A. Avron, F. Honsell, M. Miculan, and C. Paravano. Encoding modal logics in Logical Frameworks. Studia Logica, 60(1):161--208, Jan. 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in speci ed logics, such as a search based model of computation like logic programming. Representation in Logical Frameworks. The judgements as types notion of representation is described for LF informally, except for examples of particular systems, in [72, 11] It is described more carefully in [12] and, for RLF, in [86] Following [144, 145, 146] it can be summarized as follows: 1. Consider object logics as systems for deriving not bare propositions but rather judged propositions. For example, intuitionistic predicate logic is formulated as a system for deriving (or satisfying) judged ....

.... we discuss some distinctions within the judgements as types mechanism and also a di erent mechanism, called worlds as parameters: Judgements as types: A number of forms of representation, within the judgements astypes mechanism, have been discussed by Avron, Honsell, Miculan and Parvano [12]. Brie y, they are as follows: The most basic use of judgements as types is when the object logic to be represented is described by a single consequence relation. Examples of such systems include propositional and predicate classical and intuitionistic logics. Their encodings in LF employ a ....

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A. Avron, F. Honsell, M. Miculan, and C. Paravano. Encoding modal logics in logical frameworks. Studia Logica, 60(1), 1998. 40

....in specified logics, such as a search based model of computation like logic programming. Representation in Logical Frameworks. The judgements as types notion of representation is described for LF informally, except for examples of particular systems, in [72, 11] It is described more carefully in [12] and, for RLF, in [86] Following [144, 145, 146] it can be summarized as follows: 1. Consider object logics as systems for deriving not bare propositions but rather judged propositions. For example, intuitionistic predicate logic is formulated as a system for deriving (or satisfying) judged ....

.... we discuss some distinctions within the judgements as types mechanism and also a different mechanism, called worlds as parameters: Judgements as types: A number of forms of representation, within the judgements astypes mechanism, have been discussed by Avron, Honsell, Miculan and Parvano [12]. Briefly, they are as follows: The most basic use of judgements as types is when the object logic to be represented is described by a single consequence relation. Examples of such systems include propositional and predicate classical and intuitionistic logics. Their encodings in LF employ a ....

[Article contains additional citation context not shown here]

A. Avron, F. Honsell, M. Miculan, and C. Paravano. Encoding modal logics in logical frameworks. Studia Logica, 60(1), 1998.

....Our example encodings in x 5 are pathological in the sense that they require only one judgement. For example, the encoding of a fragment of intuitionistic linear logic requires the judgement of (J i = J = proof . This is in contrast to the general multi judgement representation techniques [6]. We conjecture that our studies can be applied to the general case, although we defer this development to another occasion. A certain class of uniform representations is identified by considering surjective encodings between consequences of the object logic L and consequences of the meta logic ....

A. Avron, F. Honsell, M. Miculan and C. Paravano. Encoding modal logics in logical frameworks, 1997. Manuscript.

.... of modal logic and labels ; i.e. instead of A, we consider w:A, where w is a world , and A iff 8w 2 W( w:A) This provides a language in which we can formulate 2 Different approaches to proof under assumption in modal logics, based on modified sequent systems, have been proposed, e.g. [3, 7, 9, 17, 32, 33, 57]. We do not consider such systems here, but see [56] for detailed comparison and discussion. 3 We use the vocabulary of [2] which should be consulted for a technical discussion of consequence relations, and which notes (x5.5) that every ordinary, pure single conclusioned ND system can, e.g. ....

.... 9, or we can add classical negation rules to obtain MR , the encoding of R: R = JR rules incEc (a: A = b: inc) a : A 17 Indexed judgements similar to ours have been also adopted in other encodings of some non classical logics in logical frameworks, e.g. modal and dynamic logics in [3, 28, 50]. starc R G a a end (Alternatively, we can encode R by directly extending MR . Using this encoding we can, e.g. prove G : A imp A in R as follows, cf. the proof of G: A A given in Section 2.4. goal R.thy G : A imp A ; G : A imp A 1. G : A imp A At the Isabelle prompt, ....

A. Avron, F. Honsell, M. Miculan, and C. Paravano. Encoding modal logics in logical frameworks. Studia Logica, 1997. This issue.

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A. Avron, F. Honsell, M. Miculan, and C. Paravano. Encoding modal logics in Logical Frameworks. Studia Logica, 60(1):161--208, Jan. 1998.

....list of hypotheses, and we should deal with supplementary structural rules for manipulating the list of assumptions. Instead, we represent more e#ciently the assumption set by means of the proof context provided by CIC, i.e. by taking advantage of the possibility of reasoning under assumptions [1]. First, we represent ## and # by means of two judgments bind:var o Prop and cngr:o o Prop, respectively. The former has no constructor (it declared as a Parameter) while the latter is rendered as an inductive predicate, as expected. In particular, the congruence rule for is rendered by ....

A. Avron, F. Honsell, M. Miculan, and C. Paravano. Encoding modal logics in Logical Frameworks. Studia Logica, 60(1):161--208, Jan. 1998.

....assumptions . The solution we adopt exploits again the possibility provided by Logical Frameworks of considering locally quantified premises, i.e. general judgements in the terminology of Martin Lof. For a in depth discussion on this technique for the representation of modal logics, we refer to [3]. 18 Please write authorrunninghead Author Name(s) in file First, we represent ## and # by means of two judgements bind:var o Prop and expto:o o Prop, respectively. The former has no constructor (it is declared as a Parameter) while the latter is rendered as an inductive ....

.... w:U, #X,w (#) # t 1 : T w [x : var]# X,x (#) #X,x , # X (#) b: bind x # X (#) w:U,h: T w # X,x (#) # t 2 : T w # X (#) By abstracting t 2 over b, w and x, we obtain a term t # 2 such that #X , # X (#) # t # 2 : x : var) bind x # X (#) w:U) T w # X,x (#) T w # X (#) 4 See [3] for a full description of the adequacy of encodings based on the world technique. Please write titlerunninghead (Shortened) Article Title in file 23 Moreover, since x# is well formed, by Lemma 5.2 there is a term t 3 such that #X # t 3 : iswf # X (x#) We can thus build the required ....

A. Avron, F. Honsell, M. Miculan, and C. Paravano. Encoding modal logics in Logical Frameworks. Studia Logica, 60(1):161--208, 1998.

....: # G:nat#P # p:P # c:C # # w:W # # n:nat (U w (G n) # # (U w p) # # # w:W # # n:nat (U w [c] G n) # # (U w [c] p) This is a general methodology which allows to encode adequately arbitrary proper sequent like rules. For lack of space we do not discuss adequacy formally; see [20, 12] for more details. 3 Derivation of Truth Hoare Logic In this section we outline the derivation in Coq of the rules of a ND style system for representing the truth CR for Hoare Logic #(DL) The truth CR for Hoare Logic can be obtained from Definition 1 by istantiating the appropriate parameters, ....

F. Honsell, M. Miculan, and C. Paravano. Encoding modal logics in Logical Frameworks. To appear, 1996.

....assumptions . The solution we adopt exploits again the possibility provided by Logical Frameworks of considering locally quantified premises, i.e. general judgements in the terminology of Martin Lof. For a in depth discussion on this technique for the representation of modal logics, we refer to [3]. 18 Please write authorrunninghead Author Name(s) in file First, we represent 7 and E by means of two judgements bind:var o Prop and expto:o o Prop, respectively. The former has no constructor (it is declared as a Parameter) while the latter is rendered as an inductive predicate, ....

.... t 1 : T w [x : var] X;x ( Xi X;x ; ffi X ( Delta) b: bind x X ( w:U,h: T w X;x ( t 2 : T w X ( By abstracting t 2 over b, w and x, we obtain a term t 0 2 such that Xi X ; ffi X ( Delta) t 0 2 : x : var) bind x X ( w:U) T w X;x ( T w X ( 4 See [3] for a full description of the adequacy of encodings based on the world technique. Please write titlerunninghead (Shortened) Article Title in file 23 Moreover, since x is well formed, by Lemma 5.2 there is a term t 3 such that Xi X t 3 : iswf X (x ) We can thus build the ....

A. Avron, F. Honsell, M. Miculan, and C. Paravano. Encoding modal logics in Logical Frameworks. Studia Logica, 60(1):161--208, 1998.

....of hypotheses, and we should deal with supplementary structural rules for manipulating the list of assumptions. Instead, we represent more efficiently the assumption set by means of the proof context provided by CIC, i.e. by taking advantage of the possibility of reasoning under assumptions [1]. First, we represent 7 and j by means of two judgments bind:var o Prop and cngr:o o Prop, respectively. The former has no constructor (it declared as a Parameter) while the latter is rendered as an inductive predicate, as expected. In particular, the congruence rule for is rendered by ....

A. Avron, F. Honsell, M. Miculan, and C. Paravano. Encoding modal logics in Logical Frameworks. Studia Logica, 60(1):161--208, Jan. 1998.

No context found.

A. Avron, F. Honsell, M. Miculan, and C. Paravano. Encoding modal logics in Logical Frameworks. Studia Logica, 60(1):161--208, Jan. 1998.

....Y G:nat P Y p:P Y c:C Y w:W Y n:nat (U w (G n) U w p) Y w:W Y n:nat (U w [c] G n) U w [c] p) This is a general methodology which allows to encode adequately arbitrary proper sequent like rules. For lack of space we do not discuss adequacy formally; see [20, 12] for more details. 3 Derivation of Truth Hoare Logic In this section we outline the derivation in Coq of the rules of a ND style system for representing the truth CR for Hoare Logic Sigma (DL) The truth CR for Hoare Logic can be obtained from Definition 1 by istantiating the appropriate ....

F. Honsell, M. Miculan, and C. Paravano. Encoding modal logics in Logical Frameworks. To appear, 1996.

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