J. H. Gallier. Logic for Computer Science: Foundations of Automatic Theorem Proving. Harper & Row, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....show that it is satisfiable.This method is by far the most used and it is the one on which we will focus in the rest of this thesis. Checking directly if # is a contradiction by completely simplifying it and testing if the resulting formula is empty. This method is not even known to be in NP [Gal86] 23 . Using the resolution method to show that # is unsatisfiable. This method is attributed to Robinson [Rob65] but it was first proposed by Blake in 1937 [CS88] Important complexity studies have been carried out during the late 70 s and the 80 s and include the works of Galil [Gal77] ....

Jean H. Gallier. Logic for Computer Science: Foundations of Automatic Theorem Proving. Harper & Row, 1986.

....derived in step 9 of Example 2.1 can be checked for satisfiability in the theory of LISP list structure using Nelson Oppen s procedure by considering all the variables to be constants. We have also constructed several new procedures by extending a GLSP for uninterpreted function symbols. Also, Gallier 86, ch 10.6] gives techniques for constructing GLSPs based on congruence closure for conjunctions of ground literals containing predicates. The essential idea is to introduce boolean constants True and False and to represent P(t ) True and P(t ) False. Then, if the congruence ....

Gallier, J. H., Logic for Computer Science: Foundations of Automatic Theorem Proving, Harper and Row, 1986.

....may be instantiated by any individual in the domain. Sorted Logic and Unconstrained Variables The notion of a completely unconstrained variable is a theoretical one. In practice we have in mind some restricted set of individuals even when we make sweeping generalizations. In sorted logic [Gallier, 1986, Plaisted, 1993] every variable is associated with a sort, which is analogous to a type in a computer programming language. When every variable is constrained in some manner, the notion of unconstrained is construed as unconstrained with respect to a particular set of individuals. When a partial ....

Jean H. Gallier. Logic for Computer Science: Foundations of Automatic Theorem Proving, chapter 10, pages 448--476. Harper & Row, New York, 1986.

.... Sum elim snd [ p: Sum(A,B) A type ] x) x:A ] B(x) type ] snd(p) B(fst(p) Figure 13: Derived rules for CTT 60 11 Classical first order logic The theory LK implements classical first order logic through Gentzen s sequent calculus lk (see Gallier [6]) This calculus is especially suitable for purely backwards proof it behaves almost exactly like the method of semantic tableaux. Assertions have the form # #, where # and # are lists of formulae. Associative unification handles lists; we easily get powerful proof procedures. The module ....

....R(x) ALL x. P(x) M(x) L(x) Backtracking over the choice of a safe rule (excluding the basic sequent) accomplishes nothing: applying them in any order leads to essentially the same result. This is intuitively clear enough, but a rigorous proof is essentially the Completeness Theorem [6]. Backtracking may be necessary over the choice of basic sequents. Suppose 0, 1, 2, 3 are constants in the subgoals The only assignment that satisfies all three subgoals is a 2, and this can only be discovered by search. 64 Asobj dummy type for lists Asequ type of lists, sequ = ....

J. H. Gallier, Logic for Computer Science: Foundations of Automatic Theorem Proving (Harper & Row, 1986).

....solutions. solve(queens(8,B) B= 4,2,7,3,6,8,5,1] B= 5,2,4,7,3,8,6,1] B= 3,5,2,8,6,4,7,1] yes 6. 3 Example: A Propositional calculus theorem prover Wang s Algorithm for the propositional calculus [32] works by constructing a backwards proof using the rules of the sequent calculus [21]. The following program also constructs a proof tree. Each label names some sequent calculus rule, such as andl for # left. The subtrees represent proofs of the premises of the rule. This program demonstrates function inversion. The function sizeof computes the size of proof trees. In Prolog ....

Gallier, J. H., Logic for Computer Science: Foundations of Automatic Theorem Proving, Harper & Row, 1986

....0 EX y. ALL x. P(y) P(x) 1. EX y. ALL x. P(y) P(x) by (comp tac [ size=21 size=31 size=43 Level 1 EX y. ALL x. P(y) P(x) No subgoals 2. 2 Classical first order logic The theory LK implements classical first order logic through Gentzen s sequent calculus lk (see Gallier [3] or Takeuti [13] Resembling the method of semantic tableaux, the calculus is well suited for backwards proof. Assertions have the form # where # and # are lists of formulae. Associative unification, simulated by higherorder unification, handles lists. We easily get powerful proof procedures. ....

....i.e. it is always applicable and does not exclude any productions of its arguments. Example 5.2 In mixfix notation the grammar in Example 1. 1 can be written as Mixfix( 0 , Aterm, 0 , 9) Mixfix( Aterm,Aterm] Aterm, 0,1] 0) Mixfix( Aterm,Aterm] Aterm, [3,2], 2) Mixfix( Aterm Aterm, 3] 3) The dummy type Adummy is used instead of the obvious ty to be consistent with the treatment of types of identifiers. See Section 5.5 Parsing the string 0 0 0 produces the term P (P (M Z) Z) Z where P is Const( Adummy) M is ....

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J. H. Gallier, Logic for Computer Science: Foundations of Automatic Theorem Proving (Harper & Row, 1986).

....will increase for a time and then decrease; if it gets bigger and bigger, look out 5.1.3 The completeness of propositional logic Folderol is complete for propositional logic. For a formula A,ifA is valid then it constructs a proof; if A is invalid then it constructs a model that falsifies A [Gallier 1986, page 71] Is this formula valid Folderol answers in no time. run goal (P (Q R) P Q R) empty (P (Q R) P Q R) right :right :left :left :left Q R No proof rules applicable Q P,R P (Q R) P R The first goal is not ....

....equality requires an algorithm specific to the problem at hand. Rewriting reduces an expression to normal form. Equational unification solves a set of equations in the presence of equational laws. Congruence closure, given a set of equations, e#ciently tests whether certain other equations hold [Gallier 1986]. The ideal interactive theorem prover would provide all useful algorithms known. The Boyer Moore theorem prover knows a good many algorithms and how to use them. But new algorithms are constantly being developed and some users would like to invent their own. Equality is just one example of the ....

Jean H. Gallier. Logic for Computer Science: Foundations of Automatic Theorem Proving. Harper & Row.

....that # is an element of #; the concatenation of two lists # and # # is written as either #### # or # # # ; the inclusion and intersection of lists are defined in the usual way. We will work with the usual representation of nnfs as string of symbols and its representation as syntactic tree [4]; this way, an address # in the syntactic tree of a formula A will also mean, when no confusion arises, the subformula of A corresponding to the node of address # in the tree. Let A and C be two formulas, let B be a subformula of A and let # be an address in the syntactic tree of A. By A[B C] we ....

Jean H. Gallier. Logic for Computer Science: Foundations for Automatic Theorem Proving. Wiley & Sons, 1987.

....To make this proof transparent we comment every proof step in detail. We hope that the example will make up a good background for understanding the approach to ATP in the AO style. 3. 1 The Calculus gS We treat here classical first order logic in the form of the sequent calculus G given in [14]. We use the name Gal instead of G. We treat the notion of a substitution as in [15] Any substitutional component is considered to be of the form t x, where x is a variable and t is a term of a substitution. Let L be a literal, then denotes its complement. We use the expression L(t 1 , ....

Gallier, J. : Logic for computer science: foundations of Automatic Theorem Proving. Harper and Row, Inc. New York (1986) 513 p.

....The basic object of mS is an a sequent. An a sequent may be considered as a special generalization of the standard notion of a sequent. We consider a sequents having one object ( goal ) in its succedent only. We treat here 1st order classical logic in the form of the sequent calculus G given in [22]. We treat the notion of a substitution as in [17] Any substitution component is considered to be of the form t=x, where x is a variable and t is a term of a substitution. Let L be a literal, then L denotes its complement. We use the expression L(t 1 ; t n ) to denote that t 1 ; t n ....

.... in generation of a new a sequent with only one goal (and, possibly, with new premises) The elimination of the TL1 equivalents of proposition connectives is done according to 1st order classical logic (it can be easily expressed in the terms of derivative rules of standard Gentzentype calculi [22]) and w 1 , w n ; w; E 1 , E n therewith are not changed. Essential deviation from traditional Gentzen inference search techniques is observed in the processing of quantifiers. This deviation reflects specific quantifier handling techniques investigated in [4] where variables of eliminated ....

Gallier, J. Logic for computer science: foundations of Automatic Theorem Proving. Harper and Row, Inc. New York (1986) 513 p.

....the theory of LISP list structure using Nelson Oppen s procedure considering all the variables to be constants. We have used techniques similar to Nelson Oppen to construct several new procedures by extending a GLSP for congruence closure (one such procedure is described in Section 5) Also, [8] gives techniques for constructing GLSPs based on congruence closure for conjunctions of ground literals containing predicates. The essential idea is to introduce boolean constants True and False and to represent P(t . tn) as P(t 1 . tn) True and (t 1 . tn) as (t 1 . tn) False. ....

....of 713 q such that P contains a ground literal and its negation just in case q is unsatisfiable in T. The congruence closure procedure is a closure based satisfiability procedure for the theory of equality with uninterpreted function symbols. It constructs a congruence closure graph [8] and in so doing computes a finite set of ground consequences of a conjunction of input ground equalities. As new equalities are added to a conjunction, new nodes representing terms are added to the graph and or congruence classes are merged. Many GLSPs that extend congruence closure are also ....

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Gallier, J. H., Logic for Computer Science: Foundations of Automatic Theorem Proving, Harper and Row, 1986.

....The extract terms are explicitly provided using the UseWitness tactic. After this step, the remaining proof is to show that the extract does indeed inhabit type. These proofs are surprisingly intricate. Also, proofs of well foundedness of the lexicographic measure induction are widely published [5, 10], but they rely on the least element principle which is non constructive. The proof used here is by nested inductions on the natural numbers. The method of applying well founded induction principles in proofs has a limitation; lemmas can not be used in proving well formedness goals. This is ....

J. H. Gallier. Logic for Computer Science: Foundations of Automatic Theorem Proving. Harper and Row, 1986.

....to intuitionistic provability for hereditary Harrop formulas. In other words, every hereditary Harrop formula that is a theorem in intuitionistic logic has a uniform proof. The deduction rules of the intuitionistic sequent calculus that correspond to connectives 9 G , 8G and )G are as follows [17]: P G[x t] P 9x(G) 9 G (i.e. sigma) t is an arbitrary term. P G[x c] P 8x(G) 8 G (i.e. pi) c does not occur free in P and G. P; D G P D ) G ) G (i.e. A sequent P G reads goal G is a consequence of program P . A rule Sequent Sequent reads conclusion Sequent ....

Jean H. Gallier. Logic for Computer Science: Foundations of Automatic Theorem Proving. Harper & Row, 1986.

.... node in the proof tree consists of a set of formulae. A set of tableaux rules specifies how the tree may be constructed, and these rules guide the transformation of sets of formula from a parent node to children. A tableaux may also be described as an upside down left handed sequent system [5]. The tableaux method begins with a single node (the root) containing the negation of the formula to be tested for theoremhood. If every branch of the ensuing tableaux leads to a contradiction then the root node is deemed unsatisfiable. Thus the original formula is a theorem. The dual tableaux ....

Jean H. Gallier. Logic for Computer Science: Foundations of Automatic Theorem Proving. John Wiley and Sons, 1987.

....The extract terms are explicitly provided using the UseWitness tactic. After this step, the remaining proof is to show that the extract does indeed inhabit type. These proofs are surprisingly intricate. Also, proofs of well foundedness of the lexicographic measure induction are widely published [5, 10], but they rely on the least element principle which is non constructive. The proof used here is by nested inductions on the natural numbers. The method of applying well founded induction principles in proofs has a limitation; lemmas can not be used in proving well formedness goals. This is ....

J. H. Gallier. Logic for Computer Science: Foundations of Automatic Theorem Proving. Harper and Row, 1986.

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J. H. Gallier. Logic for Computer Science: Foundations of Automatic Theorem Proving. Harper & Row, 1986.

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Gallier, J.H.: Logic for computer science: Foundations of automatic theorem proving. Harper & Row computer science and technology series. Harper & Row (1986)

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J.H. Gallier, "Logic for computer science: foundations of automatic theorem proving". Harper & Row Publishers, New York, 1986.

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Jean H. Gallier. Logic for Computer Science: Foundations of Automatic Theorem Proving. Harper & Row Publishers, New York, 1986.

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Jean H. Gallier. Logic for Computer Science: Foundations of Automatic Theorem Proving. Harper & Row Publishers, New York, 1986.

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Jean Gallier. Logic for Computer Science | Foundations of Automatic Theorem Proving, volume 5 of Harper & Row Computer Science and Technology Series. Harper & Row, 1986.

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J. Gallier. Logic for Computer Science: Foundations of Automatic Theorem Proving. Wiley, 1987.

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Gallier, J. H., Logic for Computer Science: Foundations of Automatic Theorem Proving, Harper and Row, 1986.

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Jean H. Gallier. Logic for Computer Science: Foundations of Automatic Theorem Proving. Harper & Row Publishers, Inc., 1985.

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J H Gallier. Logic for Computer Science: Foundations of Automatic Theorem Proving. John Wiley and Sons, 1987.

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