expression just denotes the formula B. Bo A... A B n i » B
15 Exercises Syntax and Semantics 5 (1) Decide what OA means under each of the above readings of D. (2) Which of the following should be regarded as true under the different readings of D? OA+A HA » cm OT DA> OA HA V DA D(A » )> (HA » OB) <X4 A <XB > O(A A B) Subformulae The finite set Sf(A) of all subformulae of A 6 Fma($) is defined inductively by Sf(p) = {p} Sf(±) = {L} Sf(A 1 > Ai) = {A l * A 2 } U Sf(A 1 ) U Sf(A 2 ) Sf(OA) = {DA} U Sf(A) Schemata We will often have occasion to refer to a schema, meaning a collection of formulae all having a common syntactic form. Thus, for example, by the schema OA+A we mean the collection of formulae {OB +B:B(E Fma($)}. Uniform Substitution The notion of a schema can be made more precise by considering uniform substitutions, as follows. Let A and B be any formulae, and p an atomic formula. By the uniform substitution of B for p in A we mean the procedure of replacing each and every occurrence of p in A by B. A formula A' is called a substitution instance of A if it arises by simultaneous uniform substitution for some of of the atomic formulae of A, i.e. if there exist some finitely many atomic formulae pi,...,p n, and formulae Bi,...,B n, such that A' is the result of
16 6 Prepositional Modal Logic simultaneously uniformly substituting?i for pi in A, and BI for p 2 in A, and..., and B n for p n in A. Let.EU = {A' : A' is a substitution instance of A}. Then a schema may be denned as a set of formulae that is equal to EA for some formula A. For example, if A is the formula Dp > p, with p atomic, then SA is what was described above as "the schema DA >.A". Frames and Models A frame is a pair f (S,R), where S is a nonempty set, and R a binary relation on 5: in symbols, R C S x S. A $model on a frame is a triple M (R, S, V), with V : $ > 2 s. Hence V is a function assigning to each atomic formula p 0 a subset V (p) of 5. Informally, V(p) is to be thought of as the set of points at which p is "true". Generally we drop the prefix $ in discussing models, provided the context is clear. The relation "A is true (holds) at point s in model M", denoted M\=,A, is denned inductively on the formation of A Fma(<!>) as follows. M K P iff s V(p) M s L (i.e. not M \= s J) M K (Ai > A 2 ) iff Ai f= s A! implies M \= s A 2 M (= DA iff for all t 5, s#i implies M\= t A Exercises 1.2 (1) A4 =..4 iff A4, A Work out the corresponding truth conditions for A AS, AVB, A <> B. (2) X (= s OA iff there exists t. S with srt and AI (=«A. Motivations 1. Necessity. Following the dictum of Leibnitz that a necessary truth is one that holds in all "possible worlds", S may be thought of as a set of such worlds, with srt when t is a conceivable alternative to s, i.e. a world in which all the necessary truths of s are realised. DA then means "A is necessarily true", while OA means "A is possible", i.e. true in some conceivable world.
17 1 Syntax and Semantics 7 2. Different notions of necessity can be entertained. Thus logical necessity may be contrasted with physical necessity, the latter taking OA to mean U A is a consequence of the laws of physics". Under this reading, srt holds when t is a scientific alternative to s, i.e. a world in which all scientific laws of s are fulfilled. Hence in our world, D(x < c) is true under the physical reading, where c is the velocity of light and x the velocity of a material body. On the other hand it is logically possible that (x < c) is false. 3. In deontic logic, D means "A ought to be true". srt then means that t is a morally ideal alternative to s, a world in which all moral laws of s are obeyed. If s is the actual world, few would maintain that srs under this interpretation. On the other hand, any world is a logical, and scientific, alternative to itself. 4. Temporal Logic. Here the members of 5 are taken to be moments of time. If srt means "t is after (later than) s", then DA means "henceforth A", i.e. "at all future times A", while OA means "eventually (at some future time) A". Dually, if srt means that t is before s, then D means "hitherto", and so on. Natural time frames (S, R) for temporal logic are given by taking S as one of the number sets w (natural numbers), Z (integers), Q (rationals), or R (reals), and R as one of the relations <,<,>,>. Another interesting possibility is to consider various orderings on the points of fourdimensional Minkowskian spacetime (cf. page 45, and Goldblatt [1980]), or even more general nonlinear "branchings" in time (Rescher and Urquhart [1971]). 5. Program states. Reading D as "after the program terminates", S is to be regarded as the set of possible states of a computation process, with srt meaning that there is an execution of the program that starts in state s and terminates in state t. A nondeterministic program may admit more than one possible "outcome" t when started in s. Then OA means "every terminating execution of the program brings about j4", while OA means that the program enables A, i.e. "there is some execution that terminates with A true". At the level of prepositional logic, the notion of state is formally taken to be primitive, as in the theory of automata, Turing machines, etc. A natural concrete interpretation of the notion is possible in quantificational logic, as will be seen in Part Three. Valuations and Tautologies Given a #model M, and a fixed s S, define true false if s otherwise
18 8 Prepositional Modal Logic Then the function V s : $» {true, false} is a valuation of the atomic formulae, a notion familiar from prepositional logic. Using the standard truthtables for prepositional connectives, V g is extended to assign a truthvalue to any formula not containing the symbol D. Thus a model on a frame gives rise to a collection {V s : s S} of valuations of <?, while, conversely, such a collection defines the model in which V(p) = {s : V 8 (p) = true}. A formula A is quasiatomic if either it is atomic (.4 #), or else it begins with a D, i.e. A = OB for some B. If $' is the set of all quasiatomic formulae, then any formula A is constructible from members of $ q U {L} using the connective >. Hence by using the truthtable for», any valuation V : $ q * {true, false} of the quasiatomic formulae extends uniquely to a valuation V : Fma($) > {true, false} of all formulae. A formula A is a tautology if V(A) = true for every valuation V of its quasiatomic subformulae. Exercise 1.3 Any tautology is a substitution instance of a tautology of prepositional logic (i.e. a Dfree tautology). Truth and Validity Formula A is true in model M, denoted M (= A, if it is true at all points in M, i.e. if M\= S A for all s S. A is valid in frame F = (S, R), denoted F (= A, if M (= A for all models M = (S, R, V) based on F. If C is a class of models (respectively, frames), then A is true (respectively, valid) in C, C \= A, if A is true (respectively, valid) in all members of C. A schema will be said to be true in a model (respectively, valid in a frame) if all instances of the schema have that property. More generally, we will use the notations M f= F and F = F, where F C Fma, to mean that all members of F are true in M, or valid in F.
19 1 Syntax and Semantics Exercises 1.4 (1) The following are true in all models, hence valid in all frames. DT D(A > B) > (HA > DB) 0(A + )> (DA » 05) > OB) O(AVB)<^(OAV OB) (2) Show that the following do not have the property of being valid in all frames. D4* A DA + DOA D(A > )> (DA » 05) OT D(Q4 » B) V D(DB » A) D(4 V 5) > DA V D5 D(DA ^ A)>DA (N.B. some instances of these schemata may be valid, e.g. when A is a tautology. What is required is to find a counterexample to validity of at least one instance of each schema.) (3) Show that OT and the schema DA» OA have exactly the same models. (4) Exhibit a frame in which DJ. is valid. (5) In any model M, (i) if A is a tautology then M \= A; (ii) if M \= A and M (=.4 > B, then At = B; (iii) if M h ^ then M \= DA. (6) Items (i)(iii) of the previous exercise hold if M is replaced by any frame f. Ancestral (Reflexive Transitive Closure) Let T = (S, R) be a frame. Define on S the relations R n C S x S, for n > 0, and R*, as follows. sr t iff s = t sr n+1 t iff 3u(sR n u & urt)
20 10 Prepositional Modal Logic Exercises 1.5 (1) R 1 = R. (2) sr*t is Bn > 0 3s 0,, 3s n e 5 with so = s, s n =, and for all i < n, SiRs i+1. (3) R* is reflexive and transitive. (4) If T is any reflexive and transitive relation on S with R C T, then.r* C T. That is,.r* is the smallest reflexive and transitive relation on S that contains R. (5) If SCI, and.r = {(s, s + 1) : s 5}, what is fl*?.r* is often known as the ancestral of.r (from the case that R is the "parent of relation). In view of exercise (4), it is also known as the reflexive transitive closure of R. The notion will play an important role in the logic of programs in Parts Two and Three. Generated Submodels If M = (S, R, V) and t S, then the submodel of M. generated by t is where M t = (S t,r*,v*), 5*  {u e S : tr*u} The structure J* = (S*, R*) is the subframe off = (S, R) generated by t. Exercises 1.6 (1) If R is transitive, then 5* = {t} U {u : tru}. (2) 5* is the smallest subset X of S that contains t and is closed under R, in the sense that u X and urv implies v e X. To evaluate the truth of formula A at point t may require investigating the truth of certain subformulae B of A at various.ralternatives v of t. But then to determine the truth value of B at v may require looking at alternatives of v. And so on. 5* comprises all points generated by this process. It is evident that evaluating truth at t will only involve points that are each obtainable from t by finitely many "^alternations". This is embodied in the
21 1 Syntax and Semantics 11 Submodel Lemma 1.7. If A 6 Fma(<l>), then for any u 5*, M* \= U A iff M K A. Proof. By induction on the formation of A. The case A = p e # follows from the definition of V*, and the case A = _L is immediate. The inductive cases A = (B» D) and yl = D.B are given as exercises. Corollary 1.8. (1) M \= A implies M* \= A. (2) M \= A iff A is true in all generated submodels of M. (3) f (= yl iff A is valid in all generated subframes of T. pmorphisms Let Mi = (Si,Ri,Vi) and M 2 = (S 2,R 2,V 2 ) be models, and / : 5i » S 2 a function satisfying srit implies f(s)r 2 f(t); f(s)r 2 u implies 3t(sR\t & f(t) u); s Vi(p) iff /(a) e Vb(p). Then / is called a pmorphism from.mi to MI A function satisfying the first two conditions is a pmorphism from frame (Si,Ri) to frame (S2,R 2 ). pmorphism Lemma 1.9. If A Fma(<I>), then for any s e Si, Proof. Exercise. Mi\=.A iff M,\= f(.)a. If there is a pmorphism / : T\ > F 2 that is surjective (onto), then frame FI is called a pmorphic image of J 7!. pmorphism Lemma If J 2 is a pmorphic image of F\, then for any formula A, Fi f= A implies F 2 (= A. Proof. Suppose A is false at some point t in some model M 2 (F 2, V 2 ) based on f 2. Take a surjective pmorphism / : Si > S 2 and define a model Mi = (J 7!, Vi) by declaring «Vi(p) iff /(*) Va(p). Then / is a pmorphism from Mi to ^2 Choosing any s with /(s) = i, the first pmorphism Lemma 1.9 gives A false at s in the model Mi based
22 12 Prepositional Modal Logic Exercise 1.11 Let T\ = ({0,1},R) and Fz = ({0},E), where in each case R is the universal relation 5x5. Show that FI\= A implies J^ = A> (o>, <) =>1 implies F± (= 4. The curious appellation "pmorphism" derives from an early use of the name "pseudoepimorphism" in this context, and seems to have become entrenched in the literature. Conditions on R The following is a list of properties of a binary relation R that are denned by firstorder sentences. 1. Reflexive: Vs(sRs) 2. Symmetric: VsVt(sRt > trs) 3. Serial: Vs3t(sRt) 4. Transitive: VsVWu(sRt A tru » sru) 5. Euclidean: VsVtVu(sRt A sru »tru) 6. Partially functional: VsVtVu(sRt A sru > t = u) 1. Functional: Vs3\t(sRt) 8. Weakly dense: VsVt(sRt > 3u(s.Ru A uflt)) 9. Weakly connected: VsVtVu(sRt A sfiw > tru Vt = uv urt) 10. Weakly directed: VsVtVu(sRt A sflu * 3v(tRv A w/zw)) Corresponding to this list is a list of schemata: 1. CU^ 2. A+OOA 3. D.4 » OA 4. D OA <> DA n(^ad^^ 10. OD^*DO^ Theorem Let T = (5, fi) be a frame. Then for each of the properties 110, ifr satisfies the property, then the corresponding schema is valid int.
23 1 Syntax and Semantics 13 Proof. We illustrate with the case of transitivity. Suppose that R is transitive. Let M be any model on f. To show that M \= HA » DDA, take any s in M with M f= g DA We have to prove M K HOA, which means or, in other words, srt implies M \=t OA, srt implies (tru implies M (= A). So, suppose srt. Then if tru, we have sru by transitivity, so M \= u A, since M (= s DA by hypothesis. The other cases are left as exercises. Theorem If a frame F (S, R) validates any one of the schemata 110, then R satisfies the corresponding property. Proof. Take the case of schema 10. To show R is weakly directed, suppose srt and sru. Let M be any model on f in which V(p) {v : urv}. Then by definition, urv implies M [= p, so M $= u Op, and hence, as sru, M (= s OOp. But then as schema 10 is valid in.f, M. h«^ ^Pi so ^ s^» M N< ^P This implies that there exists a v with trv and M (= p, i.e. w V(p), so uflz; as desired. Next, the case of schema 8. Suppose srt. Let M be a model on f with V(p) = {v : t / v}. Then M ^t P, so A'l ^a Dp Hence by validity of schema 8, M \ s OOp, so there exists a u with sru and Ai fc u Dp. Then for some v, urv and Ai ^t, p, i.e. t; = i, so that u.ri, as needed to show that R is weakly dense. Exercises 1.14 (1) Complete the proofs of Theorems 1.12 and (2) Give a property of R that is necessary and sufficient for F to validate the schema A > DA Do the same for DL.
24 14 Prepositional Modal Logic FirstOrder Definability Theorems 1.12 and 1.13 go a long way toward explaining the great success that the relational semantics enjoyed upon its introduction by Kripke [1963]. Frames are much easier to deal with than the modelling structures (Boolean algebras with a unary operator) that had been available hitherto, and many modal schemata were shown to have their frames characterised by simple firstorder properties of R. For a time it seemed that prepositional modal logic corresponded in strength to firstorder logic, but that proved not to be so. Here are a couple of illustrations. (1) The schema W : D(DA » A) > HA is valid in frame (5, R) iff (i) R is transitive, and (ii) there are no sequences SQ,...,s n,... in S with s n Rs n +i for all ra>0. (for a proof cf. Boolos [1979], p.82). Now it can be shown by the Compactness Theorem of firstorder logic that there exists a frame satisfying (i) and (ii) that is elementarily equivalent to (i.e. satisfies the same firstorder sentences as) a frame in which (ii) fails. Hence there can be no set of firstorder sentences that defines the class of frames of this schema. (2) The class of frames of the socalled McKinsey schema M: noa*ena is not defined by any set of firstorder sentences (Goldblatt [1975], van Benthem [1975]). (Both of the above schemata will figure in the discussion of incompleteness in 7, where there is also a further consideration of the question of firstorder definability.) Subsequent investigations demonstrated that prepositional modal logic corresponds to a fragment of secondorder logic (Thomason [1975]). Undefinable conditions There are some naturally occurring properties of a binary relation R that do not correspond to the validity of any modal schema. One such is irreqexivity, i.e. Vs~*(sRs). To see this, observe that the class of all frames validating a given schema is closed under pmorphic images (1.10), but the class of irreflexive frames is not so closed. For instance, it contains (u>, <), but not its pmorphic image ({0},{(0,0)}) (cf. Exercise 1.11).
25 Exercise Syntax and Semantics 15 Show that neither of the following conditions correspond to any modal schema. Antisymmetry: VsVt(sRt A trs > s = t), Asymmetry: VsVt(sRt > <trs). Historical Note The concepts of necessity and possibility have been studied by philosophers throughout history, notably by Aristotle, and in the middle ages. The contemporary symbolic analysis of modality is generally considered to have originated in the work of C. I. Lewis early this century (cf. Lewis and Langford [1932]). Lewis was concerned with a notion of strict implication. He defined "A strictly implies B" as 1(^4 A ib), where I is a primitive impossibility operator (later he expressed this as <O(A A ~<B), where O expresses possibility). He defined a series of systems, which he called SI to 55, based directly on axioms for strict implication. The standard procedure nowadays is to adjoin axioms and rules for D, or O, to the usual presentation of prepositional logic. This approach to modal logic was first used in a paper by Godel [1933]. The model theory described in this section is due to Kripke [1959, 1963]. To learn about the history of modal logic, the reader should first consult the interesting Historical Introduction to Lemmon [1977], where further references may be found.
26 2 I Proof Theory Logics Given a language based on a countable set $ of atomic formulae, a logic is denned to be any set A C Fma($) such that A includes all tautologies, and A is closed under the rule of Detachment, i.e., if A, A > B e A then B e A. Examples of Logics (1) PL = {A e Fmct($) :Aisa tautology }. (2) For any class C of models, or of frames (including the cases C = {M} and C = {f}), Ac = {A : C h A} is a logic. (3) Fma($) itself is a logic. (4) If {Ai : i & 1} is a collection of logics, then their intersection is a logic. Thus for any F C Fma(<i>) there is a smallest logic containing F, namely the intersection of the collection {A : A is a logic and r C A}. Note that PL is the smallest logic, and Fma($) the largest, in the sense that for any logic A, PLCAC Fma($). Tautological Consequence A formula A is a tautological consequence of formulae Ai,...,An if A is assigned true by every valuation that assigns true to all of AI,..., A n. In particular, a tautological consequence of the empty set of formulae is the same thing as a tautology. 16
27 2 Proof Theory 17 Exercise 2.1 Show that any logic A is closed under tautological consequence, i.e. if AI,..., A n e A, then any tautological consequence of AI,..., A n belongs to A Instead of denning a logic A to include all tautologies, it would suffice to include a set of schemata from which all tautologies can be derived by Detachment, e.g. the schemata » A. Theorems The members of a logic are called its theorems. We write \~A A to mean that A is a ^1theorem, i.e., Ki A iff A A. Soundness and Completeness Let C be a class of frames, or of models. Then logic A is sound with respect to C if for all formulae A, \~A A implies C (= A. A is complete with respect to C, if, for any A, C \= A implies \~A A. A is determined by C if it is both sound and complete with respect to C. Deducibility and Consistency If r U {4} C Fma($), then A is Adeducible from F, denoted F \ A A, if there exist BO,., 5 n i F such that \ A Bo > (Si » ( (Bni » A) )) (in the case n = 0, this means that \~A A). We write F \f A A when A is not,4deducible from F. A set r C Fma(<I>) is vlconsj'stent if F \/A L. A formula ^4 is ^1 consistent if the set {.A} is.
28 18 Prepositional Modal Logic Exercises 2.2 (1) \ A A iff 0 hi A (2) If \ A A then r h^ A. (3) If yl C yl', then F\ A A implies T h^..4. (4) If A T then F \ A A. (5) If F C A and T (^ /4, then 2\ h A A. (6) If T \ A A and {yl} h^ B, then T h^ 5. (7) Detachment: If T h^ A and r h^ yl + B, then T h^ B. (8) Deduction Theorem: F\J {A} \ A B iff F h A A> B. (9) r 1^ yl iff there exists a finite sequence A 0,..., A m = A such that for all t < m, either Ai F U A, or else ylfc = (^» ylj) for some j, k < i (i.e. ^4j follows from Aj and ^4^ by Detachment). (10) {A : F \ A A} is the smallest logic containing F U A. (11) Soundness: If M \= s T U A and T \ A A, then M \= s A. (12) If F C yl, then F is ylconsistent if, and only if, yl 7^ Fma($). (13) r 1 is ylconsistent iff there exists a formula A with F \/ A A. (14) F is ylconsistent iff there is no formula A having both F \~ A A and r \ A ^A. (15) r h^ A iff r U {>A} is not ^consistent. (16) r U {A} is ylconsistent iff F \/ A >A. (17) If F is ylconsistent, then for any formula A, at least one of F U {A} and P U {~<A} is ylconsistent. Maximal Sets Let M. = (S, R, V) be a model of a logic yl, i.e. M \= A. Associate with each s e S the set r a = {A<= Fma($) :M\=,A}. Then F s is ylconsistent (why?), and moreover, for each formula A, one of A and <A is in F s. In the next section we will be building models for certain logics. Since we have only a syntactic structure, namely yl, to begin with, we will have to use syntactic entities, such as formulae or sets of formulae, as the points of our models. It turns out that the way to proceed is to use sets of formulae that enjoy the properties possessed by those sets F s naturally associated with points of a given ylmodel. A set F C Fma(<l>) is defined to be Amaximal if
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
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 informationCARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE
CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify
More informationCorrespondence analysis for strong threevalued logic
Correspondence analysis for strong threevalued logic A. Tamminga abstract. I apply Kooi and Tamminga s (2012) idea of correspondence analysis for manyvalued logics to strong threevalued logic (K 3 ).
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationMathematics 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
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 informationReview for Final Exam
Review for Final Exam Note: Warning, this is probably not exhaustive and probably does contain typos (which I d like to hear about), but represents a review of most of the material covered in Chapters
More informationWHAT 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
More informationFinite Sets. Theorem 5.1. Two nonempty finite sets have the same cardinality if and only if they are equivalent.
MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we
More information3. Equivalence Relations. Discussion
3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,
More informationHandout #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
More informationSets and Cardinality Notes for C. F. Miller
Sets and Cardinality Notes for 620111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by MenGen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationSETS, 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
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 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 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 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 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 informationConsistency, 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
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
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 information8 Divisibility and prime numbers
8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express
More informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More informationCartesian 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
More informationChapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
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 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 informationWeek 5: Binary Relations
1 Binary Relations Week 5: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of all
More informationS(A) X α for all α Λ. Consequently, S(A) X, by the definition of intersection. Therefore, X is inductive.
MA 274: Exam 2 Study Guide (1) Know the precise definitions of the terms requested for your journal. (2) Review proofs by induction. (3) Be able to prove that something is or isn t an equivalence relation.
More informationINCIDENCEBETWEENNESS GEOMETRY
INCIDENCEBETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full
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 informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More informationCHAPTER 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
More information4 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
More information13 Infinite Sets. 13.1 Injections, Surjections, and Bijections. mcsftl 2010/9/8 0:40 page 379 #385
mcsftl 2010/9/8 0:40 page 379 #385 13 Infinite Sets So you might be wondering how much is there to say about an infinite set other than, well, it has an infinite number of elements. Of course, an infinite
More informationSOLUTIONS TO ASSIGNMENT 1 MATH 576
SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts
More informationCS 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 ChurchTuring thesis Let s recap how it all started. In 1990, Hilbert stated a
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 informationIntroducing 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
More informationResearch Note. Biintuitionistic Boolean Bunched Logic
UCL DEPARTMENT OF COMPUTER SCIENCE Research Note RN/14/06 Biintuitionistic Boolean Bunched Logic June, 2014 James Brotherston Dept. of Computer Science University College London Jules Villard Dept. of
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 informationSOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties
SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces
More informationElementary 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,
More informationApplications of Methods of Proof
CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The settheoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are
More informationBasic Concepts in Modal Logic 1
Basic Concepts in Modal Logic 1 Edward N. Zalta Center for the Study of Language and Information Stanford University Table of Contents Preface Chapter 1 Introduction 1: A Brief History of Modal Logic 2:
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 informationAn Introduction to Logics of Knowledge and Belief
Chapter 1 arxiv:1503.00806v1 [cs.ai] 3 Mar 2015 An Introduction to Logics of Knowledge and Belief Contents Hans van Ditmarsch Joseph Y. Halpern Wiebe van der Hoek Barteld Kooi 1.1 Introduction to the Book..............
More information11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(
More informationTOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS
TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS VIPUL NAIK Abstract. In this journey, we are going to explore the so called separation axioms in greater detail. We shall try to understand how these axioms
More informationAn 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
More informationCS510 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/15cs510se
More information3. 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)
More informationLogic, Algebra and Truth Degrees 2008. Siena. A characterization of rst order rational Pavelka's logic
Logic, Algebra and Truth Degrees 2008 September 811, 2008 Siena A characterization of rst order rational Pavelka's logic Xavier Caicedo Universidad de los Andes, Bogota Under appropriate formulations,
More informationThis chapter is all about cardinality of sets. At first this looks like a
CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },
More informationFixedPoint 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 informationP NP for the Reals with various Analytic Functions
P NP for the Reals with various Analytic Functions Mihai Prunescu Abstract We show that nondeterministic machines in the sense of [BSS] defined over wide classes of real analytic structures are more powerful
More informationLikewise, we have contradictions: formulas that can only be false, e.g. (p p).
CHAPTER 4. STATEMENT LOGIC 59 The rightmost column of this truth table contains instances of T and instances of F. Notice that there are no degrees of contingency. If both values are possible, the formula
More informationEMBEDDING COUNTABLE PARTIAL ORDERINGS IN THE DEGREES
EMBEDDING COUNTABLE PARTIAL ORDERINGS IN THE ENUMERATION DEGREES AND THE ωenumeration DEGREES MARIYA I. SOSKOVA AND IVAN N. SOSKOV 1. Introduction One of the most basic measures of the complexity of a
More informationKevin James. MTHSC 412 Section 2.4 Prime Factors and Greatest Comm
MTHSC 412 Section 2.4 Prime Factors and Greatest Common Divisor Greatest Common Divisor Definition Suppose that a, b Z. Then we say that d Z is a greatest common divisor (gcd) of a and b if the following
More informationFoundations 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
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 informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
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 informationA single minimal complement for the c.e. degrees
A single minimal complement for the c.e. degrees Andrew Lewis Leeds University, April 2002 Abstract We show that there exists a single minimal (Turing) degree b < 0 s.t. for all c.e. degrees 0 < a < 0,
More information2. 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
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 informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationSubsets of Euclidean domains possessing a unique division algorithm
Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness
More informationThe BanachTarski Paradox
University of Oslo MAT2 Project The BanachTarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the BanachTarski paradox states that for any ball in R, it is possible
More informationProbabilities and Random Variables
Probabilities and Random Variables This is an elementary overview of the basic concepts of probability theory. 1 The Probability Space The purpose of probability theory is to model random experiments so
More informationTHE DEGREES OF BIHYPERHYPERIMMUNE SETS
THE DEGREES OF BIHYPERHYPERIMMUNE SETS URI ANDREWS, PETER GERDES, AND JOSEPH S. MILLER Abstract. We study the degrees of bihyperhyperimmune (bihhi) sets. Our main result characterizes these degrees
More informationDEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS
DEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsionfree abelian groups, one of isomorphism
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationTHE 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.
More information1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain
Notes on realclosed fields These notes develop the algebraic background needed to understand the model theory of realclosed fields. To understand these notes, a standard graduate course in algebra is
More informationDEGREES 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
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
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 information! " # The Logic of Descriptions. Logics for Data and Knowledge Representation. Terminology. Overview. Three Basic Features. Some History on DLs
,!0((,.+#$),%$(&.& *,2($)%&2.'3&%!&, Logics for Data and Knowledge Representation Alessandro Agostini agostini@dit.unitn.it University of Trento Fausto Giunchiglia fausto@dit.unitn.it The Logic of Descriptions!$%&'()*$#)
More informationWe 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
More informationMetric Spaces Joseph Muscat 2003 (Last revised May 2009)
1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of
More informationUPDATES OF LOGIC PROGRAMS
Computing and Informatics, Vol. 20, 2001,????, V 2006Nov6 UPDATES OF LOGIC PROGRAMS Ján Šefránek Department of Applied Informatics, Faculty of Mathematics, Physics and Informatics, Comenius University,
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
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 informationCompleteness Theorems for Syllogistic Fragments
Completeness Theorems for Syllogistic Fragments Lawrence S. Moss Department of Mathematics Indiana University Bloomington, IN 47405 USA Comments/Corrections Welcome! Abstract Traditional syllogisms involve
More informationThe 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
More informationLinear Algebra. A vector space (over R) is an ordered quadruple. such that V is a set; 0 V ; and the following eight axioms hold:
Linear Algebra A vector space (over R) is an ordered quadruple (V, 0, α, µ) such that V is a set; 0 V ; and the following eight axioms hold: α : V V V and µ : R V V ; (i) α(α(u, v), w) = α(u, α(v, w)),
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu
COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationTHE 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
More informationRegular Languages and Finite Automata
Regular Languages and Finite Automata 1 Introduction Hing Leung Department of Computer Science New Mexico State University Sep 16, 2010 In 1943, McCulloch and Pitts [4] published a pioneering work on a
More informationIntroduction 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
More informationTheory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras
Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture No. # 31 Recursive Sets, Recursively Innumerable Sets, Encoding
More informationFoundations of Logic and Mathematics
Yves Nievergelt Foundations of Logic and Mathematics Applications to Computer Science and Cryptography Birkhäuser Boston Basel Berlin Contents Preface Outline xiii xv A Theory 1 0 Boolean Algebraic Logic
More informationIntroduction to Proofs
Chapter 1 Introduction to Proofs 1.1 Preview of Proof This section previews many of the key ideas of proof and cites [in brackets] the sections where they are discussed thoroughly. All of these ideas are
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it
More information