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Nonmonotonic Reasoning, Preferential Models and Cumulative Logics
, 1990
"... Many systems that exhibit nonmonotonic behavior have been described and studied already in the literature. The general notion of nonmonotonic reasoning, though, has almost always been described only negatively, by the property it does not enjoy, i.e. monotonicity. We study here general patterns of ..."
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Cited by 538 (13 self)
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Many systems that exhibit nonmonotonic behavior have been described and studied already in the literature. The general notion of nonmonotonic reasoning, though, has almost always been described only negatively, by the property it does not enjoy, i.e. monotonicity. We study here general patterns of nonmonotonic reasoning and try to isolate properties that could help us map the field of nonmonotonic reasoning by reference to positive properties. We concentrate on a number of families of nonmonotonic consequence relations, defined in the style of Gentzen [13]. Both prooftheoretic and semantic points of view are developed in parallel. The former point of view was pioneered by D. Gabbay in [10], while the latter has been advocated by Y. Shoham in [38]. Five such families are defined and characterized by representation theorems, relating the two points of view. One of the families of interest, that of preferential relations, turns out to have been studied by E. Adams in [2]. The pr...
A typed foundation for directional logic programming
 In Proc. Workshop on Extensions to Logic Programming
, 1992
"... Abstract. A long standing problem in logic programming is how to impose directionality on programs in a safe fashion. The benefits of directionality include freedom from explicit sequential control, the ability to reason about algorithmic properties of programs (such as termination, complexity and d ..."
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Cited by 12 (1 self)
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Abstract. A long standing problem in logic programming is how to impose directionality on programs in a safe fashion. The benefits of directionality include freedom from explicit sequential control, the ability to reason about algorithmic properties of programs (such as termination, complexity and deadlockfreedom) and controlling concurrency. By using Girard’s linear logic, we are able to devise a type system that combines types and modes into a unified framework, and enables one to express directionality declaratively. The rich power of the type system allows outputs to be embedded in inputs and vice versa. Type checking guarantees that values have unique producers, but multiple consumers are still possible. From a theoretical point of view, this work provides a “logic programming interpretation ” of (the proofs of) linear logic, adding to the concurrency and functional programming interpretations that are already known. It also brings logic programming into the broader world of typed languages and typesaspropositions paradigm, enriching it with static scoping and higherorder features.
Nonmonotonic Reasoning: From finitary relations to infinitary inference operations
, 1994
"... A. Tarski [22] proposed the study of infinitary consequence operations as the central topic of mathematical logic. He considered monotonicity to be a property of all such operations. In this paper, we weaken the monotonicity requirement and consider more general operations, inference operations. The ..."
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Cited by 10 (2 self)
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A. Tarski [22] proposed the study of infinitary consequence operations as the central topic of mathematical logic. He considered monotonicity to be a property of all such operations. In this paper, we weaken the monotonicity requirement and consider more general operations, inference operations. These operations describe the nonmonotonic logics both humans and machines seem to be using when infering defeasible information from incomplete knowledge. We single out a number of interesting families of inference operations. This study of infinitary inference operations is inspired by the results of [12] on nonmonotonic inference relations, and relies on some of the definitions found there.
Arithmetic and the Incompleteness Theorems
, 2000
"... this paper please consult me first, via my home page. ..."
A Generic Approach To Program Extraction
"... over "sobj"  representation of a sequence of formulae *) fun abssobj t = Abs("sobj", Type("sobj",[]), t); (*Representation of empty sequence*) val Sempty = abssobj (Bound 0); fun seqobjtr(Const("@SeqId",)$id) = id  seqobjtr(Const("@SeqVar",)$id) = id  seqobjtr(Const("@Proof",dummyT)$p) ..."
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over "sobj"  representation of a sequence of formulae *) fun abssobj t = Abs("sobj", Type("sobj",[]), t); (*Representation of empty sequence*) val Sempty = abssobj (Bound 0); fun seqobjtr(Const("@SeqId",)$id) = id  seqobjtr(Const("@SeqVar",)$id) = id  seqobjtr(Const("@Proof",dummyT)$p) = (Const("Proof",dummyT)$p)  seqobjtr(fm) = Const("Seqof",dummyT)$fm; fun seqtr($obj$seq) = seqobjtr(obj)$seqtr(seq)  seqtr() = Bound 0; fun seqtr1(Const("@MtSeq",)) = Sempty  seqtr1(seq) = abssobj(seqtr seq); fun truetr[s1,s2] = Const("Trueprop",dummyT)$seqtr1 s1$seqtr1 s2; 9.3. CONCLUSION 91 fun seqobjtr'(Const("Seqof",)$fm) = fm  seqobjtr'(Const("Proof",dummyT)$p) = Const("@Proof",dummyT)$p  seqobjtr'(id) = Const("@SeqId",dummyT)$id; fun seqtr'(obj$sq,C) = let val sq' = case sq of Bound 0 =? Const("@MtSeqCont",dummyT)  =? seqtr'(sq,Const("@SeqCont",dummyT)) in C $ seqobjtr' obj $ sq' end; fun seqtr1'(Bound 0) = Const("@MtSeq",dummyT)  seqtr1' s = seq...
Structural extensions of display calculi: a general recipe
"... Abstract. We present a systematic procedure for constructing cutfree display calculi for axiomatic extensions of a logic via structural rule extensions. The sufficient properties of the display calculus and thus the method applies to large classes of calculi and logics. As a case study, we present ..."
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Abstract. We present a systematic procedure for constructing cutfree display calculi for axiomatic extensions of a logic via structural rule extensions. The sufficient properties of the display calculus and thus the method applies to large classes of calculi and logics. As a case study, we present cutfree calculi for extensions of wellknown logics including Biintuitionistic and tense logic. 1
Labelled Tree Sequents, Tree Hypersequents and Nested (Deep) Sequents
"... We identify a subclass of labelled sequents called “labelled tree sequents ” and show that these are notational variants of treehypersequents in the sense that a sequent of one type can be represented naturally as a sequent of the other type. This relationship can be extended to nested (deep) seque ..."
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We identify a subclass of labelled sequents called “labelled tree sequents ” and show that these are notational variants of treehypersequents in the sense that a sequent of one type can be represented naturally as a sequent of the other type. This relationship can be extended to nested (deep) sequents using the relationship between treehypersequents and nested (deep) sequents, which we also show. We apply this result to transfer prooftheoretic results such as syntactic cutadmissibility between the treehypersequent calculus CSGL and the labelled sequent calculus G3GL for provability logic GL. This answers in full a question posed by Poggiolesi about the exact relationship between these calculi. Our results pave the way to obtain cutfree treehypersequent and nested (deep) sequent calculi for large classes of logics using the known calculi for labelled sequents, and also to obtain a large class of labelled sequent calculi for biintuitionistic tense logics from the known nested (deep) sequent calculi for these logics. Importing prooftheoretic results between notational variant systems in this manner alleviates the need for independent proofs in each system. Identifying which labelled systems can be rewritten as labelled tree sequent systems may provide a method for determining the expressive limits of the nested sequent formalism. Keywords: labelled tree sequents, notational variants, cutelimination, proof theory.
4. The Second Incompleteness Theorem. 5. Lengths of Proofs.
, 2007
"... Gödel's legacy is still very much in evidence. We will not attempt to properly discuss the full impact of his work and all of the ongoing important research programs that it suggests. This would require a book length manuscript. Indeed, there are several books discussing the Gödel legacy from many p ..."
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Gödel's legacy is still very much in evidence. We will not attempt to properly discuss the full impact of his work and all of the ongoing important research programs that it suggests. This would require a book length manuscript. Indeed, there are several books discussing the Gödel legacy from many points of view, including, for example, [Wa87], [Wa96], [Da05], and the historically comprehensive five volume set [Go,8603]. In sections 27 we briefly discuss some research projects that are suggested by some of his most famous contributions. In sections 811 we discuss some highlights of a main recurrent theme in our own research, which amounts to an expansion of the Gödel incompleteness phenomenon in a critical direction.