Gries, D. and F.B. Schneider. A Logical Approach to Discrete Math. Springer Verlag, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....0 or 1 according to its Boolean value. Decomposition is a syntactic operation that translates logical features into thin features by extracting bodies of unfolded clauses. An example of decomposition is shown in Figure 7. Position evaluation can be efficiently performed by composing Hasse diagram [5] on the partial order of thin features and by using a slightly modified propagation method described in Sect. 3. In this propagation, we can use depth first algorithm because thin features do not contain disjunction. It is said that using features that rarely match positions tends to cause ....

D. Gries and F. B. Schneider. A Logical Approach to Discrete Math. Springer-Verlag, New York, 1993.

....are angelic: if the behavior of the implementation can be matched, then that is the correct choice to make. Instead of performing an expensive search and the attendant problems of back tracking, we enforce enough restrictions to solve the problem simply. We take advantage of the one point rule [20]: 9(x 2 S ) P(x ) x = y) y 2 S P(y) We use a value from the implementation as the witness y . Ideally, we would like to be able to automatically identify the value, but when that is not possible, then we need a mechanism for the programmer to provide it. Resolving External Choice. When ....

D. Gries and F. B. Schneider. A Logical Approach to Discrete Math. Springer-Verlag, Berlin, 1993.

....path is contained in the pattern is the conjunction of these weakest constraints for each path separately. So, our task is to compute the constraint C (pat ; prog) s : s 2 prog : p : p 2 pat : s p) where prog : set (list stat) and pat : set (list prop) We adopt the notation of [18], and write ( x : R : T ) instead of the more common L x :R T . The obvious route is now to attempt a replay of the earlier development in this paper, by de ning a matrix whose entries are C (pat ; prog) where pat is an element of the chip chop matrix of pat . A disappointment awaits ....

D. Gries and F. B. Schneider. A Logical Approach to Discrete Math. Texts and Monographs in Computer Science. Springer Verlag, 1993.

....process # sends a message # to a correct process #, then # eventually receives #. 3.2 Sequences The algorithms presented in this paper rely on sequences. A sequence is a finite ordered list of elements. With a few minor exceptions, the notation defined here is borrowed from Gries and Schneider [17]. A sequence of three elements #, #, # is denoted by the tuple ### ## ##. The symbol # denotes the empty sequence. The length of a sequence ### is the number of elements in ### and is denoted #### . For instance, # ### ## ## ##, and ## ##. Elements can be added either at the beginning or at the ....

....empty sequence. The length of a sequence ### is the number of elements in ### and is denoted #### . For instance, # ### ## ## ##, and ## ##. Elements can be added either at the beginning or at the end of a sequence. Adding an element # at the beginning of a sequence ### is called prepending (see [17]) and is denoted by # # ### . Similarly, adding an element # at the end of a sequence ### is called appending and is denoted by ### # #. We define the operator ##for accessing a single element of the sequence. Given a sequence ### , ### ## # returns the # element of ### . The element ### ## # ....

D. Gries and F. B. Schneider. A Logical Approach to Discrete Math. Texts and monographs in computer science. Springer-Verlag, 1993.

....some detail (Sect. 5.6) Comparisons to other approaches and concluding remarks are presented in Section 6. 2. 0 Background In the sequel, we use relative quantification where Q is a quantifier ( or ) T is the type of the dummy variable x, R is the range of the dummy variable and P a predicate [14]. For example, means for all values of an integer variable i, if i is at least as large as 3 then i has property P . If no range is supplied then it is true. The notation generally means that . For example, means that we are defining by . In TTM update functions (see sequel) denotes assignment, ....

Gries, D. and F.B. Schneider. A Logical Approach to Discrete Math. Springer Verlag, 1993.

....and referring to the first element of a list) Other functions are partial according to their recursive definitions. If partial functions are admitted in formulae, a programming logic is needed that handles partial functions and undefined terms. Dijkstra and Scholten [1] Gries and Schneider [2], Feijen, van Gasteren, and other researchers in programming methodology have discovered that programming logics are useful only if they are suited to proof engineering , that is, to the design and presentation of proofs. A programming logic is acceptable only if it is a handy tool without ....

....2. 1.1. Gries s and Schneider s logic Recently Gries and Schneider gave the issue of logics for partial functions a new direction: they suggested a calculational logic for partial functions [13, 14] The original calculational logic was designed by Dijkstra and Scholten [1] Gries and Schneider [2], Feijen, van Gasteren, and others. But in its original form, calculational logic is intended for total functions only. Program semantics being defined by means of weakest preconditions in [1] partial functions do not occur at all. Therefore the original calculational logic does not handle ....

Gries, D. and Schneider, F. B. (1993) A Logical Approach to Discrete Math. Springer, New York.

....Notation and Terminology 12 a variable, then (8x; P ) P Q) P Q) and ( P ) are also predicate expressions. The predicate expressions ( P ) Q) and ( 8x; P ) can also be written in an equivalent format: P ) Q) j (P ) Q) 8x; P ) j (9x; P ) 2. 7 Set and Relational Algebra From [5], we adopted the following definition. For given set S and T , 1) Subset: S T j f8x j x 2 S ) x 2 Tg (2) Union: x 2 S [ T j x 2 S x 2 T (3) Intersection: x 2 S T j x 2 S x 2 T (4) Difference: x 2 S Gamma T j x 2 S x = 2 T (5) Cardinality: The cardinality of a set is the number ....

D. Gries and F. Schneider. A Logical Approach to Discrete Math. Spring-Verlag New York Inc., 1993.

....only minimal material in most cases. Providing a series of modules would increase the opportunities for departments to cover logical material across multiple courses in a uniform fashion; this goes beyond efforts reported at other institutions that integrate formal material into isolated courses [17, 25, 35]. Each module should consist of a short text book, on line (HTML or PowerPoint) lecture notes, presentations, programming exercises, problem sets, and guidelines for using appropriate software tools. Justification for the Proposed Solution: Several people, including both researchers and ....

Gries, D. and F. B. Schneider. A logical approach to discrete math. Springer-Verlag, 1993.

....the answer is clear the formula e = e is a tautology for any expression e. Now let us examine some popular books that use types and see how they answer this question. The books by Chandy and Misra [1988] and Manna and Pnueli [1991] despite their e#orts to be rigorous, do not provide an answer. Gries and Schneider [1993] were more careful in the description of the typed logic in their book. Their explicit typing rules tell us that A[i j ] A[i j ] is not a legal expression. Unfortunately, those same rules tell us that (i j # 0) # (A[i j ] A[i j ] is also an illegal expression. It would ....

Gries, D. and Schneider, F. B. 1993. A Logical Approach to Discrete Math. Springer-Verlag, New York.

....prove hopelessly unwieldy in comparison with two valued logics, but we shall see that this is not necessarily so. The logics that interest us are a family of equational logics that includes E, E3, E2, and E4. E is an equational version of classical predicate logic, and is described in [DS90] and [GS93]. E3 is a three valued logic with values true , false , and neither true nor false which preserves most of the theorems of E. It is, in essence, a fusion of E and the three valued typed LPF [JM94] Partiality and Nondeterminacy in Program Proofs 3 and has been described in [MB98] In this ....

.... DeltaE = definition: E = F j (E j F ) P P ) Q Q P (8x:T ffl P ) The deduction theorem holds. In practice, we do not use the inference rules directly, but employ textual substitution mechanisms that reduce proving to an algebraic style of replacing equals with equals ; see [DS90] and [GS93]. We have constructed the rather long winded axiomatisation of Figure 1 because it captures what is common to each member of the family. Using it we can construct once off a large body of theorems shared by all, and indeed it is surprising how much of traditional two valued logic is preserved. ....

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D. Gries and F. B. Schneider. A Logical Approach to Discrete Math. Springer Verlag, New York, 1993.

....(monotonicity) and AMON (antimonotonicity) which allow us to calculate appropriate conclusions # C[p A] # C[p B]or# C[p A] # C[p B] from the assumption # A # B. Introduction. This note builds further on [To] where the logical calculus of Equational (Predicate) Logic outlined in [GS1] was formalized and shown to be sound and complete. We propose here a simpler formalization than the one in [To] basing the proof apparatus solely on propositional rules of inference one of which, of course, is a version of Leibniz . This entails an unconstrained Deduction Theorem (contrast ....

....a propositional version of Leibniz, we show that there are derived rules valid in the logic, which allow the use of Leibniz style substitution within the scope of a quantifier. We also address one weakness to which David Gries has already called attention in [Gr] of the current literature ([DSc, GS1]) on equational or calculational reasoning. That is, while it is customary to mix = steps (that is, an application of a conjunctional #)and# steps (that is, an application of a conjunctional #) in a calculational proof, and while we have well documented rules to handle the former, yet the ....

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Gries, D. and Schneider, F. B. A Logical Approach to Discrete Math, New York: Springer-Verlag, 1994.

....by many scientists to write rigorous proofs for different domains of computer sciences or for teaching formal logic. In the following section we present the essentials of this calculus before we describe its structured variant. For further information on the calculational proof format we refer to [Gries and Schneider, 1994]. 2.1 Calculational Proof Format In the calculational proof format statements of theorems (logical formulas) are treated as Boolean expressions, called Boolean structures. A calculational proof is a value preserving transformation of a Boolean structure step by step into the Boolean value true, ....

Gries, D. and Schneider, F. B. (1994). A Logical Approach to Discrete Math. Texts and Monographs in Computer Science. Springer, New York, NY.

....to see that they are learning a new, powerful, mental tool. One interesting example is Portia s Suitor s Dilemma (see Table 2) It s solution is amazingly simple using our equational logic when one formalizes, manipulates, and then interprets. Its solution in natural deduction is much harder (see [9, 6] for a discussion) Another interesting problem is to make sense of the following sentence: For every value of array section b[1. 9] if that value is in a.rray section b[21. 25] then it is not in b[21. 25] This sentence may seem contradictory, but formalizing it, simplifying the formal ....

....what they had learned. Five wrote negative comments, but the other 65 were overwhelmingly positive. This is the first time we experienced or heard of an overwhelmingly enthusiastic response to a discrete math course. All their comments are summarized in the first chapter of the Instructor s Manual [6], where the gestalt of the course is discussed. We repeat a few comments here. This course was really groovy, perhaps my favorite one . I am a math major who never thought I would enjoy dging proofs, until taking this course. Stressing logic has helped my methods of thinking. I definitely ....

Gries, D., and F.B. Schneider. Instructor's Manual for "A Logical Approach to Discrete Math". Gries and Schneider, Ithaca, 1993. (To discuss obtaining a copy, email gries@cs.cornell.edu.)

....complex, hides this fact. 7 Experience with our approach In Spring 1993, we taught a discrete math course to 70 students (mainly freshman and sophomores) in the Computer Science Department at Cornell, using a draft of our text A Logical Approach to Discrete Math (Springer Verlag, New York, 1993) [5]. This experience showed that students: Lose their fear of mathematics and mathematical notation. Gain a good understanding of proofs and their development. Acquire some skill in formal manipulation and begin almost immediately to apply that skill in other courses. 2O Gain an appreciation for ....

....of proving things, and I find proofs easier to tackle now. To be perfectly honest, I enjoyed it, which is surprising considering I do not enjoy math. This course taught me the most about how to think logically in all the many courses I have taken in my Cornell career. A draft of text [5] was also used in Fall 1993 by Sam Warford at Pepperdine, who considered his course a big success . We hope that others will try out this new approach to teaching math. We think that the approach produces students with far better math skills and that it should be integrated into all math ....

Gries, D., and F.B. Schneider. A Logical Approach to Discrete Math. Springer-Verlag, New York, 1993.

....F1: Definition 1. Substitution of equals for equals: If S results from R by substitution of Q for P at one or more places in R (not necessarily at all occurrences of P in R ) and if Q , then S . Since then, 1) has become a cornerstone of calculational formulations of logic (see e.g. [3, 6]) so named because a formalization of (1) is the central inference rule. However, di#ering formalizations of (1) have been used. One [3] is in terms of function application, another [7] is in terms of substitution that doesn t avoid capture of free occurrences of variables, and a third is in ....

....the central inference rule. However, di#ering formalizations of (1) have been used. One [3] is in terms of function application, another [7] is in terms of substitution that doesn t avoid capture of free occurrences of variables, and a third is in terms of substitution that does avoid such capture [6]. This article explores the relation between these three formalizations. In Sect. 2, we present Church s first order logic F1 [2] and define two di#erent notions of substitution as well as the first formalization of (1) called Leibniz. In Sect. 3, we introduce introduce logic LF, which includes ....

Gries, D., and F.B. Schneider. A Logical Approach to Discrete Math. Springer Verlag, NY, 1993.

....Substitution preserves C validity. Textual Substitution and all the inference rules of S5 preserve C validity. Also, the axioms of S5 are C valid. Therefore, logic C is sound with respect to C validity. To illustrate C, we prove that :2p is a theorem. We use a calculational style of proof see [7] or [8] The first formula is a C theorem. Since the last formula equals the first, the last is also a C theorem. 2p ) p) p : false ] Textual Substitution in Axiom 2 Instantiation = hDefinition of textual substitution for propositional variable p i 2p ) false = hPropositional theorem Q ) ....

....Joe Halpern) The resulting axiomatization is still unsatisfactory to us, because of the need to reformulate ffi of a conjectured theorem 3ffi in disjunctive normal form. This reformulation is not in keeping with our usual way of proving theorems (using a calculational approach, where suitable [7, 8, 5]) For example, to prove 3ffi 3fl , we would be forced to prove that one of ffi and fl were satisfiable, rather than simply performing syntactic manipulations to obtain 3ffi 3fl , as is our preference. Inference rule Textual Substitution provides an alternative that is more in tune with the way ....

Gries, D., and F.B. Schneider. A Logical Approach to Discrete Math. Springer-Verlag, New York, 1993.

....herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of these organizations or the U.S. Government. Since then, 1) has become a cornerstone of calculational formulations of logic (see e.g. [3, 6]) so named because a formalization of (1) is the central inference rule. However, differing formalizations of (1) have been used. One [3] is in terms of function application, another [7] is in terms of substitution that doesn t avoid capture of free occurrences of variables, and a third is in ....

....central inference rule. However, differing formalizations of (1) have been used. One [3] is in terms of function application, another [7] is in terms of substitution that doesn t avoid capture of free occurrences of variables, and a third is in terms of substitution that does avoid such capture [6]. This article explores the relation between these three formalizations. In Sect. 2, we present Church s first order logic F1 [2] and define two different notions of substitution as well as the first formalization of (1) called Leibniz. In Sect. 3, we introduce introduce logic LF, which includes ....

Gries, D., and F.B. Schneider. A Logical Approach to Discrete Math. Springer Verlag, NY, 1993.

....truths from old ones in a simple and efficient way. Why not make such a calculational style available to users of logic A first cut at an equational logic for computer scientists appeared in the monograph [4] Three years later, a freshman sophomore level text incorporating the approach appeared [5]. The new approach offers hope for a new view of logic and an entirely different method of teaching logic and proof. One of the principles guiding research by those studying the formal development of programs has been simplicity and uniformity of concept and notation. Tools should be as simple as ....

....legitimate concerns when constructing proofs, just as they are in other creative activities. In this article, we illustrate the use of simplicity as a goal by discussing the treatment of undefined terms and partial functions in equational propositional logic E and its extension to predicate logic [5]. Partial functions are ubiquitous in programming some basic mathematical operations are partial (e.g. division) some basic programming operations are partial (e.g. array subscripting b[n] and many functions that arise through recursive definitions are partial. Therefore, a logic for ....

[Article contains additional citation context not shown here]

Gries, D., and F.B. Schneider. A Logical Approach to Discrete Math. SpringerVerlag, New York, 1993.

....they are learning a new, powerful, mental tool. One interesting example is Portia s Suitor s Dilemma (see Table 2) Its solution is amazingly simple using our equational logic when one formalizes, manipulates, and then interprets. Its solution in natural deduction is awkward and much longer (see [9, 6] for a discussion) 12 Table 3: Informal Proof Techniques Informal proof technique Basis for the informal technique Case analysis (p Mutual implication p = q p) Contradiction p # false Contrapositive p # q # p Another interesting problem is to make sense of the ....

....they had learned. Five wrote negative 23 comments, but the other 65 were overwhelmingly positive. This is the first time we experienced or heard of an overwhelmingly enthusiastic response to a discrete math course. All their comments are summarized in the first chapter of the Instructor s Manual [6], where the gestalt of the course is discussed. We repeat a few comments here. This course was really groovy, perhaps my favorite one . I am a math major who never thought I would enjoy doing proofs, until taking this course. Stressing logic has helped my methods of thinking. I definitely ....

Gries, D., and F.B. Schneider. Instructor's Manual for "A Logical Approach to Discrete Math". Gries and Schneider, Ithaca, 1993. (To discuss obtaining a copy, email gries@cs.cornell.edu.)

....the proof is based, namely the ability to put predicates and sets in correspondence using the relation P.z S . The corresponding English proof, because its structure is so complex, hides this fact. 7 Experience with our approach We have written a text, A Logical Approach to Discrete Math [5], that embodies our approach to teaching discrete math. The text covers the conventional topics of discrete math. In Spring 1993, we taught a discrete math course to 70 students (mainly freshmen and sophomores) in the Computer Science Department at Cornell, using a draft of the text. The students ....

....fear of proving things, and I find proofs easier to tackle now. To be perfectly honest, I enjoyed it, which is surprising considering I do not enjoy math. This course taught me the most about how to think logically in all the many courses I have taken in my Cornell career. A draft of text [5] was also used in Fall 1993 by Sam Warford at Pepperdine, who considered his course a big success . We hope that others will try out this new approach to teaching math. We think that the approach produces students with far better math skills and that it would help to integrate the approach into ....

Gries, D., and F.B. Schneider. A Logical Approach to Discrete Math. Springer-Verlag, New York, 1993.

....be replaced using it. Textual Substitution and all the inference rules of S5 preserve S5c validity. Also, the axioms of S5 are S5c valid. Therefore, logic S5c is sound with respect to S5c validity. To illustrate S5c, we prove that :2p is a theorem. We use a calculational style of proof see [4] or [5] The first formula is an S5c theorem. Since the last formula equals the first, the last is also an S5c theorem. 2p ) p) p : false] Textual Substitution in Axiom 2 Instantiation = hDefinition of textual substitution for propositional variable p i 2p ) false = hPropositional theorem ....

Gries, D., and F.B. Schneider. A Logical Approach to Discrete Math. Springer-Verlag, New York, 1993.

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Gries, D. and F.B. Schneider. A Logical Approach to Discrete Math. Springer Verlag, 1993.

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D. Gries and F. Schneider, A Logical Approach to Discrete Maths, Springer-Verlag, (1994)

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D. Gries and F. Schneider. A Logical Approach to Discrete Math. SpringerVerlag 1993.

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D. Gries, F.B. Schneider, A Logical Approach to Discrete Math, Springer, Berlin, 1993.

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